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.

The Markov graph of a self-map on a combinatorial tree is a directed graph that encodes the covering relations between edges of the tree under the map. This work explores the dynamical structure of self-maps on trees with weakly connected Markov graphs. The main result of the paper is a complete characterization of self-maps on finite sets that yield weakly connected Markov graphs for all trees. Additionally, we describe the dynamical structure of self-maps whose Markov graphs take specific forms, including complete digraphs, cycles, paths, in-stars, and out-stars.

In this paper, we construct two infinite families of graphs \(G(d,c)\) and \(G^+(d,c)\), where, in both cases, a vertex label is \(x_1x_2\ldots x_c\) with \(x_i\in\{1,2,\ldots, d\}\). We provide a tight lower bound on the metric dimension of \(G^+(d, c)\). Moreover, we give the definition and properties of the supertoken graphs, a generalization of the well-known token graphs. Finally, we provide an upper bound on the metric dimension of supertoken graphs.

Let \(A\) be a commutative ring with nonzero identity and \(n\geq 2\) be a positive integer. With the ring \(R=A\times\cdots\times A\) (\(n\) times), one can associate graphs \(TD(R)\) and \(ZD(R)\) respectively called the total dot product graph and the zero-divisor dot product graph of \(R\). In this paper, we study some topological properties of these two dot product graphs of \(R.\) In particular, it is shown that, the zero-divisor dot product graph \(ZD(R)\) is a projective graph if and only if \(R\) is isomorphic to \(\frac{Z_2\left[x\right]}{\left\langle x^2+x+1\right\rangle}\times\frac{Z_2\left[x\right]}{\left\langle x^2+x+1\right\rangle}.\) Moreover, we prove that no total dot product graph can be projective. With these observations, we classify all commutative rings for which dot product graphs \(ZD(R)\) and \(TD(R)\) have crosscap two.

Some methods of decomposing \(v(=mn)\times b\) incidence matrix of regular group divisible (RGD) designs into square submatrices of order \(m\) are described. Such designs are known as tactical decomposable designs. As a by–product, resolvable solutions of some RGD designs are obtained. A relationship between tactical decomposable designs and \(\left(2,\ n\right)-\)threshold schemes is also given.

Let \(G=(V,E,F)\) be a planar graph with vertex set \(V\), edge set \(E\), and set of faces \(F.\) For nonnegative integers \(a,b,\) and \(c\), a type \((a,b,c)\) face-magic labeling of \(G\) is an assignment of \(a\) labels to each vertex, \(b\) labels to each edge, and \(c\) labels to each face from the set of integer labels \(\{1,2,\dots a|V|+b|E|+c|F|\}\) such that each label is used exactly once, and for each \(s\)-sided face \(f \in F,\) the sum of the label of \(f\) with the labels of the vertices and edges incident with \(f\) is equal to some fixed constant \(\mu_s\) for every \(s.\) We find necessary and sufficient conditions for every quadruple \((a,b,c,n)\) such that the \(n\)-prism graph \(Y_n \cong K_2 \square C_n\) admits a face-magic labeling of type \((a,b,c)\).

A special type of algebraic intersection graph called the \(n\)-inordinate invariant intersection graph has been constructed based on the symmetric group, and its structural properties are studied in the literature. In this article, we discuss the different types of dominator coloring schemes of the \(n\)-inordinate invariant intersection graphs and their complements, \(n\)-inordinate invariant non-intersection graphs, by obtaining the required coloring pattern and determining the graph invariant associated with the coloring.

Let \(G\) be a connected graph and let \(d_G\) be the geodesic distance on \(V(G)\). The metric spaces \((V(G), d_{G})\) were characterized up to isometry for all finite connected \(G\) by David C. Kay and Gary Chartrand in 1965. The main result of this paper expands this characterization on infinite connected graphs. We also prove that every metric space with integer distances between its points admits an isometric embedding in \((V(G), d_G)\) for suitable \(G\).

