We introduce the Leonardo \(k\)-triangle and derive the explicit formula for generalized Leonardo numbers by using some properties of this triangle. These include elegant formulas for the generalized Leonardo numbers, although with our suggested notation as a tool of thought, we claim that Fibonacci numbers are a particular case of Leonardo numbers, rather than the other way around. Moreover, we introduce the dual Leonardo \(k\)-triangle to generalize the explicit formula for dual Leonardo \(k\)-numbers.

A vertex \(v\) is called an AR-vertex, if \(v\) has distinct sum of edge labels for each distinct subset of edges incident on \(v\). i.e., if \(\{x_1,x_2,\dots,x_k\}\) are the edge labels of the edges incident on \(v\), then the \(2^k\) subset sums are all distinct. An injective edge labeling \(f\) of a graph \(G\) is said to be an AR-labeling of \(G\) if \(f:E \rightarrow \mathbb{N}\) is such that every vertex in \(G\) is an AR-vertex under \(f\). A graph \(G\) is said to be an AR-graph if there exists an AR-labeling \(f:E\rightarrow \{1,2,\dots,m\}\), where \(m\) denotes the number of edges of \(G\). A study of AR-labeling and AR-graphs is initiated in this paper.

For a finite simple undirected graph \(G=(V,E)\), a subset \(C\subseteq V\) is called an identifying code of \(G\) if the closed neighborhood of every vertex has a nonempty and unique intersection with \(C\). The minimum cardinality of such a set is denoted by \(\gamma^{ID}(G)\). In this paper, we strengthen the previously known results of Nadimi Dafrazi and Vatandoost on the identifying-code number of middle graphs, which were established for bipartite graphs and certain other classes, by proving the exact value for every finite simple graph \(G\) of order at least one. By establishing a new lower bound, we show that the identifying-code number of the middle graph \(M(G)\) is equal to its independence number; specifically, \(\gamma^{ID}(M(G))=\alpha(M(G))=|V(G)|.\)

A brief survey on tactical decomposable families of rectangular designs (RDs) is presented. Rectangular designs have well–known applications in statistics. Here, applications of RDs in cryptography and coding theory are described. Earlier, \((2,\ n) -\)threshold schemes were proposed from tactical decomposable regular group divisible designs. Threshold schemes are proposed here from tactical decomposable RDs. Further, an application of RDs in low–density parity–check (LDPC) codes is also given. Tactical decomposable RDs had not been previously used in the constructions of threshold schemes and LDPC codes.

Recently, it was proved that if \(K_{2,2n}\) admits an edge \(k\)-product cordial labeling, then either \(k=2n+1\) or \(k\ge 4n+1\), and two related existence problems were posed. In this paper, we prove that \(K_{2,2n}\) is edge \((2n+1)\)-product cordial if and only if \(2n+1\) is a prime. We further show that \(K_{2,4}\) is edge \(k\)-product cordial if and only if \(k=5\) or \(k\ge 9\), while \(K_{2,6}\) is edge \(k\)-product cordial if and only if \(k=7\) or \(k\ge 13\). These results completely answer one of the posed questions and provide further evidence for a general existence conjecture.

Let \(G\) be a finite simple graph with \(E(G)\neq\emptyset\), let \(I(G)\) be its edge ideal, and let \(R(G)=K[x_1,\dots,x_n]/I(G)\). We develop a support-signature approach to the zero-divisor graph \(\Gamma(R(G))\) using the minimal vertex covers of \(G\). For each \(z\in R(G)\), the minimal primes avoiding \(z\) determine a support signature, and the realizable signatures of nonzero zero-divisors define a finite support graph \(\Sigma(G)\), with adjacency given by disjointness. We show that \(\Gamma(R(G))\) is a blow-up of \(\Sigma(G)\) by its support classes. Consequently, the twin classes are explicitly characterized, the twin-class quotient is canonically identified with \(\Sigma(G)\), and the girth and diameter admit finite-quotient descriptions under suitable nontriviality hypotheses. Over an infinite field, \(\operatorname{Aut}(\Gamma(R(G)))\) is noncanonically isomorphic to a semidirect product of the internal symmetric groups of the support classes by \(\operatorname{Aut}(\Sigma(G))\). If \(m\geq2\) is the number of minimal vertex covers, every nonempty proper subset of \([m]\) occurs as a support signature, yielding \(\operatorname{Aut}(\Sigma(G))\cong S_m\). Examples involving paths, stars, and \(4\)-cycles illustrate the method and its finite symmetry quotient.

Hall’s theorem on differences of bijections characterizes the multisets \(\{a_1,\ldots,a_{|G|}\}\) in a finite abelian group \(G\) that can be written in the form \( a_i=b_i-c_i, \) where both \(b_1,\ldots,b_{|G|}\) and \(c_1,\ldots,c_{|G|}\) are enumerations of \(G\). The necessary and sufficient condition is the zero-sum condition \( a_1+\cdots+a_{|G|}=0. \) This paper studies the corresponding problem for finite nonabelian groups, with differences replaced by quotients. Thus we ask when a multiset \(A\) of cardinality \(|G|\) can be represented as \( A=\{b(i)c(i)^{-1}:1\le i\le |G|\}, \) where \(b\) and \(c\) are bijections onto \(G\). Passing to the abelianization gives a necessary condition, namely that the product of the images of the elements of \(A\) is trivial in \(G_{\rm ab}\). We show that this condition is not sufficient in general, even when the elements of \(A\) admit an ordering whose product is the identity in \(G\). The main structural result is a cycle-tiling criterion: quotient-realizability is equivalent to a decomposition of \(A\) into product-one words whose partial-product sets tile \(G\) by right translates. The use of permutation cycles is standard, but the criterion translates quotient-realizability into an exact tiling condition. We then use this criterion to construct a counterexample in \(S_3\), and we extend the same obstruction to infinitely many finite nonabelian groups.

We investigate diagonal equations \(ax^{m}+by^{m}-cz^{m}=1\) over finite fields \(F\) using combinatorial designs naturally associated with \(F\). Building on prior work that resolved the case \(a=b=c=1\), we obtain exact formulas for the solutions when \(a=1\) and \(b=c\), under circularity assumptions. For general coefficients, we present an algorithm that determines whether a given instance can be reduced to the settled cases, or else identifies it as requiring brute-force computation.

In this paper, we expand our interest in the 16th Hilbert’s problem to acquire a comprehensive understanding of the maximum number of crossing limit cycles in \(\mathbb{R}^3\), specifically within a class of three- dimensional discontinuous piecewise differential system generated by two arbitrary Euler systems separated by the unit sphere \(\mathbb{S}^2=\{ (x,y,z) \in\mathbb{R}^3; x^2 + y^2 + z^2 = 1\}\).

Let \(G\) be a graph with no isolated vertices. A \(k\)-coupon coloring of \(G\) is an assignment of colors from \([k]=\{1,2,\ldots,k\}\) to the vertices of \(G\) such that the neighborhood of every vertex contains all colors from \([k]\). The maximum integer \(k\) for which a \(k\)-coupon coloring exists is called the coupon coloring number of \(G\), and is denoted by \(\chi_c(G)\). In this paper, we investigate coupon coloring in inflated graphs arising from various classes of graphs. In addition, we introduce new graph operations based on inflation and study their effect on the existence and behavior of coupon colorings. Our results contribute to a deeper understanding of how inflation based graph operations influence coupon coloring.

We recall the definition and properties of a moment sequence and show that all real sequences whose Hankel matrices have finite rank (see definition in the sequel) satisfy a homogeneous linear equation with constant coefficients. Then we analyze the cases in which a difference equation with constant coefficients and suitably chosen initial conditions and having as an input a positive moment sequence has a solution that is a positive moment sequence. We give one general simple result and give many examples illustrating the theory. The main result states that the roots of the odd multiplicity of the characteristic equation must lie outside the support of the measure that produces the moment sequence that is in the input and the initial conditions suitably chosen.

We present constructions of semi-magic squares of side \(n=2k\), whose entries are elements of a dihedral group \(D_{2k^2}\), for every \(n\equiv0\pmod4\).

A zero divisor graph on a finite commutative ring \(\mathfrak{R}\) is a graph with set of vertices consists of zero divisor elements \(Z(\mathfrak{R})\) of the ring, and we have an edge between any two elements in \(\mathfrak{R}\) if their product is the zero element. In this work, we will explore some zero divisor graph invariants constructed on the rings of the form \(\mathfrak{R}=\mathbb{Z}_n,\) when \(n\) is a product of square free primes. In particular, we will find the radio number for zero-divisor graphs constructed on \(\mathbb{Z}_{\mathfrak{p}_1 \mathfrak{p}_2 \mathfrak{p}_3}\) where \(\mathfrak{p}_1, \mathfrak{p}_2,\) and \(\mathfrak{p}_3\) are distinct primes with \(2 \leq \mathfrak{p}_3 < \mathfrak{p}_2 < \mathfrak{p}_1\) and combining with the known results for \(\mathbb{Z}_{\mathfrak{p}^3}\) and \(\mathbb{Z}_{\mathfrak{p}_1^2 \mathfrak{p}_2}\), It covers all possible cases when \(n\) is divisible by at most three primes.

