We present a proof of a conjecture of Goh and Wildberger on the factorization of the spread polynomials. We indicate how the factors can be effectively calculated and exhibit a connection to the factorization of Fibonacci numbers into primitive parts.

An edge-coloring of a graph \(G\) with natural numbers \(1,2,\ldots\) is called a sum edge-coloring if the colors of edges incident to any vertex of \(G\) are distinct and the sum of the colors of the edges of \(G\) is minimum. The edge-chromatic sum of a graph \(G\) is the sum of the colors of edges in a sum edge-coloring of \(G\). In general, the problem of finding the edge-chromatic sum of an \(r\)-regular (\(r\geq 3\)) graph is \(NP\)-complete. In this paper we provide some bounds on the edge-chromatic sums of various products of graphs. In particular, we give tight upper bounds on the edge-chromatic sums of tensor, strong tensor, Cartesian, strong products and composition of graphs. We also determine the edge-chromatic sums and edge-strengths of the Cartesian products of regular graphs and paths (cycles) with an even number of vertices. Finally, we determine the edge-chromatic sums and edge-strengths of grids, cylinders, and tori.

Generalised nice sets are defined as subsets of edges of the extended Fano plane satisfying a certain absorbing property. The classification up to collineations, purely combinatorial in nature, provides 245 generalised nice sets. In turn, this gives rise to new Lie algebras obtained by modifying the bracket of homogeneous elements on an initial \(\mathbb Z_2^3\)-graded Lie algebra.

Let \(\alpha(n)\) denote the number of perfect square permutations in the symmetric group \(S_n\). The conjecture \(\alpha(2n+1) = (2n+1) \alpha(2n)\), provided by Stanley [4], was proved by Blum [1] using generating functions. However, several structural questions about these special permutations remained open. This paper presents an alternative and constructive proof for this conjecture, which highlights the deeper interplay between cycle structures and square properties. At the same time, we demonstrate that all permutations with an even number of even cycles in both \(S_{2n}\) and \(S_{2n+1}\) can be categorized into three disjoint types that correspond to each other.

In this paper, we generalize the \(k\)-Jacobsthal sequences and call them the generalized \((k,t)\)-Jacobsthal \(p\)-sequences. Also, we obtain combinatorial identities. Then, the generalized\((k,t)\)-Jacobsthal \(p\)-matrix is used to factorize the Pascal matrix. Finally, using the Riordan method, we obtain two factorizations of the Pascal matrix involving the generalized \((k,t)\)-Jacobsthal \(p\)-sequences.

An open-dominating set \(S\) for a graph \(G\) is a subset of vertices where every vertex has a neighbor in \(S\). An open-locating-dominating set \(S\) for a graph \(G\) is an open-dominating set such that each pair of distinct vertices in \(G\) have distinct set of open-neighbors in \(S\). We consider a type of a fault-tolerant open-locating dominating set called error-detecting open-locating-dominating sets. We present more results on the topic including its NP-completeness proof, extremal graphs, and a characterization of cubic graphs that permit an error-detecting open-locating-dominating set.

In this paper, we explore some interesting applications of the matrix tree theorem. In particular, we present a combinatorial interpretation of a distribution of \((n-1)^{n-1}\), in the context of uprooted spanning trees of the complete graph \(K_{n}\), which was previously obtained by Chauve–Dulucq–Guibert. Additionally, we establish a combinatorial explanation for the distribution of \(m^{n-1}n^{m-1}\), related to spanning trees of the complete bipartite graph \(K_{m,n}\), which seems new. Furthermore, we extend this study to the graph \(K_{n}\setminus \{e_{1,n}\}\), obtained by deleting an edge from \(K_n\), and derive a new identity for the number of its uprooted spanning trees.

The domatic number of a graph is the maximum number of pairwise disjoint dominating sets admitted by the graph. We introduce a game based around this graph invariant. The domatic number game is played on a graph \(G\) by two players, Alice and Bob, who take turns selecting a vertex and placing it into one of \(k\) sets. Alice is trying to make each of these sets into a dominating set of \(G\) while Bob’s goal is to prevent this from being accomplished. The maximum \(k\) for which Alice can achieve her goal when both players are playing optimal strategies, is called the game domatic number of \(G\). There are two versions of the game and two resulting invariants depending on whether Alice or Bob is the first to play. We prove several upper bounds on these game domatic numbers of arbitrary graphs and find the exact values for several classes of graphs including trees, complete bipartite graphs, cycles and some narrow grid graphs. We pose several open problems concerning the effect of standard graph operations on the game domatic number as well as a vexing question related to the monotonicity of the number of sets available to Alice.

Sparse magic squares are a generalization of magic squares and can be used to the magic labeling of graphs. An \(n\times n\) array based on \(\mathcal{X}\)\(=\{0,1,\cdots,nd\}\) is called a sparse magic square of order \(n\) with density \(d\) (\(d<n\)), denoted by SMS\((n,d)\), if each non-zero element of \(\mathcal{X}\) occurs exactly once in the array, and its row-sums, column-sums and two main diagonal sums is the same. An SMS\((n,d)\) is called pandiagonal (or perfect) denoted by PSMS\((n,d)\), if the sum of all elements in each broken diagonal is the same. A PSMS\((n,d)\) is called regular if there are eactly \(d\) positive entries in each row, each column and each main diagonal. In this paper, some construction of regular pandigonal sparse magic squares is provided and it is proved that there exists a regular PSMS\((n,6)\) for all positive integer \(n\equiv 5 \pmod{6}\), \(n>6\).