MacMahon extensively studied integer compositions, including the notion of conjugation. More recently, Agarwal introduced \(n\)-color compositions and their cyclic versions were considered by Gibson, Gray, and Wang. In this paper, we develop and study a conjugation rule for cyclic \(n\)-color compositions. Also, for fixed \(\ell\), we identify and enumerate the subset of self-conjugate compositions of \(\ell\), as well as establish a bijection between these and the set of cyclic regular compositions of \(\ell\) with only odd parts.

The covering cover pebbling number, \(\sigma(G)\), of a graph \(G\), is the smallest number such that some distribution \(D \in \mathscr{K}\) is reachable from every distribution starting with \(\sigma(G)\) (or more) pebbles on \(G\), where \(\mathscr{K}\) is a set of covering distributions. In this paper, we determine the covering cover pebbling number for two families of graphs those do not contain any cycles.

Jeff Remmel introduced the concept of a \(\mathit{k}\)-11-representable graph in 2017. This concept was first explored by Cheon et al. in 2019, who considered it as a natural extension of word-representable graphs, which are exactly 0-11-representable graphs. A graph \(G\) is \(k\)-11-representable if it can be represented by a word \(w\) such that for any edge (resp., non-edge) \(xy\) in \(G\) the subsequence of \(w\) formed by \(x\) and \(y\) contains at most \(k\) (resp., at least \(k+1\)) pairs of consecutive equal letters. A remarkable result of Cheon et al. is that  any graph is 2-11-representable, while it is still unknown whether every graph is 1-11-representable. Cheon et al. showed that the class of 1-11-representable graphs is strictly larger than that of word-representable graphs, and they introduced a useful toolbox to study 1-11-representable graphs, which was extended by additional powerful tools suggested by Futorny et al. in 2024. In this paper, we prove that all graphs on at most 8 vertices are 1-11-representable hence extending the known fact that all graphs on at most 7 vertices are 1-11-representable. Also, we discuss applications of our main result in the study of multi-1-11-representation of graphs we introduce in this paper analogously to the notion of multi-word-representation of graphs suggested by Kenkireth and Malhotra in 2023.

Topological Indices (TIs) are quantitative measures derived from molecular geometry and are utilized to predict physicochemical properties. Although more than 3000 TIs have been documented in the published literature, only a limited number of TIs have been effectively employed owing to certain limitations. A significant drawback is the higher degeneracy resulting from the lower discriminative power. TIs utilize simple graphs in which atoms and bonds are conceptualized as the vertices and edges of mathematical graphs. As multiple edges are not supported in these graphs, double and triple bonds are considered single. Consequently, the molecular structure undergoes alterations during the conversion process, which ultimately affects the discriminative power. In this investigation, indices for double-bond incorporation were formulated to preserve structural integrity. This study addresses, demonstrates, and verifies a set of double-bonded indices. The indices demonstrated promising results, exhibiting enhanced discriminative power when validated for polycyclic aromatic hydrocarbons using regression analysis. These indices and their potential applications will significantly contribute to QSAR/QSPR studies.

With the rapid development of wireless communication networks, it brings more and more convenience to users. However, with the expansion of network size, the limitation of channel resources in network communication is becoming more obvious. Effective channel assignment has a great impact on the quality of communication networks. However, in real communication networks, underutilization of channels and excessive number of channels produce large interference, so it is necessary to find a reasonable channel assignment method. In this paper, we study the optimal channel assignment strategy for the Cartesian product of an \(m\)-vertex complete bipartite graph and an \(m\)-order cycle, where \(m\geq 5\) is odd. Determines the exact value and lower bound of its radio number.

This study introduces a novel approach to investigating Sombor indices and applying machine learning methods to assess the similarity of non-steroidal anti-inflammatory drugs (NSAIDs). The research aims to predict the structural similarities of nine commonly prescribed NSAIDs using a machine learning technique, specifically a linear regression model. Initially, Sombor indices are calculated for nine different NSAID drugs, providing numerical representations of their molecular structures. These indices are then used as features in a linear regression model trained to predict the similarity values of drug combinations. The model’s prediction performance is evaluated by comparing the predicted similarity values with the actual similarity values. Python programming is employed to verify accuracy and conduct error analysis.