A graph \(G(V, E)\) is word-representable if there exists a word \(w\) over the alphabet \(V\) such that for distinct letters \(x,y\in V\), \(x\) and \(y\) alternate in \(w\) if and only if they are adjacent in \(G\). In general, determining whether a graph is word-representable is an NP-complete problem. A graph is co-bipartite if its complement is bipartite. Therefore, the vertex set of a co-bipartite graph can be partitioned into two disjoint subsets \(X\) and \(Y\) such that the subgraphs induced by \(X\) and \(Y\) are cliques. Necessary and sufficient conditions for a co-bipartite graph to be word-representable in terms of a vertex ordering are known. Based on this ordering, we study the representation number of word-representable co-bipartite graphs and analyse the speed and entropy of this graph class. We show that the representation number of any word-representable co-bipartite graph is at most \(3\), and permutation graphs are the only co-bipartite graphs with representation number \(2\). We prove that the speed is at most \(2^{O(n \log n)}\) and the entropy is \(0\). In particular, we obtain an upper bound on the number of labelled graphs in this class, which is significantly smaller than the known bound for the class of all co-bipartite graphs. These results provide a better understanding of the structure and enumeration of word-representable co-bipartite graphs and show that vertex ordering is an effective tool for studying this class.

Spread of information within a wide variety of systems can be represented as evolving processes on digraphs. Starting from a random subset of initially active vertices, the measures we introduce assess the probability, speed, or number of steps it takes to spread information to the entire digraph, thus achieving digraph synchrony. Some of these measures may be viewed as generalizations of digraph connectivity or as generalizations of the diameter of a digraph to higher-order diameters. The paper places considerable emphasis on the regular case of Cayley digraphs associated to finite groups. It is demonstrated that, with appropriate assumptions on the growth of the generating sets, all the higher-order diameters of random Cayley digraphs are almost surely at most 2, as the digraph order goes to infinity. Certain results on the velocity of spread of information in digraphs are also presented.

We introduce and study a new family of circulant digraphs associated with the cyclic group \({\mathbb Z}_N\), obtained by restricting admissible combinations of two generators \(a\) and \(b\) to three coordinate sectors. The resulting distance-like function differs from the standard directed distance in circulant digraphs and gives rise to new geometric and combinatorial phenomena. Using planar lattice representations and periodic tessellations, we analyze the growth of reachable sets and derive Moore-type upper bounds for the corresponding order/diameter problem. We construct explicit infinite families of three-quarters circulant structures with the prescribed diameter and provide lattice-based methods for determining admissible generator pairs. Separate constructions are obtained for even and odd diameters. In addition, computational experiments for small and moderate orders suggest improved families for even diameters and motivate a conjectural asymptotic formula for the maximum attainable order. The paper highlights the interplay between constrained lattice representations, periodic tilings, and extremal problems for circulant networks.

Let \(G\) be a simple connected graph and \(A(G)\) and \(D(G)\) represent the adjacency matrix and the diagonal matrix of degrees of graph G, respectively. The normalized Laplacian of \(G\) is defined by \(\mathcal{L}(G)=I_n-D(G)^{-1/2}A(G)D(G)^{-1/2},\) where \(I_n\) is the identity matrix of order \(n\). The normalized Laplacian plays an important role in spectral graph theory. In this paper, we characterize the connected graphs that minimize the spectral radius of the normalized Laplacian in the class of graphs with exactly one vertex of degree greater than two and in the class of graphs with exactly two vertices of degree greater than two. In each class, we determine the extremal graphs and the exact minimum normalized Laplacian spectral radius.

The inequality chain \(ir(G)\le \gamma(G)\le i(G)\le \alpha(G) \le \Gamma(G) \le I\!R(G)\) is known as the domination chain, where \(ir(G), \gamma(G), i(G), \alpha(G), \Gamma(G)\) and \(I\!R(G)\) are the lower irredundance number, the domination number, the independence domination number, the independence number, the upper domination number and the upper irredundance number of \(G\), respectively. The Ramsey-type problem seeks to characterize the family \({\mathcal H}\) of graphs such that every \({\mathcal H}\)-free graph \(G\) has a bounded parameter \(\mu\). The classical Ramsey’s theorem states that every \(\{K_n, E_n\}\)-free graph has a bounded number of vertices. Furuya (Discrete Math.Theor 2018) characterized \({\mathcal H}\) such that every connected \({\mathcal H}\)-free graph \(G\) has a bounded domination number. The characterization of the graph family \({\mathcal H}\) for which every connected \({\mathcal H}\)-free graph \(G\) has a bounded independence number was due to Choi, Furuya, Kim, Park (Discrete math. 2020) and Chiba, Furuya (Electron. J. Combin., 2022). In this paper, we further characterize \({\mathcal H}\) such that every connected \({\mathcal H}\)-free graph \(G\) has bounded \(\mu(G)\) for \(\mu\) belonging to the set \(\{ir(G), i(G), \Gamma(G), \text{IR}(G)\}\). This completes the characterization of \({\mathcal H}\) for which every connected \({\mathcal H}\)-free graph \(G\) has bounded \(\mu(G)\) for \(\mu(G)\) along the domination chain. Additionally, we characterize \({\mathcal H}\) such that every connected \({\mathcal H}\)-free graph \(G\) has bounded \(\mu(G)\) for \(\mu\) related to the domination number. Specifically, we consider the following parameters of \(G\): open irredundance number \(O\!I\!R(G)\), independence saturation number \(I\!S(G)\) and irredundance saturation number \(I\!R\!S(G)\).

Let \(D\) be a finite simple digraph with vertex set \(V(D)\). For \(v\in V(D)\), the set \(N^-[v]\) consists of \(v\) and all vertices of \(D\) from which arcs go into \(v\). Let \(k\ge 1\) be an integer. A signed double Roman \(k\)-dominating function (SDR\(k\)DF) on a digraph \(D\) is a function \(f:V(D)\rightarrow\{-1,1,2,3\}\) satisfying the following conditions: (i) \(\sum\limits_{x\in N^-[v]}f(x)\ge k\) for each \(v\in V(D)\); (ii) every vertex \(u\) with \(f(u)=-1\) has an in-neighbor \(z\) with \(f(z)=3\) or two in-neighbors \(x\) and \(y\) with \(f(x)=f(y)=2\); (iii) every vertex \(u\) with \(f(u)=1\) has an in-neighbor \(z\) with \(f(z)\ge 2\). The weight of an SDR\(k\)DF \(f\) is \(\omega(f)=\sum\limits_{v\in V(D)}f(v)\). The signed double Roman \(k\)-domination number \(\gamma_{sdR}^k(D)\) is the minimum weight of an SDR\(k\)DF on \(D\). In this paper, we study the signed double Roman \(k\)-domination number of digraphs and present various bounds on \(\gamma_{sdR}^k(D)\). In addition, we determine this parameter for several classes of digraphs. Some of our results extend well-known properties of the signed double Roman \(k\)-domination number \(\gamma_{sdR} ^k(G)\) of graphs \(G\).

We give combinatorial interpretations of some Rogers\(-\)Ramanujan type identities, also known as sum-product identities in terms of \((n+t)-\)color partitions and split \((n+t)-\)color partitions. The identities discussed in this study contains negative exponent of \(q\). These interesting results reveal rich structure and great potential for further research because they reveal intricate mathematical structures, and link various other fields.

Let \(G\) be a graph of a network system with vertices, \(V(G)\), representing physical locations and edges, \(E(G)\), representing informational connectivity. A locating-dominating (LD) set \(S \subseteq V(G)\) is a subset of vertices representing detectors capable of sensing an “intruder” at precisely their location or at some unknown point in their open-neighborhood. An LD set must be capable of locating an intruder anywhere in the graph using this collection of detectors. We explore three types of fault-tolerant LD sets: redundant LD sets, which allow at most one detector to be removed or disabled, error-detecting LD sets, which allow at most one false negative, and error-correcting LD sets, which allow at most one error (false positive or false negative). In particular, we determine lower and upper bounds for the minimum density of these three fault-tolerant locating-dominating sets in the infinite king grid.

Several necessary properties of König–Egerváry graphs involving the core, the corona, and critical independent sets are by now part of the folklore of the theory, and have motivated different lines of research within the same framework. In particular, every König–Egerváry graph satisfies the core–corona identity \(|core(G)|+|corona(G)|=2\alpha(G),\) the covering relation \(corona(G)cup N(core(G))=V(G),\) and the fact that \(core(G)\) is a critical independent set. Each of these conditions captures a different aspect of the interaction between maximum independent sets and matchings, but none of them alone characterizes the König–Egerváry property. In this note we show that their conjunction does: a graph \(G\) is König–Egerváry if and only if the above two core–corona conditions hold and \(core(G)\) is critical. Equivalently, the class of König–Egerváry graphs is precisely the intersection of the three graph families determined by these conditions. We also provide examples showing that the characterization is sharp: any two of the three conditions may hold in a graph which is not König–Egerváry.