Two graphs are said to be \(Q\)-cospectral (respectively, \(A\)-cospectral) if they have the same signless Laplacian (respectively, adjacency) spectrum. A graph is said to be \(DQS\) (respectively, \(DAS\)) if there is no other non-isomorphic graphs \(Q\)-cospectral (respectively, \(A\)-cospectral) with it. A tree on \(n\) vertices with maximum degree \(d_1\) is called starlike and denoted by \(ST(n, d_1)\), if it has exactly one vertex with the degree greater than 2. A tree is called double starlike if it has exactly two vertices of degree greater than 2. If we attach \(p\) pendant vertices (vertices of degree 1) to each of pendant vertices of a path \(P_n\), the the resulting graph is called the double starlike tree \(H_n(p,p)\). In this article, we prove that all double starlike trees \(H_n(p,p)\) are \(DQS\), where \(p\geq 1, n\geq 2\) and \(p\) denotes . In addition, by a simple method, we show that all starlike trees are \(DQS\) excluding \(K_{1,3}=ST(4,3)\).

A split graph is a graph in which the vertices can be partitioned into an independent set and a clique. We show that every nonsplit graph has at most four split maximal proper edge induced subgraphs. The exhaustive list of fifteen classes of nonsplit graphs having a split maximal proper edge induced subgraph is determined in this paper.

A kernel \(J\) of a digraph \(D\) is an independent set of vertices of \(D\) such that for every \(z\in V(D)\backslash J\) there exists an arc from \(z\) to \(J.\) A digraph \(D\) is said to be kernel-perfect if every induced subdigraph of it has a kernel. We characterise kernel-perfectness in special families of digraphs, namely, the line digraph, the subdivision digraph, the middle digraph, the digraph \(R(D)\) and the total digraph. We also obtain some results on kernel-perfectness in the generalised Mycielskian of digraphs. Moreover, we find some new classes of kernel-perfect digraphs by introducing a new product on digraphs.

The first, second Zagreb connection indices and modified first Zagreb connection index are defined as \(Z{C_1}(G)={\sum\limits_{{v\in V(G)}} {{\tau _G}^2(v)} }\), \(ZC{}_{2}(G)=\displaystyle\sum_{uv\in E(G)}^{}\tau{}_{G}(u)\tau{}_{G}(v)\) and \(ZC{}_{1}^{\ast }(G)=\displaystyle\sum_{v\in V(G)}^{}d{}_{G}(v)\tau{}_{G}(v)\), respectively. In this paper, we consider the maximum values of \(Z{C_1}(G)\), \(Z{C_2}(G)\), \(Z{C_1}^{*}(G)\) of \(n\)-vertex trees with fixed matching number \(m\) and the extremal graphs are also characterized.

An improper interval (edge) coloring of a graph \(G\) is an assignment of integer colors to the edges of \(G\) satisfying the condition that, for every vertex \(v \in V(G)\), the set of colors assigned to the edges incident with \(v\) forms an integral interval. An interval coloring is \(k\)-improper if at most \(k\) edges with the same color all share a common endpoint. The minimum integer \(k\) such that there exists a \(k\)-improper interval coloring of the graph \(G\) is the interval coloring impropriety of \(G\), denoted by \(\mu_{int}(G)\). In this paper, we provide a construction of an interval coloring of a subclass of complete multipartite graphs. Additionally, we determine improved upper bounds on the interval coloring impropriety of several classes of graphs, namely 2-trees, iterated triangulations, and outerplanar graphs. Finally, we investigate the interval coloring impropriety of the corona product of two graphs, \(G\odot H\).

A decomposition \(\mathcal{C}\) of a graph \(G\) is primitive if no proper, nontrivial subset of \(\mathcal{C}\) is a decomposition of an induced subgraph of \(G\). The existence of primitive decompositions has been studied for several decompositions, including path and cycle decompositions for complete and cocktail party graphs. In this work, we classify the existence of primitive star decompositions for complete graphs.

In this paper, we study the distribution on \([k]^n\) for the parameter recording the number of indices \(i \in [n-1]\) within a word \(w=w_1\cdots w_n\) such that \(|w_{i+1}-w_i|\ \leq 1\) and compute the corresponding (bivariate) generating function. A circular version of the problem wherein one considers whether or not \(|w_n-w_1|\ \leq 1\) as well is also treated. As special cases of our results, one obtains formulas involving staircase and Hertzsprung words in both the linear and circular cases. We make use of properties of special matrices in deriving our results, which may be expressed in terms of Chebyshev polynomials. A generating function formula is also found for the comparable statistic on finite set partitions with a fixed number of blocks represented sequentially.

Let \(G=(V,E)\) be a graph of order \(n\) without isolated vertices. A bijection \(f\colon V\rightarrow \{1,2,\dots,n\}\) is called a local distance antimagic labeling, if \(w(u)\not=w(v)\) for every edge \(uv\) of \(G\), where \(w(u)=\sum_{x\in N(u)}f(x)\). The local distance antimagic chromatic number \(\chi_{ld}(G)\) is defined to be the minimum number of colors taken over all colorings of \(G\) induced by local distance antimagic labelings of \(G\). The concept of Generalized Mycielskian graphs was introduced by Stiebitz [20]. In this paper, we study the local distance antimagic labeling of the Generalized Mycielskian graphs.

A graph is called \(t\)-existentially closed (\(t\)-e.c.) if it satisfies the following adjacency property: for every \(t\)-element subset \(A\) of the vertices, and for every subset \(B \subseteq A\), there exists a vertex \(x \in A\) that is adjacent to all vertices in \(B\) and to none of the vertices in \(A \setminus B\). A \(t\)-e.c. graph is critical if removing any single vertex results in a graph that is no longer \(t\)-e.c. This paper investigates \(2\)-e.c. critical Cayley graphs and vertex-transitive graphs, providing explicit constructions of \(2\)-e.c. critical Cayley graphs on cyclic groups. It is shown that a \(2\)-e.c. critical Cayley graph (as well as vertex-transitive graphs) of order \(n\) exists if and only if \(n \geq 9\) and \(n \notin \{10, 11, 14\}\). Additionally, this paper examines the numbers of \(2\)-e.c. (critical) vertex-transitive graphs among all vertex-transitive graphs for small orders, and presents detailed observations on some \(2\)-e.c. and \(3\)-e.c. vertex-transitive graphs.