Total dominator total coloring of a graph is a total coloring of the graph such that each object of the graph is adjacent or incident to every object of some color class. The minimum namber of the color classes of a total dominator total coloring of a graph is called the total dominator total chromatic number of the graph. Here, we will find the total dominator chromatic numbers of wheels, complete bipartite graphs and complete graphs.

We initiate to study a \(D\)-irregular labeling, which generalizes both non-inclusive and inclusive \(d\)-distance irregular labeling of graphs. Let \(G=(V(G),E(G))\) be a graph, \(D\) a set of distances, and \(k\) a positive integer. A mapping \(\varphi\) from \(V(G)\) to the set of positive integers \(\{1,2,\dots,k\}\) is called a \(D\)-irregular \(k\)-labeling of \(G\) if every two distinct vertices have distinct weights, where the weight of a vertex \(x\) is defined as the sum of labels of vertices whose distance from \(x\) belongs to \(D\). The least integer \(k\) for which \(G\) admits a \(D\)-irregular labeling is the \(D\)-irregularity strength of \(G\) and denoted by \(\mathrm{s}_D(G)\). In this paper, we establish several fundamental properties on \(D\)-irregularity strength for arbitrary graphs. We also determine this parameter exactly for families of graphs with small diameter or small maximum degree.

A proper coloring assigns distinct colors to the adjacent vertices of a graph. An equitable near proper coloring of a graph \(G\) is an improper coloring in which neighbouring vertices are allowed to receive the same color such that the cardinalities of two distinct color classes differ by not more than one and the number of monochromatic edges is minimised by giving certain restrictions on the number of color classes that can have an edge between them. This paper discusses the equitable near proper coloring of line, middle, and total graphs of certain graph classes, such as paths, cycles, sunlet graphs, star graphs, and gear graphs.

Directed hypergraphs represent a natural extension of directed graphs, while soft set theory provides a method for addressing vagueness and uncertainty. This paper introduces the notion of soft directed hypergraphs by integrating soft set principles into directed hypergraphs. Through parameterization, soft directed hypergraphs yield a sequence of relation descriptions derived from a directed hypergraph. Additionally, we present several operations for soft directed hypergraphs, including extended union, restricted union, extended intersection, and restricted intersection, and explore their characteristics.

For a connected graph \(G\), the edge Mostar index \(Mo_e(G)\) is defined as \(Mo_e(G)=\sum\limits_{e=uv \in E(G)}|m_u(e|G) – m_v(e|G)|\), where \(m_u(e|G)\) and \(m_v(e|G)\) are respectively, the number of edges of \(G\) lying closer to vertex \(u\) than to vertex \(v\) and the number of edges of \(G\) lying closer to vertex \(v\) than to vertex \(u\). We determine a sharp upper bound for the edge Mostar index on bicyclic graphs and identify the graphs that achieve the bound, which disproves a conjecture proposed by Liu et al. [Iranian J. Math. Chem. 11(2) (2020) 95–106].

Given a connected graph \(G\) and a configuration \(D\) of pebbles on the vertices of \(G\), a pebbling transformation involves removing two pebbles from one vertex and placing one pebble on its adjacent vertex. A monophonic path is defined as a chordless path between two non-adjacent vertices \(u\) and \(v\). The monophonic cover pebbling number, \(\gamma_{\mu}(G)\), is the minimum number of pebbles required to ensure that, after a series of pebbling transformations using monophonic paths, all vertices of \(G\) are covered with at least one pebble each. In this paper, we determine the monophonic cover pebbling number (\(MCPN\)) for the gear graph, sunflower planar graph, sun graph, closed sun graph, tadpole graph, lollipop graph, double star-path graph, and a class of fuses.

By means of the generating function method, a linear recurrence relation is explicitly resolved. The solution is expressed in terms of the Stirling numbers of both the first and the second kind. Two remarkable pairs of combinatorial identities (Theorems 3.1 and 3.3) are established as applications, that contain some well–known convolution formulae on Stirling numbers as special cases.