Let \(\alpha(G)\), \(\mu(G)\) and prk\((G)\) denote the independence number, the matching number and the permanental rank of \(G\), respectively. Here prk\((G)\) is the maximum order of a principal submatrix with nonzero permanent of the adjacency matrix of \(G\). Let \(d(G)=\max_{S\subseteq V(G)}\{|S|-|N(S)|\}\) be the critical difference of \(G\). Let core\((G)\) and ker\((G)\) be the intersection of all maximum independent sets and all critical independent sets, respectively. In this note we use Larson’s critical independence decomposition to split the graph into two induced subgraphs, \(L_G\) and \(L_G^c\), where \(L_G\) is Kőnig–Egerváry and \(L_G^c\) is 2-bicritical. We prove that for every graph \(G\) one has \(\alpha(G)-\mu(G) = |L_G|-prk(L_G)+\alpha(L_G^c)-\mu(L_G^c) = d(L_G)+\alpha(L_G^c)-\mu(L_G^c).\) Moreover, we show that \(\alpha(L_G^c)\le \mu(L_G^c)\) and establish the refined kernel bound \(d(L_G)+k\le |ker(G)|,\) where \(k\) is the number of nontrivial connected components of \(L_G\) without a perfect matching. Consequently, \(\alpha(G)-\mu(G)+k\le |ker(G)|.\) In particular, when \(\alpha(G)>\mu(G)\), one has \(|L_G|>prk(L_G)\). The bound is sharp for every prescribed value of \(k\). Since ker\((G)\subseteq core(G)\) for every graph, we recover as a consequence the known Boros–Golumbic–Levit inequality \(\alpha(G)-\mu(G)+1\le |core(G)|\) for connected graphs with at least two vertices and \(\alpha(G)>\mu(G)\). This result improves on related results by Hammer et al. (1982) and by Levit and Mandrescu (1999).

A pair of letters \(x\) and \(y\) are said to alternate in a word \(w\) if, after removing all letters except for the copies of \(x\) and \(y\) from \(w\), the resulting word is of the form \(xyxy\ldots\) (of even or odd length) or \(yxyx\ldots\) (of even or odd length). A graph \(G = (V(G), E(G))\) is word-representable if there exists a word \(w\) over the alphabet \(V(G)\) such that two distinct vertices \(x, y \in V(G)\) are adjacent in \(G\) (i.e., \(xy \in E(G)\)) if and only if the letters \(x\) and \(y\) alternate in \(w\). A split graph is a graph in which the vertices can be partitioned into a clique and an independent set. Word-representability of split graphs has been studied in a series of papers in recent years. Partial progress has been made in characterizing word-representable split graphs through minimal forbidden induced subgraphs, but a complete classification remains open. In this work, we study a specific subclass: split graphs with an independent set of size four, and we provide a minimal forbidden induced subgraph characterization of word-representable graphs in this class as a step towards addressing the broader classification problem. The subclass we study also corresponds to an open problem posed by Kitaev and Pyatkin. In addition, we outline possible approaches and proof strategies that may lead to a complete characterization of word-representable split graphs.

We consider the encoding of graph problems as Quadratic Unconstrained Binary Optimization (QUBO) problems, solvable by quantum or classical annealers. However, nowhere-zero flows have not previously been included among graph problems encoded as QUBO problems. Nowhere-zero flows are related to Tutte’s \(5\)-flow conjecture and occur in many contexts in graph theory. We provide a QUBO Hamiltonian encoding of nowhere-zero flows and prove the correctness of the construction. The resulting Hamiltonian \(H_{\mathrm{mod},k}\) has zero ground-state energy if and only if the graph \(G\) has a nowhere-zero \(\mathbb{Z}_k\)-flow. By Tutte’s equivalence theorem, zero ground energy is equivalent to \(\varphi(G)\le k\), and the zero-energy degeneracy is given by the flow polynomial \(F(G;k)\). The construction uses one-hot variables for edge flow residues modulo \(k\) and auxiliary variables for the per-vertex modular quotient. We prove that correctness is independent of the choice of orientation, root vertex, and positive penalty weights. We verify the construction on \(59\) graph and \(k\) examples, including both yes-instances and no-instances. We also sweep orientations and root choices on selected robustness instances and test a finite suite of positive penalty weights. The Hamiltonian is implemented using the dimod.BinaryQuadraticModel class, compatible with the D-Wave Ocean SDK. Quantum-hardware runs and claims about potential speedup are left to future work.

Measures of spread of information are introduced, with applications to neuronal activity in regions of a brain or to the design of artificial robotic networks in which efficient transmission of information is sought. We make links to spectral connectivity measures in graphs, such as spanning trees and higher-order diameters (as defined here). The exposition is then specialized to regular graphs by developing a formula that expresses the number of spanning trees in terms of walks in the complementary graph. Using traces, we then develop bounds for the number of spanning trees. Two approaches are used to establish such bounds: the first involves a logarithmic series expansion of the number of spanning trees in the complementary graph, while the second relies on certain \(l_p\) norm inequalities. Consequences to bipartite graphs are then examined.

The Padovan sequence \((P_n)_{n\geq 0}\) is defined by the third-order linear recurrence \(P_n=P_{n-2}+P_{n-3}\) for \(n\geq 3\), with initial terms \(P_0=1\) and \(P_1=P_2=0\). We derive closed forms for the weighted finite sums \(\sum\limits_{i=1}^{n} i^mP_i\) for all integers \(m\geq 0\) and \(n\geq 1\). The construction introduces an alternating integer sequence \((\mathcal{A}^{(m)})_{m\geq 0}\) and a family of coefficient polynomials \(\mathcal{C}^{(m)}(x)\) whose shifted evaluations determine the coefficients of \(P_n\), \(P_{n+1}\), and \(P_{n+2}\). The resulting formula unifies the cases \(m=0,1,2,\ldots\) and provides an effective recurrence, together with an exponential generating function, for the coefficients. The same polynomial family also gives explicit weighted-sum identities for arbitrary sequences satisfying the Padovan recurrence, including the Perrin and Van der Laan sequences.

The concept of the integrated adjacency matrix for mixed graphs was first introduced in [9], where its spectral properties were analyzed in relation to the structural characteristics of the mixed graph. Building upon this foundation, this paper introduces the integrated Laplacian matrix, the integrated signless Laplacian matrix, and the normalized integrated Laplacian matrix for mixed graphs. We further explore how the spectra of these matrices relate to the structural properties of the mixed graph.

The study of piecewise differential systems appears in various scientific topics and serves as an important tool for modeling many phenomena in contemporary research. Moreover, the existence and maximum number of limit cycles in such systems represent one of the most difficult problems in mathematics. This paper examines the existence and the maximum number of crossing limit cycles for the 3\(D\)-discontinuous piecewise differential system formed by a linear differential center and relay system separated by cylinder. Firstly we consider the right circular cylinder \(\mathcal{C}_1 =\{(x,y,z)\in \mathbb{R}^3:x^2+y^2=1\}\) as a switching manifold. Secondly we separate the entire space by the parabolic cylinder \(\mathcal{C}_2 =\{(x,y,z)\in \mathbb{R}^3:z=y^2\}\).

For a graph \(G\) and a positive integer \(k\), the \(k\)-Bell colour graph of \(G\) is the graph whose vertices are the partitions of \(V\) into at most \(k\) independent sets, with two of these being adjacent if there exists a vertex \(x\) such that the partitions are identical when restricted to \(V – \{x\}\). The \(k\)-Stirling colour graph of \(G\) is defined similarly, but for partitions into exactly \(k\) independent sets. Building on the existing result that for each \(k \geq 3\), the \(k\)-Bell colour graph of a tree with at least 4 vertices is Hamiltonian, we show that every graph on \(n\) vertices, except \(K_n\) and \(K_n – e\), has a Hamiltonian \(n\)-Bell colour graph, and this result is best possible. It is also shown that, for \(k \geq 4\), the \(k\)-Stirling colour graph of a tree with at least \(k+1\) vertices is Hamiltonian.

Several graph decompositions that factorize the determinant of the adjacency matrix isolate a Kőnig–Egerváry part, such as the SD–KE decomposition and the critical independence decomposition of Larson. This suggests that the study of graph unimodularity can be approached, to a large extent, through the structure of Kőnig–Egerváry graphs. In this paper, we advance this point of view by introducing a new determinant factorization within the class of Kőnig–Egerváry graphs. More precisely, given a Kőnig–Egerváry graph \(G\), we consider the partition of \(V(G)\) into its perfect-flower part \(PF(G)\) and its perfect-flower-free part \(PFF(G)\), and prove that \[\det(G)=\det(G[PF(G)])\det(G[PFF(G)]).\] We also obtain the analogous factorization for the permanent. This decomposition provides a new tool for the study of unimodularity, reducing the problem to two induced subgraphs of very different nature: the graph \(G[PF(G)]\), whose structure is closely related to Sterboul–Deming configurations with perfect matchings, and the graph \(G[PFF(G)]\), which is governed by the theory of critical independent sets. In this way, the paper provides a new structural framework for the study of unimodular graphs through Kőnig–Egerváry theory.