For a family \(\mathcal F\) of graphs, a graph \(G\) is said to be \(\mathcal F\)-free if \(G\) contains no member of \(\mathcal F\) as an induced subgraph. We let \(\mathcal G_{3}(\mathcal F)\) be the family of \(3\)-connected \(\mathcal F\) -free graphs. Let \(P_{n}\) and \(C_{n}\) denote the path and the cycle of order \(n\), respectively. Let \(T_{0}\) be the tree of order nine obtained by joining a pendant edge to the central vertex of \(P_{7}\). Let \(T_{1}\) and \(T_{2}\) be the trees of order ten obtained from \(T_{0}\) by joining a new vertex to a vertex of \(P_{7}\) adjacent to an endvertex, and to a vertex of \(P_{7}\) adjacent to the central vertex, respectively. We show that \(\mathcal G_{3}(\{C_{3}, C_{4}, T_{1}\})\) and \(\mathcal G_{3}(\{C_{3}, C_{4}, T_{2}\})\) are finite families.

In this paper, we study additional aspects of the capacity distribution on the set ℬ_n of compositions of n consisting of 1’s and 2’s extending recent results of Hopkins and Tangboonduangjit. Among our results are further recurrences for this distribution as well as formulas for the total capacity and sign balance on ℬ_n. We provide algebraic and combinatorial proofs of our results. We also give combinatorial explanations of some prior results where such a proof was requested. Finally, the joint distribution of the capacity statistic with two further parameters on ℬ_n is briefly considered.

We consider a ring \(R_{u^3} = \mathbb{F}_2+u\mathbb{F}_2+u^2\mathbb{F}_2+u^3\mathbb{F}_2, u^4=0\). We discuss the structure of self-dual codes, Type I codes and Type II codes over the ring \(R_{u^3}\) in terms of the structure of their Torsion and Residue codes. We construct Type I and Type II optimal codes over the ring \(R_{u^3}\) for different lengths.

Magic squares are known to mankind since ages. With their eye catching properties, they have attracted and inspired researchers to further explore and work with them. The present paper is written with an aim to explore the usefulness of magic squares in the construction of partially balanced incomplete block (PBIB) designs. In this regard, we have proposed a new method for the construction of magic squares and studied their properties. We have also established a link between these properties and some existing association schemes. We have then introduced four new association schemes using these properties which have been later used for the construction of PBIB designs.

We consider a vertex-coloring problem where the amount one pays for using a color is a function of how many times the color is used. For a cost-function \(f\), we define the \(f\)-chromatic number of graph \(G\) as the minimum cost of a (proper) coloring of \(G\), and focus on the case that the marginal costs \(f(i+1)-f(i)\) are non-increasing. We provide bounds for general graphs, for specific classes of graphs, and for some operations on graphs. We also consider the number of colors used in an optimal coloring, and for example, characterize the trees where the bipartite coloring is not always optimal.

Let \(q\) be a positive integral power of some prime \(p\) and \(\mathbb{F}_{q^m}\) be a finite field with \(q^m\) elements for some \(m \in \mathbb{N}\). Here we establish a sufficient condition for the existence of primitive normal pairs of the type \((\epsilon, f(\epsilon))\) in \(\mathbb{F}_{q^m}\) over \(\mathbb{F}_{q}\) with two prescribed traces, \(\text{Tr}_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(\epsilon)=a\) and \(\text{Tr}_{{\mathbb{F}_{q^m}}/{\mathbb{F}_q}}(f(\epsilon))=b\), where \(f(x) \in \mathbb{F}_{q^m}(x)\) is a rational function with some restrictions and \(a, b \in \mathbb{F}^*_q\). Furthermore, for \(q=5^k\), \(m \geq 9\) and rational functions with degree sum 4, we explicitly find at most 13 fields in which the desired pair may not exist.

Let \(G = (V(G), E(G))\) be a simple connected graph. The inverse sum indeg index of \(G\), denoted by \(\text{ISI}(G)\), is defined as the sum of the weights \(\frac{d(u)d(v)}{d(u) + d(v)}\) of all edges \(uv\) of \(G\), where \(d(u)\) denotes the degree of a vertex in \(G\). In this paper, we first present some lower and upper bound for \(ISI\) index in terms of graph parameters such as maximum degree, minimum degree and clique number. Moreover, we compute \(ISI\) index of several graph operations like join, cartesian product, composition, corona and strong product of graphs.

We consider the eternal distance-2 domination problem, recently proposed by Cox, Meger, and Messinger, on trees. We show that finding a minimum eternal distance-2 dominating set of a tree is linear time in the order of the graph by providing a fast algorithm. Additionally, we characterize trees that have eternal distance-2 domination number equal to their domination number or their distance-2 domination number, {along with trees that are} eternal distance-2 domination critical. We conclude by providing general upper and lower bounds for the eternal distance-k domination number of a graph. We construct an infinite family of trees which meet said upper bound and another infinite family of trees whose eternal distance-k domination number is within a factor of 2 of the given lower bound.

We apply the splitting operation defined on binary matroids (Raghunathan et al., 1998) to \(p\)– matroids, where \(p\)-matroids refer to matroids representable over \(GF(p).\) We also characterize circuits, bases, and independent sets of the resulting matroid. Sufficient conditions to yield Eulerian \(p\)-matroids from Eulerian and non-Eulerian \(p\)-matroids by applying the splitting operation are obtained. A class of connected \(p\)-matroids that gives connected \(p\)-matroids under the splitting operation is characterized. In Application, we characterize a class of paving \(p\)-matroids, which produces paving matroids after the splitting operation.