For a graph G and for non-negative integers p, q, and r, the triplet \((p, q, r)\) is said to be an admissible triplet if \(3p + 4q + 6r = |E(G)|\). If G admits a decomposition into p cycles of length 3, q cycles of length 4, and r cycles of length 6 for every admissible triplet \((p, q, r)\), then we say that G has a \(\{C_{3}^{p}, C_{4}^{q}, C_{6}^{r}\}\)-decomposition. In this paper, the necessary conditions for the existence of \(\{C_{3}^{p}, C_{4}^{q}, C_{6}^{r}\}\)-decomposition of \(K_{\ell, m, n} (\ell \leq m \leq n)\) are proved to be sufficient. This affirmatively answers the problem raised in Decomposing complete tripartite graphs into cycles of lengths 3 and 4, Discrete Math. 197/198 (1999), 123-135. As a corollary, we deduce the main results of Decomposing complete tripartite graphs into cycles of lengths 3 and 4, Discrete Math., 197/198, 123-135 (1999) and Decompositions of complete tripartite graphs into cycles of lengths 3 and 6, Austral. J. Combin., 73(1), 220-241 (2019).

A graph G(V, E) is Γ-harmonious when there is an injection f from V to an Abelian group Γ such that the induced edge labels defined as w(xy) = f(x) + f(y) form a bijection from E to Γ. We study Γ-harmonious labelings of several cycles-related classes of graphs, including Dutch windmills, generalized prisms, generalized closed and open webs, and superwheels.

If Γ is a finite group and G a graph such that Aut(G) ≡ Γ acts regularly on V(G), then we say that G is a graphical regular representation (GRR) of Γ. The question asking which finite groups have at least one GRR was an important question in algebraic graph theory and it was completely solved as a result of work done by several researchers. However, it remains a challenge to discern whether a group known to have GRRs has GRRs with specific properties, such as being trivalent. In this paper, we shall be deriving simple conditions on the parameters of a subset of a dihedral group for easily constructing trivalent graphical regular representations (GRR) of the group. Specifically, we shall prove the following:

Let n be an odd integer greater than 5 and let r, s, and t be integers less than n such that the difference of any two of them is relatively prime to n. If 3r – 2s = t (mod n), then Cay(D_n, {ab^r, ab^s, ab^t}) is a GRR of D_n.

We will also be looking at very convenient corollaries of this result. But another main aim of this paper is to show how a simple use of Schur rings can be used to derive such results. This paper therefore also serves as a review of some basic results about Schur rings which we feel should be among the standard armory of an algebraic graph theorist.

This paper presents an investigation of a modified Leslie-Gower predator-prey model that incorporates fractional discrete-time Michaelis-Menten type prey harvesting. The analysis focuses on the topology of nonnegative interior fixed points, including their existence and stability dynamics. We derive conditions for the occurrence of flip and Neimark-Sacker bifurcations using the center manifold theorem and bifurcation theory. Numerical simulations, conducted with a computer package, are presented to demonstrate the consistency of the theoretical findings. Overall, our study sheds light on the complex dynamics that arise in this model and highlights the importance of considering fractional calculus in predator-prey systems with harvesting.

For a set \( S \) of vertices in a connected graph \( G \), the multiplicative distance of a vertex \( v \) with respect to \( S \) is defined by \(d_{S}^{*}(v) = \prod\limits_{x \in S, x \neq v} d(v,x).\) If \( d_{S}^{*}(u) \neq d_{S}^{*}(v) \) for each pair \( u,v \) of distinct vertices of \( G \), then \( S \) is called a multiplicative distance-locating set of \( G \). The minimum cardinality of a multiplicative distance-locating set of \( G \) is called its multiplicative distance-location number \( loc_{d}^{*}(G) \). If \( d_{S}^{*}(u) \neq d_{S}^{*}(v) \) for each pair \( u,v \) of distinct vertices of \( G-S \), then \( S \) is called an external multiplicative distance-locating set of \( G \). The minimum cardinality of an external multiplicative distance-locating set of \( G \) is called its external multiplicative location number \( loc_{e}^{*}(G) \). We prove the existence or non-existence of multiplicative distance-locating sets in some well-known classes of connected graphs. Also, we introduce a family of connected graphs such that \( loc_{d}^{*}(G) \) exists. Moreover, there are infinite classes of connected graphs \( G \) for which \( loc_{d}^{*}(G) \) exists as well as infinite classes of connected graphs \( G \) for which \( loc_{d}^{*}(G) \) does not exist. A lower bound for the multiplicative distance-location number of a connected graph is established in terms of its order and diameter.