In a graph, the distance between two vertices is the length of a shortest path connecting them. The distance-2 domination number \(\gamma_2(G)\) of a graph \(G\) is the minimum size of a vertex subset such that every vertex outside it is within distance two of some subset vertex. In a connected graph, a connected dominating set is a subset \(S\) whose induced subgraph is connected and in which every vertex not in \(S\) is adjacent to some vertex in \(S\); the connected domination number \(\gamma_c(G)\) is the size of a smallest such set. For \(k\ge 1\), let 𝒯_k be the set of trees \(T\) satisfying \(\gamma_c(T)=2\gamma_2(T)=2k\). The collection 𝒯₁ is the set of all double stars. In this paper, we provide a constructive characterization of 𝒯_k for all \(k\ge 2\).

This paper introduces row-major natural ordering labeling (hereafter referred to as the “Natural Square”) to the domino tiling space of the \(4\times4\) square grid and elucidates the structural correspondence between geometric group actions and algebraic invariants. First, based on the 36 complete tilings derived from Kasteleyn theory, these tilings are classified into 9 equivalence classes (hereafter, families) under the action of the dihedral group \(D_4\). Next, phenomena observed through exhaustive computational experiments are analyzed, and it is shown that the sum of the block-product sums for any tiling and its 90-degree rotation is invariably 1,428 (90-Degree Rotation Complement Theorem for Block-Product Sums). Furthermore, algebraic complementary pairs that satisfy power-sum equalities (Prouhet–Tarry–Escott type) across multiple degrees are identified, and the structure of higher-moment preservation phenomena that transcend geometric constraints is discussed.

A tree could be defined as follows. An edge is a tree. If \(T_{k-1}=\cup_{i=1}^{k-1}e_i\) is a tree with \(k-1\) edges \(e_i\), and \(e_k\) an edge, then \(T_k=T_{k-1}\cup e_k\) is a tree if \(T_{k-1}\cap e_k\) is a point. We generalize this construction: A simplex \(S_1\) of dimension \(\ge1\) is a thick tree. If \(G_{k-1}=\cup_{i=1}^{k-1}S_i\) is a thick tree, where \(S_i\) are simplices of dimension \(\ge1\), and \(S_k\) a new simplex of dimension \(\ge1\), then \(G_{k-1}\cup S_k\) is a thick tree if \(G_{k-1}\cap S_k\) is a point. All homological properties of Stanley-Reisner rings of thick trees are well known. We determine the Hilbert series and Betti numbers for Stanley-Reisner rings of skeletons of thick trees. From this one can read of projective dimension, regularity, and judge when they are Cohen-Macaulay.

In this paper we introduce and study the hyper-Mersenne numbers, a class of integer sequences extending the classical Mersenne numbers which arise in a combinatorial and algebraic context. Using generating functions, we derive explicit formulae and identities for these sequences. In particular, we find relations to binomial coefficients and figurate numbers. We also provide a closed-form expression for the determinant of associated matrices, valid in full generality.

A graph \(G\) is \(H\)-saturated if \(G\) does not have \(H\) as a subgraph but \(G + uv\) has at least one copy of \(H\) for any edge \(uv \notin E(G)\). The smallest number of edges of all \(H\)-saturated graphs of order \(n\) is called \(H\)-saturation number and is denoted by \(sat(n; H)\). In this paper, we establish the existence of \(C_{4}\)-saturated graphs with prescribed the number of edges in some length that is close to \(sat(n; C_{4})\).

The Hall number \(h(G)\) of a graph \(G\) is the minimum integer \(k\) such that every \(k\)-list assignment satisfying Hall’s condition on all induced subgraphs of \(G\) admits a proper coloring. In this paper, we investigate graphs for which the Hall number strictly captures list colorability, satisfying the equality \(h(G)=ch(G)\). We confirm a conjecture of Allagan by proving that this equality holds for every complete multipartite graph without singleton parts. For complete \(k\)-partite graphs of the form \(K(m,n,1,\dots,1)\), we establish that \(h(G)=ch(G)\) for all sufficiently large \(n\). Furthermore, we also determine \(h(G)\) for \(2\)-trees and wheel graphs \(W_n\). We show that for a \(2\)-tree \(G\), \(h(G) \in \{1, 2, 3\}\) for \(|V(G)| = 3, 4\), and \(\ge 5\), respectively. For wheel graphs, we demonstrate that \(h(W_n)\) is dictated by the parity of the rim: \(h(W_n)=3\) for odd \(n\ge5\), and \(h(W_n)=4\) for even \(n\ge6\).

For a connected graph \(G\) of order at least two, a total monophonic set of a graph \(G\) is a monophonic set \(S\) such that the subgraph \(G[S]\) induced by \(S\) has no isolated vertices. The minimum cardinality of a total monophonic set of \(G\) is the total monophonic number of \(G\) and is denoted by \(m_{t}(G)\). We determine bounds for it and characterize graphs which realize the lower bound. Also, some general properties satisfied by this concept are studied. It is shown that for positive integers \(a, b\) such that \(3 \leq a \leq b\) with \(b \leq 2a\), there exists a connected graph \(G\) such that \(m(G) = a\) and \(m_t(G) = b\). Further, if \(p, a, b\) are positive integers such that \(4 \leq a \leq b \leq p\), then there exists a connected graph \(G\) of order \(p\) with \(m_t(G) = a\) and \(m_c(G) = b\), where \(m_c(G)\) is the connected monophonic number of \(G\).

The concept of k-extendability is a fundamental notion in matching theory and is closely related to factor-criticality. In a seminal work, Zhang et al. established sharp conditions under which these two concepts are equivalent. In this paper, we study the equivalence between extendability and factor-criticality from the perspective of Sachs subgraphs and discuss conditions under which these notions are equivalent.

A graph \(G=(V,E)\) is said to be an absolute mean graceful graph if there exists a one-to-one function \(f:V(G)\to \{0,\pm1,\pm2,\ldots,\pm|E(G)|\}\) such that the induced edge-labeling function \(f^*:E(G)\to \{1,2,\ldots,|E(G)|\}\), defined by \[f^*(xy)=\left\lceil{\dfrac{|f(x)-f(y)|}{2}}\right\rceil,\] is bijective. The labeling function \(f\) is called an absolute mean graceful labeling of the graph \(G\). In this paper, we obtain absolute mean graceful labelings for \(m\)-splitting and \(m\)-shadow graphs of various graphs.

There are a number of variations of proper edge-colourings of graphs with restrictions on the subgraphs induced by two colour classes. Deciding whether a graph has a 3-edge-colouring with no 2-coloured 4-cycle is NP-complete for cubic graphs with chromatic index 3. But for bipartite cubic graphs the problem is solved completely in this paper. Furthermore, the minimum number of 2-coloured 4-cycles in a 3-edge-colouring is determined for any subcubic bipartite multigraph.

Let \(F_n:=K_1+nK_2\) be a fan of order \(2n+1\). For \(1\le s<t\), we consider the weakened Gallai-Ramsey number \(gr^t_s(F_n)\), defined to be the least \(p\in \mathbb{N}\) such that every Gallai \(t\)-coloring of \(K_p\) contains a subgraph isomorphic to \(F_n\) whose edges use at most \(s\) colors. Our main results include the evaluations \(gr^t_2(F_2)=t+3\), \(gr^3_2(F_3)=9\), and \(gr^t_{2n-1}(F_n)=2n+1\).

We introduce a parity–sum statistic on permutations that assigns to each position a weight determined by the parity of the entry occupying it. When restricted to alternating permutations, this statistic yields two \(q\)–analogues of the Euler numbers, corresponding to the up–down and down–up types. We establish symmetry and reciprocity properties of these polynomials. Specializing at \(q=1\), the resulting recursions reduce to the classical enumerative relations and recover Andr’e’s convolution identity for the Euler numbers. The distribution of the parity–sum statistic over the full symmetric group is also determined.

Previous work by Lesniak (1975) and recently by Qiao and Zhan (2022) established a sharp lower bound for the number of leaves of a tree as a function of its order and diameter. In this note, we derive a lower bound for the number of leaves as a function of the entire sequence of eccentricities, and provide a complete characterisation of all trees attaining equality. We also obtain a new but simpler proof for the diameter-bound. Furthermore, we establish the analogous result for the maximum with respect to the entire eccentric sequence.

Soft set theory, first introduced by Molodtsov, is a flexible approach for handling uncertainty-related problems and modeling uncertain information. Since soft set operations form the core of the theory, their algebraic properties and related structures have attracted considerable research interest. Several forms of symmetric difference operations have been proposed, including restricted and extended symmetric difference operations. Although restricted symmetric difference has already been defined, its definition is not fully consistent with the general formula of restricted soft set operations. In this paper, we first provide an alternative definition of restricted symmetric difference that follows the general form of restricted soft set operations. We then investigate its algebraic properties together with the extended symmetric difference operation, both for soft sets with a fixed parameter set and for soft sets over \(U\). We also establish new properties analogous to the symmetric difference operation in classical set theory, including the case where parameter sets may be disjoint. By deriving distributive rules, we show that important algebraic structures arise when restricted or extended symmetric difference operations are combined with other soft set operations. This study fills a gap in the literature, guides readers new to the theory, and presents a comprehensive analysis of restricted and extended symmetric difference operations, including corrected theorems and proofs from earlier studies.