For a connected graph \(G=(V,E)\) of order at least two, a \(u-v\) chordless path in \(G\) is a \(monophonic\) \(path\). The edge monophonic closed interval \(I_{em}[u,v]\) consists of all the edges lying on some \(u-v\) monophonic path. For \(S'\subseteq V(G),\) the set \(I_{em}[S']\) is the union of all sets \(I_{em}[u,v]\) for \(u,v\in S'.\) A set \(S'\) of vertices in \(G\) is called an \(edge\) \(monophonic\) \(set\) of \(G\) if \(I_{em}[S']=E(G).\) The edge monophonic number \({m_1}(G)\) of G is the minimum cardinality of its edge monophonic sets of \(G\). In this paper the monophonic number and the edge monophonic number of corona product graphs are obtained. Exact values are determined for several classes of corona product graphs.

The stretched Littlewood-Richardson coefficient \(c^{t\nu}_{t\lambda,t\mu}\) was conjectured by King, Tollu, and Toumazet to be a polynomial function in \(t\). It was shown to be true by Derksen and Weyman using semi-invariants of quivers. Later, Rassart used Steinberg’s formula, the hive conditions, and the Kostant partition function to show a stronger result that \(c^{\nu}_{\lambda,\mu}\) is indeed a polynomial in variables \(\nu, \lambda, \mu\) provided they lie in certain polyhedral cones. Motivated by Rassart’s approach, we give a short alternative proof of the polynomiality of \(c^{t\nu}_{t\lambda,t\mu}\) using Steinberg’s formula and a simple argument about the chamber complex of the Kostant partition function.

In this work, we study type B set partitions for a given specific positive integer \(k\) defined over \(\langle n \rangle = \{-n, -(n-1), \cdots, -1, 0, 1, \cdots, n-1, n\}\). We found a few generating functions of type B analogues for some of the set partition statistics defined by Wachs, White and Steingrímsson for partitions over positive integers \([n] = \{1, 2, \cdots, n\}\), both for standard and ordered set partitions respectively. We extended the idea of restricted growth functions utilized by Wachs and White for set partitions over \([n]\), in the scenario of \(\langle n \rangle\) and called the analogue as Signed Restricted Growth Function (SRGF). We discussed analogues of major index for type B partitions in terms of SRGF. We found an analogue of Foata bijection and reduced matrix for type B set partitions as done by Sagan for set partitions of \([n]\) with specific number of blocks \(k\). We conclude with some open questions regarding the type B analogue of some well known results already done in case of set partitions of \([n]\).

Suppose that \(\phi\) is a proper edge-\(k\)-coloring of the graph \(G\). For a vertex \(v \in V(G)\), let \(C_\phi(v)\) denote the set of colors assigned to the edges incident with \(v\). The proper edge-\(k\)-coloring \(\phi\) of \(G\) is strict neighbor-distinguishing if for any adjacent vertices \(u\) and \(v\), \(C_\phi(u) \varsubsetneq C_\phi(v)\) and \(C_\phi(v) \varsubsetneq C_\phi(u)\). The strict neighbor-distinguishing index, denoted \(\chi’_{snd}(G)\), is the minimum integer \(k\) such that \(G\) has a strict neighbor-distinguishing edge-\(k\)-coloring. In this paper we prove that if \(G\) is a simple graph with maximum degree five, then \(\chi’_{snd}(G) \leq 12\).

Let \(2 \le k \in \mathbb{Z}\). A total coloring of a \(k\)-regular simple graph via \(k+1\) colors is an efficient total coloring if each color yields an efficient dominating set, where the efficient domination condition applies to the restriction of each color class to the vertex set. In this work, focus is set upon graphs of girth \(k+1\). Efficient total colorings of finite connected simple cubic graphs of girth 4 are constructed starting at the 3-cube. It is conjectured that all of them are obtained by means of four basic operations. In contrast, the Robertson 19-vertex \((4,5)\)-cage, the alternate union \(Pet^k\) of a (Hamilton) \(10k\)-cycle with \(k\) pentagon and \(k\)-pentagram 5-cycles, for \(k > 1\) not divisible by 5, and its double cover \(Dod^k\), contain TCs that are nonefficient. Applications to partitions into 3-paths and 3-stars are given.

Using generating functions, we are proposing a unified approach to produce explicit formulas, which count the number of nodes in Smolyak grids based on various univariate quadrature or interpolation rules. Our approach yields, for instance, a new formula for the cardinality of a Smolyak grid, which is based on Chebyshev nodes of the first kind and it allows to recover certain counting-formulas previously found by Bungartz-Griebel, Kaarnioja, Müller-Gronbach, Novak-Ritter and Ullrich.

Topological indices have become an essential tool to investigate theoretical and practical problems in various scientific areas. In chemical graph theory, a significant research work, which is associated with the topological indices, is to deduce the ideal bounds and relationships between known topological indices. Mathematical development of the novel topological index is valid only if the topological index shows a good correlation with the physico-chemical properties of chemical compounds. In this article, the chemical applicability of the novel GQ and QG indices is calibrated over physico-chemical properties of 22 benzenoid hydrocarbons. The GQ and QG indices predict the physico-chemical properties of benzenoid hydrocarbons, significantly. Additionally, this work establishes some mathematical relationships between each of the GQ and QG indices and each of the graph invariants: size, degree sequences, maximum and minimum degrees, and some well-known degree-based topological indices of the graph.