Earlier optimal key pre\(-\)distribution schemes (KPSs) for distributed sensor networks (DSNs) were proposed using combinatorial designs via transversal designs, affine, and partially affine resolvable designs. Here, nearly optimal KPSs are introduced and a class of such KPSs is obtained from resolvable group divisible designs. These KPSs are nearly optimal in the sense of local connectivity. A metric for the efficiency of KPSs is given. Further, an optimal KPS has also been proposed using affine resolvable \( L_{2} \)-type design.

We study the nonzero algebraic real algebras \( A \) with no nonzero joint divisor of zero. We prove that if \( Z(A) \neq 0 \) and \( A \) satisfies one of the Moufang identities, then \( A \) is isomorphic to \( \mathbb{R} \), \( \mathbb{C} \), \( \mathbb{H} \), or \( \mathbb{O} \). We show also that if \( A \) is power-associative, flexible, and satisfies the identity \( (a,a,[a,b])=0 \), then \( A \) is isomorphic to \( \mathbb{R} \), \( \mathbb{C} \), \( \mathbb{H} \), or \( \mathbb{O} \). Finally, we prove that \( \mathbb{R} \), \( \mathbb{C} \), \( \mathbb{H} \), and \( \mathbb{O} \) are the only algebraic real algebras with no nonzero divisor of zero satisfying the middle Moufang identity, or the right and left Moufang identities.

The metric dimension of a graph is the smallest number of vertices such that all vertices are uniquely determined by their distances to the chosen vertices. The corona product of graphs \( G \) and \( H \) is the graph \( G \odot H \) obtained by taking one copy of \( G \), called the center graph, \( |V(G)| \) copies of \( H \), called the outer graph, and making the \( j^{th} \) vertex of \( G \) adjacent to every vertex of the \( j^{th} \) copy of \( H \), where \( 1 \leqslant j \leqslant |V(G)| \). The Join graph \( G + H \) of two graphs \( G \) and \( H \) is the graph with vertex set \( V(G + H)=V(G) \cup V(H) \) and edge set \( E(G + H)=E(G) \cup E(H) \cup \{uv :u \in V(G),v \in V(H)\} \). In this paper, we determine the Metric dimension of Corona product and Join graph of zero divisor graphs of direct product of finite fields.

Generally, all the models discussed so far are continuous time models. The continuous time models are quite apt at explaining the phenomena they are trying to predict and have known methods to get information from these type of models. But these models are not accurate for the physical systems which are observed over discreet time periods or which have non-continuous phenomena embedded in them, like production of new generation. Some species like salmon have non-overlapping generation characteristics since they have an annual spawning season and are born each year at a certain time. The discrete models are much more apt in describing the nature’s complex dynamics than the continuous models. A discrete-time modified Leslie-Gower system with double Allee effect is studied in this paper. The stability analysis of interior fixed points is performed. Using center manifold theorem it is shown that the system under consideration exhibits period-doubling and Neimark-Sacker bifurcations. The numerical simulations are provided to illustrate the consistency of the theoretical results.

We investigate the Sombor indices for a diverse group of nonsteroidal anti-inflammatory drugs (NSAIDs) to understand their molecular architecture and physicochemical properties. By utilizing quantitative structure-property relationship (QSPR) modeling, we establish mathematical models linking Sombor indices to key pharmacodynamic and toxicological parameters. Our study sheds light on how the molecular composition of NSAIDs influences their drug profiles and biological behavior, offering valuable insights for drug development and safety assessment.

In this paper, the relations of maximum degree energy and maximum reserve degree energy of a complete graph after removing a vertex have been shown to be proportional to the energy of the complete graph. The results of splitting the graph and shadow graphs are also presented for the complete graph after removing a vertex.