How efficiently can a closed curve of unit length in \(\mathbb{R}^d\) be covered by \(k\) closed curves so as to minimize the maximum length of the \(k\) curves? We show that the maximum length is at most \(2k^{-1} – \frac{1}{4} k^{-4}\) for all \(k\geq 2\) and \(d \geq 2\). As a first byproduct, we show that \(k\) agents can traverse a Euclidean TSP instance significantly faster than a single agent. We thereby sharpen recent planar results by Berendsohn, Kim, and Kozma (2025) and extend these improvements to all dimensions. As a second byproduct, we obtain a linear time approximation algorithm with ratio \(2 – \frac{1}{4} k^{-3}\) for covering any closed polygonal curve in \(\mathbb{R}^d\) by \(k\) closed curves so that the maximum length of an individual curve is minimized.

A Fibonacci cordial (FC) labeling of a graph G is an injective function f : V(G) → {F₀, F₁, …, F_n}, where F_i is the i^th Fibonacci number, such that the induced edge labeling f^*(uv) = (f(u) + f(v)) (mod  2) satisfies |e_f(0) − e_f(1)| ≤ 1. A graph admitting such a labeling is called a Fibonacci cordial. First introduced by Rokad and Ghodasara (2016), FC labeling has been studied for several graph families. Mitra and Bhoumik (2020) extended this to complete graphs, cycles, and their corona (C_n and K_p for p ≤ 3). Motivated to build upon their work, we investigate C_n ⊙ K_p for p ≥ 4. Additionally, we examine whether the aforementioned family of corona graphs retains Fibonacci cordiality when alterations are made to the order of the corona, as observed in the family K_n ⊙ C_m. Moreover, we investigate the conditions under which two additional corona graph families, namely K_m ⊙ K_m and K_n, n ⊙ K_p, exhibit Fibonacci cordial labeling.

A graph is near-bipartite if its vertex set can be partitioned into two parts such that one part is an independent set and the other induces a forest. It is clear that near-bipartite graphs are 3-colorable. It was proved that planar graphs without 4-, 5– and 8-cycles are 3-colorable [Discuss. Math. Graph Theory, 31(2011): 775-789]. It is asked by Kang et al that whether it is true that the planar graphs without 4-, 5– and 8-cycles are near-bipartite [Discuss. Math. Graph Theory 45 (2025) 129-145]. A 6-cycle is called a special 6-cycle if this 6-cycle shares an edge with a triangle of G. In this paper, we prove that planar graphs without 4-, 5-, 8-cycles and special 6-cycles are near-bipartite which is a step toward the problem.

We study a modular generalization of the (σ, ρ)-Dominating Set problem on graphs of bounded treewidth, where vertices must satisfy neighborhood constraints modulo a fixed integer m. We present a dynamic programming algorithm over tree decompositions that achieves a runtime of O(n ⋅ (2m)^2tw + 2) and space usage O(n ⋅ (2m)^tw + 1), establishing fixed-parameter tractability when both treewidth tw and log m are treated as parameters. We prove this bound exhibits the correct exponential dependence on twlog m, as this factor is inherent to modular constraint satisfaction under the Strong Exponential Time Hypothesis (SETH). Experimental evaluation on synthetic graphs confirms the algorithm’s efficiency for small values of tw and m, highlighting its applicability to network design, logic circuits, and distributed systems with modular constraints.

The solutions of some new triangular designs not tabulated in Clatworthy (1973) are presented here. These designs are known in the literature but re–presented here with more explicit block lists. As a by–product, resolvable solutions of some designs are also obtained. The work addresses a recognized gap in combinatorial design theory and appears to extend classical catalogs.

Let W = {w₁, w₂, w₃, …, w_k} be an ordered set of vertices in a connected graph G. The representation of a vertex v ∈ V(G) with respect to W is the k-tuple r(v|W) = (d(v, w₁), d(v, w₂), …, d(v, w_k)), where d(v, w_i) is the length of the shortest path from v to w_i. If each vertex in G is uniquely identified by the distance vector, r(v|W) = (d(v, w₁), d(v, w₂),…,d(v, w_k)), then W is called a resolving set for G. If the resolving set is also independent, it is referred to as an independent resolving set. The independent metric dimension of G, denoted by idim(G), is the smallest cardinality of an independent resolving set. This study explores the independent metric dimension of the circulant graphs C_n(1, 2), C_n(1, 2, 3), C_n(1, 2, 3, 4) for sufficiently large n.

In this paper, given a homeomorphism f of a compact metric space X, we show that the set of all asymptotic average shadowable points of f is an open and invariant set and f has the asymptotic average shadowing property if and only if the set of all asymptotic average shadowable points of f is X if and only if any Borel probability measure μ of X has the asymptotic average shadowing property.

A supermagic labeling (often also called vertex-magic edge labeling) of a graph \(G(V,E)\) with \(|E|=q\) is a bijection from \(E\) to the set of first \(k\) positive integers such that the sum of labels of all incident edges of every vertex \(x\in V\) is equal to the same integer \(c\). An existence of a supermagic labeling of Cartesian product of two cycles, \(C_{n}\Box C_m\) for \(n,m\geq4\) and both \(n,m\) even and for any \(C_n\Box C_n\) with \(n\geq3\) was proved by Ivančo. Ivančo also conjectured that such labeling is possible for any \(C_n\Box C_m\) with \(n,m\geq3\). We prove his conjecture for all \(n,m\) odd that are not relatively prime.

Let \(G=(V,E)\) be a graph. For a vertex \(u\) of \(V(G)\), its closed neighborhood, \(N[S]\), is defined as \(N[u]=\{u\}\cup\{v|v\in V(G), v\neq u, u\) and \(v\) are adjacent in \(G \}\). A vertex subset \(S\) of \(V(G)\) is called a subversion strategy of \(G\) if each of the vertices in \(N[S]\) is deleted from \(G\). By \(G/S\) we denote the survival subgraph \(G-N[S]\). A subversion strategy \(S\) is called a cut strategy of \(G\) if \(G/S\) is disconnected, or is a clique, or is empty. In this paper, we revise the definition of neighbor-isolated scattering number, which was introduced by Aslan, as \(NIS(G)=\max\{i(G/S)-|S|\}\), where \(S\) represents a cut strategy of \(G\) such that every component of \(G/S\) is an isolated vertex or a clique, and \(i(G/S)\) represents the number of the components of \(G/S\). We discuss the relationship between this parameter and the structure of graphs. Some tight bounds and extremal graphs with given order and neighbor-isolated scattering number are determined.

Let \(k\) be an odd prime and choose \(s\in\mathbb{Z}_k^\times\) with \(s^2\not\equiv \pm1\pmod{k}\) (hence \(k\ge7\)). We give a deterministic, purely algebraic construction of compound pandiagonal (Nasik) magic squares of order \(k^{2}\) with consecutive entries \(\{0,1,\dots,k^{4}-1\}\). The input is the \(k\times k\) Modular Inverse Shift (MIS) kernel \(M_s(i,j)=si+s^{-1}j\in\mathbb{Z}_k\), a classical linear Latin square. Our contribution is not a new Latin-square object, but a closed-form integration of: (i) orthogonality of \((M_s,M_s^{\mathsf T})\), (ii) toroidal diagonal-regularity, and (iii) a two-level base-\(k\) digit superposition producing a \(k^2\times k^2\) square with closed-form evaluation of entries. We encode four \(\mathbb{Z}_k\)-digits coming from \((M_s,M_s^{\mathsf T})\) at both the block level and the within-block level, obtaining an explicit formula \(P_s(I,J)\in\{0,\dots,k^{4}-1\}\). Orthogonality yields bijectivity, while a carry-sensitive diagonal decomposition proves that every broken diagonal of both slopes sums to the magic constant. Finally, evaluating block sums shows that the induced \(k\times k\) block-sum array is itself pandiagonal magic, establishing the compound property.

We provide a hierarchy of “nonconventional ergodic theorems” in quantum setting involving operators and unitaries acting on the Hilbert space of the Gelfand-Naimark-Segal covariant representation associated to a reference \(C^*\)-dynamical system. The first two levels correspond to the Mean Ergodic Theorem by J. von Neumann involving a unitary, and the ergodic theorem by Kovács and Szúcs relative to unitarily implemented automorphisms of von Neumann algebras, respectively. As a consequence, we provide multiple correlation results concerning the ergodic behaviour of “three-operator” expectations.

As a generalization of vector spaces over finite fields, we study vector spaces over finite commutative rings, and obtain Anzahl formulas and a dimensional formula for subspaces. By using these results, we discuss normalized matching (NM) property of a class of subspace posets.