In 2003, the frequency assignment problem in a cellular network motivated Even et al. to introduce a new coloring problem: Conflict-Free coloring. Inspired by this problem and by the Gardner-Bodlaender’s coloring game, in 2020, Chimelli and Dantas introduced the Conflict-Free Closed Neighborhood \(k\)-coloring game (CFCN \(k\)-coloring game). The game starts with an uncolored graph \(G\), \(k\geq 2\) different colors, and two players, Alice and Bob, who alternately color the vertices of \(G\). Both players can start the game and respect the following legal coloring rule: for every vertex \(v\), if the closed neighborhood \(N[v]\) of \(v\) is fully colored then there exists a color that was used only once in \(N[v]\). Alice wins if she ends up with a Conflict-Free Closed Neighborhood \(k\)-coloring of \(G\), otherwise, Bob wins if he prevents it from happening. In this paper, we introduce the game for open neighborhoods, the Conflict-Free Open Neighborhood \(k\)-coloring game (CFON \(k\)-coloring game), and study both games on graph classes determining the least number of colors needed for Alice to win the game.

This paper investigates the number of rooted biloopless nonseparable planar near-triangulations and presents some formulae for such maps with three parameters: the valency of root-face, the number of edges and the number of inner faces. All of them are almost summation-free.

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we confirm the total-coloring conjecture for 1-planar graphs without 4-cycles with maximum degree \(\Delta\geq10\).

For a graph \(G=(V,E)\) of size \(q\), a bijection \(f : E \to \{1,2,\ldots,q\}\) is a local antimagic labeling if it induces a vertex labeling \(f^+ : V \to \mathbb{N}\) such that \(f^+(u) \ne f^+(v)\), where \(f^+(u)\) is the sum of all the incident edge label(s) of \(u\), for every edge \(uv \in E(G)\). In this paper, we make use of matrices of fixed sizes to construct several families of infinitely many tripartite graphs with local antimagic chromatic number 3.

An outer independent double Roman dominating function (OIDRDF) of a graph \( G \) is a function \( f:V(G)\rightarrow\{0,1,2,3\} \) satisfying the following conditions:
(i) every vertex \( v \) with \( f(v)=0 \) is adjacent to a vertex assigned 3 or at least two vertices assigned 2;
(ii) every vertex \( v \) with \( f(v)=1 \) has a neighbor assigned 2 or 3;
(iii) no two vertices assigned 0 are adjacent.
The weight of an OIDRDF is the sum of its function values over all vertices, and the outer independent double Roman domination number \( \gamma_{oidR}(G) \) is the minimum weight of an OIDRDF on \( G \). Ahangar et al. [Appl. Math. Comput. 364 (2020) 124617] established that for every tree \( T \) of order \( n \geq 4 \), \( \gamma_{oidR}(T)\leq\frac{5}{4}n \) and posed the question of whether this bound holds for all connected graphs. In this paper, we show that for a unicyclic graph \( G \) of order \( n \), \( \gamma_{oidR}(G) \leq \frac{5n+2}{4} \), and for a bicyclic graph, \( \gamma_{oidR}(G) \leq \frac{5n+4}{4} \). We further characterize the graphs attaining these bounds, providing a negative answer to the question posed by Ahangar et al.

Let \(G\) be a \((p,q)\) graph. Let \(f\) be a function from \(V(G)\) to the set \(\{1,2,\ldots, k\}\) where \(k\) is an integer \(2< k\leq \left|V(G)\right|\). For each edge \(uv\) assign the label \(r\) where \(r\) is the remainder when \(f(u)\) is divided by \(f(v)\) (or) \(f(v)\) is divided by \(f(u)\) according as \(f(u)\geq f(v)\) or \(f(v)\geq f(u)\). \(f\) is called a \(k\)-remainder cordial labeling of \(G\) if \(\left|v_{f}(i)-v_{f}(j)\right|\leq 1\), \(i,j\in \{1,\ldots , k\}\) where \(v_{f}(x)\) denote the number of vertices labeled with \(x\) and \(\left|\eta_{e}(0)-\eta_{o}(1)\right|\leq 1\) where \(\eta_{e}(0)\) and \(\eta_{o}(1)\) respectively denote the number of edges labeled with even integers and number of edges labeled with odd integers. A graph with admits a \(k\)-remainder cordial labeling is called a \(k\)-remainder cordial graph. In this paper we investigate the \(4\)-remainder cordial labeling behavior of Prism, Crossed prism graph, Web graph, Triangular snake, \(L_{n} \odot mK_{1}\), Durer graph, Dragon graph.

Given a connected graph \(H\), its first Zagreb index \(M_{1}(H)\) is equal to the sum of squares of the degrees of all vertices in \(H\). In this paper, we give a best possible lower bound on \(M_{1}(H)\) that guarantees \(H\) is \(\tau\)-path-coverable and \(\tau\)-edge-Hamiltonian, respectively. Our research supplies a continuation of the results presented by Feng et al. (2017).

The degree of an edge \(uv\) of a graph \(G\) is \(d_G(u)+d_G(v)-2.\) The degree associated edge reconstruction number of a graph \(G\) (or dern(G)) is the minimum number of degree associated edge-deleted subgraphs that uniquely determines \(G.\) Graphs whose vertices all have one of two possible degrees \(d\) and \(d+1\) are called \((d,d+1)\)-bidegreed graphs. It was proved, in a sequence of two papers [1,17], that \(dern(mK_{1,3})=4\) for \(m>1,\) \(dern(mK_{2,3})=dern(rP_3)=3\) for \(m>0, ~r>1\) and \(dern(G)=1\) or \(2\) for all other bidegreed graphs \(G\) except the \((d,d+1)\)-bidegreed graphs in which a vertex of degree \(d+1\) is adjacent to at least two vertices of degree \(d.\) In this paper, we prove that \(dern(G)= 1\) or \(2\) for this exceptional bidegreed graphs \(G.\) Thus, \(dern(G)\leq 4\) for all bidegreed graphs \(G.\)

A proper total coloring of a graph \( G \) such that there are at least 4 colors on those vertices and edges incident with a cycle of \( G \), is called an acyclic total coloring. The acyclic total chromatic number of \( G \), denoted by \( \chi^{”}_{a}(G) \), is the smallest number of colors such that \( G \) has an acyclic total coloring. In this article, we prove that for any graph \( G \) with \( \Delta(G)=\Delta \) which satisfies \( \chi^{”}(G)\leq A \) for some constant \( A \), and for any integer \( r \), \( 1\leq r \leq 2\Delta \), there exists a constant \( c>0 \) such that if \( g(G)\geq\frac{c\Delta}{r}\log\frac{\Delta^{2}}{r} \), then \( \chi^{”}_{a}(G)\leq A+r \).

We study a discrete-time model for the spread of information in a graph, motivated by the idea that people believe a story when they learn of it from two different origins. Similar to the burning number, in this problem, information spreads in rounds and a new source can appear in each round. For a graph \(G\), we are interested in \(b_2(G)\), the minimum number of rounds until the information has spread to all vertices of graph \(G\). We are also interested in finding \(t_2(G)\), the minimum number of sources necessary so that the information spreads to all vertices of \(G\) in \(b_2(G)\) rounds. In addition to general results, we find \(b_2(G)\) and \(t_2(G)\) for the classes of spiders and wheels and show that their behavior differs with respect to these two parameters. We also provide examples and prove upper bounds for these parameters for Cartesian products of graphs.

An hourglass \(\Gamma_0\) is the graph with degree sequence \(\{4,2,2,2,2\}\). In this paper, for integers \(j\geq i\geq 1\), the bull \(B_{i,j}\) is the graph obtained by attaching endvertices of two disjoint paths of lengths \(i,j\) to two vertices of a triangle. We show that every 3-connected \(\{K_{1,3},\Gamma_0,X\}\)-free graph, where \(X\in \{ B_{2,12},\,B_{4,10},\,B_{6,8}\}\), is Hamilton-connected. Moreover, we give an example to show the sharpness of our result, and complete the characterization of forbidden induced bulls implying Hamilton-connectedness of a 3-connected {claw, hourglass, bull}-free graph.

Let \(G=(V,E)\) be a simple connected graph with vertex set \(V\) and edge set \(E\). The Randić index of graph \(G\) is the value \(R(G)=\sum_{uv\in E(G)} \frac{1}{\sqrt{d(u)d(v)}}\), where \(d(u)\) and \(d(v)\) refer to the degree of the vertices \(u\) and \(v\). We obtain a lower bound for the Randić index of trees in terms of the order and the Roman domination number, and we characterize the extremal trees for this bound.

In this paper, it is pointed out that the definition of `Fibonacci \((p,r)\)-cube’ in many papers (denoted by \(I\Gamma_{n}^{(p,r)}\)) is incorrect. The graph \(I\Gamma_{n}^{(p,r)}\) is not the same as the original one (denoted by \(O\Gamma_{n}^{(p,r)}\)) introduced by Egiazarian and Astola. First, it is shown that \(I\Gamma_{n}^{(p,r)}\) and \(O\Gamma_{n}^{(p,r)}\) have different recursive structure. Then, it is proven that all the graphs \(O\Gamma_{n}^{(p,r)}\) are partial cubes. However, only a small part of graphs \(I\Gamma_{n}^{(p,r)}\) are partial cubes. It is also shown that \(I\Gamma_{n}^{(p,r)}\) and \(O\Gamma_{n}^{(p,r)}\) have different medianicity. Finally, several questions are listed for further investigation.

A \(q\)-total coloring of \(G\) is an assignment of \(q\) colors to the vertices and edges of \(G\), so that adjacent or incident elements have different colors. The Total Coloring Conjecture (TCC) asserts that a total coloring of a graph \(G\) has at least \(\Delta+1\) and at most \(\Delta+2\) colors. In this paper, we determine that all members of new infinite families of snarks obtained by the Kochol superposition of Goldberg and Loupekine with Blowup and Semiblowup snarks are Type~1. These results contribute to a question posed by Brinkmann, Preissmann and D. Sasaki (2015) by presenting negative evidence about the existence of Type~2 cubic graphs with girth at least 5.

In this note, we establish six Gallai theorems involving twelve minority and majority parameters. Accordingly, the complexity problems corresponding to some of these parameters are obtained.

A \(k\)-tree is a graph that can be formed by starting with \(K_{k+1}\) and iterating the operation of making a new vertex adjacent to all the vertices of a \(k\)-clique of the existing graph. A structural characterization of 3-trees with diameter at most 2 is proven. This implies a corollary for planar 3-trees which leads to a description of their degree sequences.

In this paper, we present a new combinatorial characterization of Hermitian cones in \(\mathrm{PG}(3,q^2)\).

In this paper we consider some new weighted and alternating weighted generalized Fibonomial sums and the corresponding \(q-\)forms. A generalized form of weight sequences which contains squares in subscripts is discussed for the first time in the literature. The main key to get success in sums is an ability to change one sum into another that is simpler in some way. Thus, in order to prove these sums by doing some manipulations and tricks, our approach is to use classical \(q-\)analysis, in particular a formula of Rothe, a version of Cauchy binomial theorem and Gauss identity.

A new series of four-associate class partially balanced incomplete block designs in two replications has been proposed. The blocks of these designs are of two different sizes. The blocks can be divided into two groups such that every treatment appears in each group exactly once, and any two blocks belonging to two different groups have a constant number of treatments in common, i.e., these designs are affine resolvable.