IOur focus is on the set of lower-triangular, infinite matrices that have natural operations like addition, multiplication by a number, and matrix multiplication. With respect to addition this set forms and abelian group while with respect to matrix multiplication, the invertivle elements of the set form a group. The set becomes an algebra (non-commutative in fact) with unity when all three operations are considered together. We indicate important properties of the algebraic structures obtained in this way. In particular, we indicate several sub-groups or sub-rings. Among sub-groups, we consider the group of Riordan matrices and indicate its several sub-groups. We show a variety of examples (approximately 20) of matrices that are composed of the sequences of important polynomial or number families as entries of certain lower-triangular infinite matrices. New, significant relationships between these families can be discovered by applying well-known matrix operations like multiplication and inverse calculation to this representation. The paper intends to compile numerous simple facts about the lower-triangular matrices, specifically the family of Rionian matrices, and briefly review their properties.

In a graph, a vertex is said to dominate itself and all its neighbors. A subset \(S\subseteq V(G)\) is a double dominating set of a graph \(G\), if every vertex of \(V\) is dominated by at least two vertices of \(S\). The double domination number denoted by \(\gamma_{2\times(G)}\) is the minimum cardinality of a double dominating set. Graph operations are fundamental in graph theory and have various applications across different fields including network analysis, parallel computing and electrical circuit design. This paper studies the problem of finding the double domination number under unary and binary operations of graphs. We investigate the double domination number of graphs under unary operations such as inflation and cubic inflation. Also, we introduced two new unary operations inspired from inflation operation and studied the impact of these operations on double domination number. Further, we explore the double domination number of edge corona and neighborhood corona of two graphs. Additionally, we study the double domination number of various corona operations of two graphs combined with subdivision of a graph and \(R-\)graph.

In Theorem 8.7 of  [22] eight centralities on trees are presented that all coincide with the median. In this paper we explore a functional extension for three of these Centralities, viz. the Centroid, the Security Center, and The Telephone Center of a tree. In the functional extension model, instead of using the whole vertex set to determine `central’ vertices we allow any multiset of vertices to determine the central vertices. The centroid and security center allow straightforward functional extensions, and both coincide with the well-kown median function. The functional extension of the Telephone center is a different story, and we present three versions, each of which catches most but not all the features of the original Telephone center. These all have a close relationship with the median function. As a bonus we obtain a deeper insight in the median function on trees.

In the present paper, we are interested in the distribution of the elements lying along the Raab direction in the binomial coefficients triangle. More precisely, we prove that the sequence \(\{\binom{n-rk}{k}\}_{0\leq k \leq \lfloor n/(r+1)\rfloor}\) is asymptotically distributed according to a Gaussian law. We also provide some experimental evidences.

We determine the maximum number of edges of a graph without containing the 2-power of a Hamilton path. Using this result, we establish a spectral condition for a graph containing the 2-power of a Hamilton path. Furthermore, we characterized the extremal graphs with the largest spectral radius that do not contain the 2-power of a Hamilton path.

We introduce the ID-index of a finite simple connected graph. For a graph \(G=(V,\ E)\) with diameter \(d\), we let \(f:V\longrightarrow \mathbb{Z}\) assign ranks to the vertices. Then under \(f\), each vertex \(v\) gets a string, which is a \(d\)-vector with the \(i\)-th coordinate being the sum of the ranks of the vertices that are of distance \(i\) from \(v\). The ID-index of \(G\), denoted by \(IDI(G)\), is defined to be the minimum number \(k\) for which there is an \(f\) with \(|f(V)|=k\), such that each vertex gets a distinct string under \(f\). We present some relations between ID-graphs, which were defined by Chartrand, Kono, and Zhang, and their ID-indices; give a lower bound on the ID-index of a graph; and determine the ID-indices of paths, grids, cycles, prisms, complete graphs, some complete multipartite graphs, and some caterpillars.

We prove that the class of trees with unique minimum edge-vertex dominating sets is equivalent to the class of trees with unique minimum paired dominating sets.

We investigate properties and structure of \(zero \ divisor \ graph\) of endomorphism ring of direct product of cyclic groups \(\mathbb{Z}_n\). We provide a method to determine the number of zero divisors of \(End(\mathbb{Z}_2 \times \mathbb{Z}_{2p})\), for some prime \(p\). We proved that minimum distance between any two vertices of \(zero \ divisor \ graph\) of \(End(\mathbb{Z}_m \times \mathbb{Z}_m)\) is 2.

Let \(G = (V, E)\) be a graph. The Gutman-Milovanović index of a graph \(G\) is defined as \(\sum\limits_{uv \in E} (d(u) d(v))^{\alpha}(d(u) + d(v))^{\beta}\), where \(\alpha\) and \(\beta\) are any real numbers and \(d(u)\) and \(d(v)\) are the degrees of vertices \(u\) and \(v\) in \(G\), respectively. In this note, we present sufficient conditions based on the Gutman-Milovanović index with \(\alpha > 0\) and \(\beta >0\) for some Hamiltonian properties of a graph. We also present upper bounds for the Gutman-Milovanović index of a graph for different ranges of \(\alpha\) and \(\beta\).

Suppose \(G_1=(V_1, E_1)\) is a graph and \(G_2=(V_2, E_2)\) is a strong digraph of \(G_1\), where \(V_1\) and \(V_2\) represent the vertex sets, \(E_1\) and \(E_2\) represent the edge sets. Let \(u\) and \(v\) be any two vertices of \(G_2\). The strong distance \(sd(u,v)\) is the minimum value of edges in a strong subdiagraph of \(G_2\) that contains \(u\) and \(v\). The minimum strong diameter of \(G_2\) is defined as the maximum eccentricity \(se(u)\) from \(u\) to all other vertices in \(G_2\). In this paper, we propose different strong orientation methods to explore the minimum strong diameter of the strong product graph of \(K_{m_1,m_2,\ldots,m_k}\otimes P_n\), where \(K_{m_1,m_2,\ldots,m_k}\) and \(P_n\) represent respectively complete multipartite graph and path. ‌‌In addition, based on strong orientation methods, a new algorithm is proposed to model the presence or absence of a minimum strong diameter in a strong product graph. Simulation experiments show a trend of simultaneous decrease and concentration in the minimum strong diameter of the strong product graph, as the value of parts in \(K_{m_1,m_2,\ldots,m_k}\) increases while the length of \(P_n\) remains constant.

We consider a joint ordered multifactorisation for a given positive integer \(n\geq 2\) into \(m\) parts, where \(n=n_1~\times~\ldots~\times~n_m\), and each part \(n_j\) is split into one or more component factors. Our central result gives an enumeration formula for all such joint ordered multifactorisations \(\mathcal{N}_m(n)\). As an illustrative application, we show how each such factorisation can be used to uniquely construct and so count the number of distinct additive set systems (historically referred to as complementing set systems). These set systems under set addition generate the first \(n\) non-negative consecutive integers uniquely and, when each component set is centred about 0, exhibit algebraic invariances. For fixed integers \(n\) and \(m\), invariance properties for \(\mathcal{N}_m(n)\) are established. The formula for \(\mathcal{N}_m(n)\) is comprised of sums over associated divisor functions and the Stirling numbers of the second kind, and we conclude by deducing sum over divisor relations for our counting function \(\mathcal{N}_m(n)\). Some related integer sequences are also considered.

In this work we study the acyclic orientations of graphs. We obtain an encoding of the acyclic orientations of the complete \(p\)-partite graph with size of its parts \(n_1,n_2,\ldots,n_p\) via a vector with \(p\) symbols and length \(n=n_1+n_2+\ldots+n_p\) when the parts are fixed but not the vertices in each part. We also give a recursive way to construct all acyclic orientations of a complete multipartite graph, this construction can be done by computer easily in order \(\mathcal{O}(n)\). Furthermore, we obtain a closed formula for non-isomorphic acyclic orientations of both the complete multipartite graphs and the complete multipartite graphs with a directed spanning tree. Moreover, we obtain a closed formula for the number of acyclic orientations of a complete multipartite graph \(K_{n_1,\ldots,n_p}\) with labelled vertices. Finally, we obtain a way encode all acyclic orientations of an arbitrary graph as a permutation code. Using the codification mentioned above we obtain sharp upper and lower bounds of the number of acyclic orientations of a graph.

In this work, we defined almost neo balancing numbers and determined the general terms of them in terms of balancing and Lucas-balancing numbers. We also deduced some results on relationship with triangular, square triangular, Pell, Pell-Lucas numbers and these numbers. Further we formulate the sum of first \(n\)-terms of these numbers.

Let \(G\) be a graph with vertex set \(V(G) = \{v_1, v_2, \dots, v_n\}\). We associate to \(G\), a matrix \(P(G)\) whose \((i, j)\)-th entry is the maximum number of vertex-disjoint paths between the corresponding vertices if \(i\neq j\), and is zero otherwise. We call this matrix the path matrix of \(G\), and its eigenvalues are referred to as the path eigenvalues of \(G\). In this paper, we investigate the path eigenvalues of graphs resulting from certain graph operations and specific graph families.