Let \( 0<k\in\mathbb{Z} \). Let the star 2-set transposition graph \( ST^2_k \) be the \( (2k-1) \)-regular graph whose vertices are the \( 2k \)-strings on \( k \) symbols, each symbol repeated twice, with its edges given each by the transposition of the initial entry of one such \( 2k \)-string with any entry that contains a different symbol than that of the initial entry. The pancake 2-set transposition graph \( PC^2_k \) has the same vertex set of \( ST^2_k \) and its edges involving each the maximal product of concentric disjoint transpositions in any prefix of an endvertex string, including the external transposition being that of an edge of \( ST^2_k \). For \( 1<k\in\mathbb{Z} \), we show that \( ST^2_k \) and \( PC^2_k \), among other intermediate transposition graphs, have total colorings via \( 2k-1 \) colors. They, in turn, yield efficient dominating sets, or E-sets, of the vertex sets of \( ST^2_k \) and \( PC^2_k \), and partitions into \( 2k-1 \) such E-sets, generalizing Dejter-Serra work on E-sets in such graphs.

This paper investigates the Turan-like problem for \(\mathcal{K}^-_{r + 1}\)-free \((r \geq 2)\) unbalanced signed graphs, where \(\mathcal{K}^-_{r + 1}\) is the set of unbalanced signed complete graphs with \(r+1\) vertices. The maximum number of edges and the maximum index for \(\mathcal{K}^-_{r + 1}\)-free unbalanced signed graphs are given. Moreover, the extremal \(\mathcal{K}^-_{r + 1}\)-free unbalanced signed graphs with the maximum index are characterized.

In this paper, we give a classification of all Mengerian \(4\)-uniform hypergraphs derived from graphs.

The \( n \)-dimensional Möbius cube \( MQ_n \) is an important variant of the hypercube \( Q_n \), which possesses some properties superior to the hypercube. This paper investigates the fault-tolerant edge-pancyclicity of \( MQ_n \), and shows that if \( MQ_n \) (\( n \geq 5 \)) contains at most \( n-2 \) faulty vertices and/or edges then, for any fault-free edge \( uv \) in \( MQ_n^i (i=0,1) \) and any integer \( \ell \) with \( 7-i \leqslant \ell \leqslant 2^n – f_v \), there is a fault-free cycle of length \( \ell \) containing the edge \( uv \), where \( f_v \) is the number of faulty vertices. The result is optimal in some senses.

In a recent paper Cameron, Lakshmanan and Ajith [6] began an exploration of hypergraphs defined on algebraic structures, especially groups, to investigate whether this can add a new perspective. Following their suggestions, we consider suitable hypergraphs encoding the generating properties of a finite group. In particular, answering a question asked in their paper, we classified the finite solvable groups whose generating hypergraph is the basis hypergraph of a matroid.

Let \( G \) be a graph, the zero forcing number \( Z(G) \) is the minimum of \( |Z| \) over all zero forcing sets \( Z \subseteq V(G) \). In this paper, we are interested in studying the zero forcing number of quartic circulant graphs \( C_{p}\left(s,t\right) \), where \( p \) is an odd prime. Based on the fact that \( C_{p}\left(s,t\right) \cong C_{p}\left(1,q\right) \), we give the exact values of the zero forcing number of some specific quartic circulant graphs.

Behera and Panda defined a balancing number as a number b for which the sum of the numbers from \(1\) to \(b – 1\) is equal to the sum of the numbers from \(b + 1\) to \(b + r\) for some r. They also classified all such numbers. We define two notions of balancing numbers for Farey fractions and enumerate all possible solutions. In the stricter definition, there is exactly one solution, whereas in the weaker one all sufficiently large numbers work. We also define notions of balancing numbers for levers and mobiles, then show that these variants have many acceptable arrangements. For an arbitrary mobile, we prove that we can place disjoint consecutive sequences at each of the leaves and still have the mobile balance. However, if we impose certain additional restrictions, then it is impossible to balance a mobile.

The secure edge dominating set of a graph \( G \) is an edge dominating set \( F \) with the property that for each edge \( e \in E-F \), there exists \( f \in F \) adjacent to \( e \) such that \( (F-\{f\})\cup \{e\} \) is an edge dominating set. In this paper, we obtained upper bounds for edge domination and secure edge domination number for Mycielski of a tree.

In this paper we contribute to the literature of computational chemistry by providing exact expressions for the detour index of joins of Hamilton-connected (\(HC\)) graphs. This improves upon existing results by loosening the requirement of a molecular graph being Hamilton-connected and only requirement certain subgraphs to be Hamilton-connected.

The geometrical properties of a plane determine the tilings that can be built on it. Because of the negative curvature of the hyperbolic plane, we may find several types of groups of symmetries in patterns built on such a surface, which implies the existence of an infinitude of possible tiling families. Using generating functions, we count the vertices of a uniform tiling from any fixed vertex. We count vertices for all families of valence \(5\) and for general vertices with valence \(6\), with even-sized faces. We also give some general results about the behavior of the vertices and edges of the tilings under consideration.

This study extends the concept of competition graphs to cubic fuzzy competition graphs by introducing additional variations including cubic fuzzy out-neighbourhoods, cubic fuzzy in-neighbourhoods, open neighbourhood cubic fuzzy graphs, closed neighbourhood cubic fuzzy graphs, cubic fuzzy (k) neighbourhood graphs and cubic fuzzy [k]-neighbourhood graphs. These variations provide further insights into the relationships and competition within the graph structure, each with its own defined characteristics and examples. These cubic fuzzy CMGs are further classified as cubic fuzzy k-competition graphs that show competition in the \(k\)th order between components, \(p\)-competition cubic fuzzy graphs that concentrate on competition in terms of membership degrees, and \(m\)-step cubic fuzzy competition graphs that analyze competition in terms of steps. Further, some related results about independent strong vertices and edges have been obtained for these cubic fuzzy competition graph classes. Finally, the proposed concept of cubic fuzzy competition graphs is supported by a numerical example. This example showcases how the framework of cubic fuzzy competition graphs can be practically applied to the predator-prey model to illustrate the representation and analysis of ambiguous information within the graph structures.

A graph \( X \) is \( k \)-spanning cyclable if for any subset \( S \) of \( k \) distinct vertices there is a 2-factor of \( X \) consisting of \( k \) cycles such that each vertex in \( S \) belongs to a distinct cycle. In this paper, we examine the \( k \)-spanning cyclability of 4-valent Cayley graphs on Abelian groups.