Null graphs (respectively, 1-regular graphs) are the only regular graphs with local antimagic chromatic number 1 (respectively, undefined). In this paper, we proved that the join of 1-regular graph and a null graph has local antimagic chromatic number 3. As a by-product, we also obtained many families of (possibly disconnected or regular) bipartite and tripartite graph with local antimagic chromatic number 3.

Let \(G = (V, E)\) be a graph with vertex set \(V\) and edge set \(E\). A vertex set \(S \subset V\) is a perfect dominating set if every vertex in \(V – S\) is adjacent to exactly one vertex in \(S\). A perfect dominating set \(S\) is furthermore: (i) an efficient dominating set or a \(1\)-efficient dominating set if no two vertices in \(S\) are adjacent, (ii) a total efficient dominating set or a \(2\)-efficient dominating set if every vertex in \(S\) is adjacent to exactly one other vertex in \(S\), and (iii) a \(1,2\)-efficient dominating set if every vertex in \(S\) either adjacent to no vertices in \(S\) or to exactly one other vertex in \(S\). In this paper we introduce the concept of \(1,2\)-efficiency in graphs and apply it to the existence of \(1,2\)-efficient sets in grid graphs \(G_{m,n}\), that is, graphs resembling chessboards having a rectangular array of \(m \times n\) vertices arranged into \(m\) rows of \(n\) vertices, or \(n\) columns of \(m\) vertices. It is well known that almost no grid graphs are \(1\)-efficient, and relatively few grid graphs are \(2\)-efficient. However, in this paper, we show that all but a relatively small percentage of grid graphs are \(1,2\)-efficient.

onsidering regions in a map to be adjacent when they have nonempty intersection (as opposed to the traditional view requiring intersection in a linear segment) leads to the concept of a facially complete graph: a plane graph that becomes complete when edges are added between every two vertices that lie on a face. Here we present a complete catalog of facially complete graphs: they fall into seven types. A consequence is that if \(q\) is the size of the largest face in a plane graph \(G\) that is facially complete, then \(G\) has at most \(\left\lfloor 3q/2\right\rfloor\) vertices. This bound was known, but our proof is completely different from the 1998 approach of Chen, Grigni, and Papadimitriou (Planar map graphs, Proc. 30th ACM Symp. Th. of Computing, 514–523). Our method also yields a count of the 2-connected facially complete graphs with \(n\) vertices. We also show that if a plane graph has at most two faces of size 4 and no larger face, then the addition of both diagonals to each 4-face leads to a graph that is 5-colorable.

A bowtie graph is the union of two edge disjoint 3-cycles which share a single vertex. A mixed bowtie is a partial orientation of a bowtie graph. In this paper, we consider decompositions of the complete mixed graph into mixed bowties consisting of a union of two isomorphic copies of mixed triples.

In this paper, we study the signless Laplacian eigenvalues of the subgraph \(Z^*(\Gamma(\mathbb{Z}_n))\) of the total graph of the integers modulo \(n\), \(\mathbb{Z}_n\), for certain values of \(n\). We also identify specific values of \(n\) for which the graph is \(Q\)-integral. Finally, we discuss the normalized Laplacian spectrum of \(Z^*(\Gamma(\mathbb{Z}_n))\).

We obtain several results towards the proof that the necessary conditions are sufficient for the existence of a GDD\((n_1 = 1, n_2 = n, 4; \lambda_1, \lambda_2)\) where \(\lambda_1 \ge \lambda_2\). We also have some general results including the constructions for larger block sizes as well as when the first group size \(n_1\) is not 1 or \(\lambda_1 < \lambda_2\).

We introduce a new arc in directed graphs of integers. Our arcs are determined by the values of the popular arithmetic functions such as the divisor function \(\tau\), the prime divisors counting functions \(\omega\) and \(\Omega\), and the sum of digits function \(s_b\), evaluated at the multiples \(N\) of a particular integer. Among other things, we determine the positive integers that have arcs to all except a finite number of positive integers. We also propose some possible research problems at the end of this article.

Let \(G\) be a plane graph with vertex, edge, and region sets \(V(G), E(G), F(G)\) respectively. A zonal labeling of a plane graph \(G\) is a labeling \(\ell: V(G)\rightarrow \{1,2\}\subset \mathbb{Z}_3\) such that for every region \(R\in F(G)\) with boundary \(B_R\), \(\sum\limits_{v\in V(B_R)}\ell(v)=0\) in \(\mathbb{Z}_3\). We extend this to general abelian groups, defining a \(\Gamma\)-zonal labeling as a labeling \(\ell:V(G)\rightarrow \Gamma\setminus \{0\}\) such that for every region \(R\in F(G)\), \(\sum\limits_{v \in V(B_R)}\ell(v)\) is \(0\). We explore existence of \(\Gamma\)-zonal labelings for various families of graphs. We also introduce two variations: generative and strong \(\Gamma\)-zonal labelings. A generative \(\Gamma\)-zonal labeling is one in which the elements used to label the vertices generate the group \(\Gamma\). A strong \(\Gamma\)-zonal labeling is a labeling in which the additive order of \(\ell(v)\) is equal to \(\deg(v).\) Examples and existence results are provided for both variations. It is shown that strong \(\Gamma\)-zonal labelings have a connection to edge colorings that generalizes the connection between zonal labelings and proper edge \(3\)-colorings of cubic maps.

In this paper, \(k\)-domination is considered for the king’s, queen’s, knight’s, and bishop’s graphs for square boards of any dimension size. We also consider \(k\)-tuple total domination for the queen’s and bishop’s graphs for square boards as well.

Finite games in normal form and their mixed extensions are a corner stone of noncooperative game theory. Often  generic finite games and their mixed extensions are considered. But the properties which one expects in generic games and the existence of games with these properties are often treated only in passing. The paper considers strong properties and proves that generic games have these properties. The space of mixed strategy combinations is embedded in a natural way into a product of real projective spaces. All relevant hypersurfaces extend to this bigger space. The paper shows that for all games in the complement of a semialgebraic subset of codimension at least one all relevant hypersurfaces in the bigger space are smooth and maximally transversal. The proof uses the theorem of Sard and follows an argument of Khovanskii.

A proper \(k\)-coloring \(\alpha\) of a graph \(G\) induces a partition \(\Pi = \{C_1, C_2, \dots, C_k\}\), where \(C_i = \{v \in V(G) \mid \alpha(v) = i\}\). The color code of a vertex \(v \in V(G)\) with respect to \(\Pi\) is defined as the tuple \(c_{\Pi}(v) = (d(v, C_1), d(v, C_2), \dots, d(v, C_k))\), where \(d(v, C_i)\) represents the distance from \(v\) to the set \(C_i\). A proper \(k\)-coloring \(\alpha\) is called a locating \(k\)-coloring of \(G\) if \(\alpha\) induces a partition \(\Pi\) such that for any two distinct vertices \(u, v \in V(G)\), it holds that \(c_{\Pi}(u) \neq c_{\Pi}(v)\). The locating chromatic number of \(G\), denoted \(\chi_L(G)\), is the smallest \(k\) for which a locating \(k\)-coloring of \(G\) exists. In this paper, we establish a connection between the locating \(k\)-coloring of \(C_n(1,2,\dots,t) + K_m\) and the union of graphs \(\bigcup_{i=1}^p C_{n_i} + K_m\), leveraging properties of simple cycles in directed graphs. Using this connection, we determine the locating chromatic number of \(C_n(1,2,\dots,t) + K_m\) for \(t = 2\) and \(n \in [6, 28]\), as well as for \(t = 3\) and \(n \in [8, 24]\).

We characterize line digraphs of polytrees, including several of their well-known subclasses. For a given undirected tree, we characterize its orientations with weak line digraphs, and count the exact number. Furthermore, we find the minimum, maximum, and average sizes of these line digraphs. We provide an explicit formula for the number of weak components in line digraphs of polytrees in terms of the inner sources and sinks. Additionally, we count the average number of weak components in them among all orientations of a fixed tree. Finally, we propose an algorithm for finding weak components in line digraphs of polytrees.

Let \(k\ge 1\) be an integer. Let \(G=(V,E)\) be a connected graph with \(n\) vertices and \(m\) edges. Suppose fires break out at two adjacent vertices. In each round, a firefighter can protect \(k\) vertices, and then the fires spread to all unprotected neighbors. For \(uv\in E(G)\), let \(sn_{k}(uv)\) denote the maximum number of vertices the firefighter can save when fires break out at the ends of \(uv\). The \(k\)-edge surviving rate \(\rho'_k(G)\) of \(G\) is defined as the average proportion of vertices saved when the starting vertices of the fires are chosen uniformly at random over all eages, i.e., \(\rho'_k(G)=\sum\limits_{uv\in E(G)}sn_{k}(uv)/nm\). In particular, we write \(\rho'(G)=\rho'_1(G)\). For a given class of graphs \(\mathcal{G}\) and a constant \(\varepsilon>0\), we seek the minimum value \(k\) such that \(\rho'_k(G)>\varepsilon\) for all \(G\in \mathcal{G}\). In this paper, we prove that for Halin graphs, this minimum value is exactly 1. Specifically, every Halin graph \(G\) satisfies \(\rho'(G)> 1/12\).

A bipartite labeling of a tree of order \(n\) is a bijective function that identifies the vertices of \(T\) with the elements of \(\{0, 1, \dots, n-1\}\) in such a way that there exists an integer \(\lambda\) such that the set of labels on the stable sets of \(T\) are \(\{0,1, \dots, \lambda\}\) and {\(\lambda + 1, \lambda +2. \dots, n-1\}.\) The most restrictive and versatile bipartite labeling is the variety called \(\alpha\text{-labeling}\). In this work we present a new construction of \(\alpha\text{-labeled}\) trees where any two adjacent vertices of a path-like tree, or a similar caterpillar, can be amalgamated with selected vertices of two equivalent trees.

A set \(D\) of vertices of a graph \(G=(V_G,E_G)\) is a dominating set of \(G\) if every vertex in \(V_G-D\) is adjacent to at least one vertex in \(D\). The domination number of a graph \(G\), denoted by \(\gamma(G)\), is the cardinality of a smallest dominating set of \(G\). A subset \(D\subseteq V_G\) is called a certified dominating set of \(G\) if \(D\) is a dominating set of \(G\), and every vertex in \(D\) has either zero or at least two neighbours in \(V_G-D\). The cardinality of a smallest certified dominating set of \(G\) is called the certified domination number of \(G\), and it is denoted by \(\gamma_{\rm cer}(G)\). A vertex \(v\) of \(G\) is certified critical if \(\gamma_{\rm cer}(G -v) < \gamma_{\rm cer}(G)\), and a graph \(G\) is vertex certified domination critical or \(\gamma_{cer}\)–critical if the removal of any vertex reduces its certified domination number. In this paper, we give examples and properties of certified critical vertices and vertex certified domination critical graphs. As an example of an application of certified critical vertices, we give a constructive characterisation of trees for which the smaller partite set is a minimum certified dominating set.

In this paper, we consider Ramsey and Gallai-Ramsey numbers for a generalized fan \(F_{t,n}:=K_1+nK_t\) versus triangles. Besides providing some general lower bounds, our main results include the evaluations of \(r(F_{3,2}, K_3)=13\) and \(gr(F_{3,2}, K_3, K_3)=31\).

Seymour’s Second Neighborhood Conjecture (SSNC) asserts that every finite oriented graph has a vertex \(v\) whose second out-neighborhood is at least as large as its first out-neighborhood. Such a vertex is called a Seymour vertex. In this note, we introduce pseudo-Seymour set such that Seymour’s conjecture becomes: Every oriented graph has a singleton pseudo-Seymour set. We prove that any oriented graph has a pseudo-Seymour set \(S\) with \(|S|=2\). Furthermore, we show that there are pseudo-Seymour sets of any size at least 2. We define \(\rho\)-Seymour vertex where \(0 < \rho \leq 1\), and give an approach such that finding \(\rho=1\) is equivalent to the existence of Seymour vertex. Attempting to maximize its value, we prove that for any oriented graph with minimum out-degree \(\delta\), there is \(\rho=\frac{2}{3}(1+\frac{1}{2\delta})\).

In this paper, given a homeomorphism \(f\) of a compact metric space \(X\), we show that the set of all asymptotic average shadowable points of \(f\) is an open and invariant set and \(f\) has the asymptotic average shadowing property if and only if the set of all asymptotic average shadowable points of \(f\) is \(X\) if and only if any Borel probability measure \(\mu\) of \(X\) has the asymptotic average shadowing property.

This paper is concerned with the pullback attractors for the Kirchhoff type BBM equations defined on unbounded domains. Sobolev embeddings are invalid on unbounded domains. We obtain the pullback asymptotic compactness of such non-autonomous BBM equations by using the method of uniform tail-estimates.

We say that a graph \(G\) has a path-system with respect to a set \(W\) of even number of vertices in \(G\) if \(G\) has vertex-disjoint paths \(P_1,P_2, \ldots, P_m\) such that (i) each path \(P_i\) connects two vertices of \(W\) and (ii) the set of end-vertices of the paths \(P_i\) is exactly \(W\). In particular, \(m=|W|/2\). Moreover, if \(G\) has a path-system with respect to every set \(W\) of even number of vertices in \(G\), we say that \(G\) has a path system. We prove the following theorems: (i) if \(G\) is an \(r\)-edge-connected \(r\)-regular graph, then for any \(r-1\) edges \(e_1,\ldots, e_{r-1}\), \(G-\{e_1,\ldots, e_{r-1}\}\) has a path-system, (ii) every \(k\)-connected \(K_{1,k+1}\)-free graph has a path-system, and (iii) if a connected bipartite graph \(G\) with bipartition \((A,B)\) satisfies \(|A| \le 2|B|\), \(|N_G(X)| \ge 2|X|\) or \(N_G(X)=B\) for all \(X\subseteq A\), and \(|N_G(Y)| \ge |Y|\) or \(N_G(Y)=A\) for all \(Y\subseteq B\), then \(G\) has a path-system with respect to every set \(W\) of even number of vertices of \(A\).

For a graph \(G\), an Italian dominating function (IDF) is a function \(f: V(G) \rightarrow \{0,1,2\}\) such that all vertices labeled with 0 must have at least two neighbors assigned the label 1 or at least one neighbor assigned the label 2. The weight of \(f\), denoted by \(w(f)\), is calculated by summing all the labels assigned by the function. Let \(f\) be an IDF on \(G\) with a minimum weight, denoted as \(\gamma_I(G)\). If \(S\) is the set of vertices where \(f(v) > 0\), then an Inverse Italian Dominating Function (IIDF) \(f'\) is defined as an IDF on \(G\) such that \(f'(v) = 0\) for all \(v \in S\). The notation \(\gamma_{iI}(G)\) represents the Inverse Italian Domination Number of the graph \(G\), which is the minimum weight among all IIDFs on \(G\). In this paper, we find \(\gamma_{iI}(G)\) of graphs and characterize the graphs for which \(\gamma_I(G) = 2\) and \(3\), as well as those with \(\gamma_{iI}(G) = 2\) and \(3\). Additionally, we provide a characterization of trees and graphs that achieve the largest possible \(\gamma_{iI}(G)\).

The strongly chordal graph literature has recently expanded to include the sequentially smaller classes of \(s\)-strongly chordal graphs for \(s = 1, 2, 3,\ldots\) (and the limiting class of majorly chordal graphs). These stronger classes preserve — while simultaneously intensifying —  the conventional chords-of-cycles inspiration of chordal graph theory. This leads to characterizing corresponding \(s\)-strongly chordal bipartite graphs and majorly chordal bipartite graphs. Our new analysis does this by using chains of quadrangles, with each adjacent pair of quadrangles having a unique edge in common. This leads to constructive characterizations that exploit a somewhat unexpected resemblance to earlier characterizations of \(s\)-strongly chordal graphs involving chains of triangles sharing common edges to characterize \(s\)-strongly chordal tripartite (and, similarly, multipartite) graphs.

In this paper, we present a method for constructing a new sequence, which we call \(k-\)division sequence and denoted by \(h_n(k)\). Using Fibonacci and Pell sequences, we define the \(k-\)division Fibonacci-Pell sequence obtain its properties, and prove that this sequence is periodic. Then, as an application of the sequence, we define \(1-\)division Fibonacci-Pell sequence on finite groups and study it in the groups \(G_m\) and \(H_{(u,l,m)}\) groups.

We consider the minimum weight and smallest weight minimum-size dominating set problems in vertex-weighted graphs and networks. The latter problem is a two-objective optimization problem, which is different from the classic minimum weight dominating set problem that requires finding a dominating set of the smallest weight in a graph without trying to optimize its cardinality. In other words, the objective of minimizing the size of the dominating set in the two-objective problem can be considered as a constraint, i.e. a particular case of finding Pareto-optimal solutions. First, we show how to reduce the two-objective optimization problem to the minimum weight dominating set problem by using Integer Linear Programming formulations. Then, under different assumptions, the probabilistic method is applied to obtain upper bounds on the minimum weight dominating sets in graphs. The corresponding randomized algorithms for finding small-weight dominating sets in graphs are described as well. Computational experiments are used to illustrate the results for two different types of random graphs.

Let \(G\) be a plane graph. If two edges are adjacent and consecutive on the boundary walk of a face of \(G\), then they are said to be facially adjacent. We call \(G\) facially entire \(k\)-colorable if there is a mapping from \(V(G)\cup E(G)\cup F(G)\) to a \(k\) color set so that any two facially adjacent edges, adjacent vertices, adjacent faces, and incident elements receive different colors. The facial entire chromatic number of \(G\) is defined to be the smallest integer \(k\) such that \(G\) is facially entire \(k\)-colorable. In 2016, Fabrici, Jendrol’ and Vrbjarová conjectured that every connected, loopless, bridgeless plane graph is facially entire \(7\)-colorable. In this paper, we give a positive answer to this conjecture for \(K_4\)-minor-free graphs. More specifically, we shall prove that every \(K_{4}\)-minor-free graph is facially entire \(7\)-colorable.