A path \(x_1, x_2, \dots, x_n\) in a connected graph \( G \) that has no edge \( x_i x_j \) \((j \geq i+3)\) is called a monophonic-triangular path or mt-path. A non-empty subset \( M \) of \( V(G) \) is a monophonic-triangular set or mt-set of \( G \) if every member in \( V(G) \) exists in a mt-path joining some pair of members in \( M \). The monophonic-triangular number or mt-number is the lowest cardinality of an mt-set of \( G \) and it is symbolized by \( mt(G) \). The general properties satisfied by mt-sets are discussed. Also, we establish \( mt \)-number boundaries and discover similar results for a few common graphs. Graphs \( G \) of order \( p \) with \( mt(G) = p \), \( p – 1 \), or \( p – 2 \) are characterized.

This note presents a counterexample to Propositions 7 and 8 in the paper [1], where the authors determine the values of \( V \) and \( W \). These values are crucial in determining the Hamming distance and MDS codes in the family of certain constacyclic codes over \(\mathbb{F}_{p^m}[u]/\langle u^3 \rangle\), which implies that the results found in [2] are incorrect. Furthermore, we provide corrections to the aforementioned results.

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 \emph{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 \emph{Decomposing complete tripartite graphs into cycles of lengths \( 3 \) and \( 4 \), Discrete Math., 197/198, 123-135 (1999)} and \emph{Decompositions of complete tripartite graphs into cycles of lengths \( 3 \) and \( 6 \), Austral. J. Combin., 73(1), 220-241 (2019)}.

For a graph \( G \) and a subgraph \( H \) of a graph \( G \), an \( H \)-decomposition of the graph \( G \) is a partition of the edge set of \( G \) into subsets \( E_i \), \( 1 \leq i \leq k \), such that each \( E_i \) induces a graph isomorphic to \( H \). In this paper, it is proved that every simple connected unicyclic graph of order five decomposes the \( \lambda \)-fold complete equipartite graph whenever the necessary conditions are satisfied. This generalizes a result of Huang, *Utilitas Math.* 97 (2015), 109–117.

We classify the geometric hyperplanes of the Segre geometries, that is, direct products of two projective spaces. In order to do so, we use the concept of a generalised duality. We apply the classification to Segre varieties and determine precisely which geometric hyperplanes are induced by hyperplanes of the ambient projective space. As a consequence we find that all geometric hyperplanes are induced by hyperplanes of the ambient projective space if, and only if, the underlying field has order \(2\) or \(3\).

A modification of Merino-Mǐcka-Mütze’s solution to a combinatorial generation problem of Knuth is proposed in this survey. The resulting alternate form to such solution is compatible with a reinterpretation by the author of a proof of existence of Hamilton cycles in the middle-levels graphs. Such reinterpretation is given in terms of a dihedral quotient graph associated to each middle-levels graph. The vertices of such quotient graph represent Dyck words and their associated ordered trees. Those Dyck words are linearly ordered via a rooted tree that covers all their tight, or irreducible, forms, offering an universal reference point of view to express and integrate the periodic paths, or blocks, whose concatenation leads to Hamilton cycles resulting from the said solution.

The hub cover pebbling number, \(h^{*}(G)\), of a graph $G$, is the least non-negative integer such that from all distributions of \(h^{*}(G)\) pebbles over the vertices of \(G\), it is possible to place at least one pebble each on every vertex of a set of vertices of a hub set for \(G\) using a sequence of pebbling move operations, each pebbling move operation removes two pebbles from a vertex and places one pebble on an adjacent vertex. Here we compute the hub cover pebbling number for wheel related graphs.

An outer independent double Roman dominating function (OIDRDF) on a graph \( G \) is a function \( f : V(G) \to \{0, 1, 2, 3\} \) having the property that (i) if \( f(v) = 0 \), then the vertex \( v \) must have at least two neighbors assigned 2 under \( f \) or one neighbor \( w \) with \( f(w) = 3 \), and if \( f(v) = 1 \), then the vertex \( v \) must have at least one neighbor \( w \) with \( f(w) \ge 2 \) and (ii) the subgraph induced by the vertices assigned 0 under \( f \) is edgeless. The weight of an OIDRDF is the sum of its function values over all vertices, and the outer independent double Roman domination number \( \gamma_{oidR}(G) \) is the minimum weight of an OIDRDF on \( G \). The \( \gamma_{oidR} \)-stability (\( \gamma^-_{oidR} \)-stability, \( \gamma^+_{oidR} \)-stability) of \( G \), denoted by \( {\rm st}_{\gamma_{oidR}}(G) \) (\( {\rm st}^-_{\gamma_{oidR}}(G) \), \( {\rm st}^+_{\gamma_{oidR}}(G) \)), is defined as the minimum size of a set of vertices whose removal changes (decreases, increases) the outer independent double Roman domination number. In this paper, we determine the exact values on the \( \gamma_{oidR} \)-stability of some special classes of graphs, and present some bounds on \( {\rm st}_{\gamma_{oidR}}(G) \). In addition, for a tree \( T \) with maximum degree \( \Delta \), we show that \( {\rm st}_{\gamma_{oidR}}(T) = 1 \) and \( {\rm st}^-_{\gamma_{oidR}}(T) \le \Delta \), and characterize the trees that achieve the upper bound.

We introduce a two-player game where the goal is to illuminate all edges of a graph. At each step the first player, called Illuminator, taps a vertex. The second player, called Adversary, reveals the edges incident with that vertex (consistent with the edges incident with the already tapped vertices). Illuminator tries to minimize the taps needed, and the value of the game is the number of taps needed with optimal play. We provide bounds on the value in trees and general graphs. In particular, we show that the value for the path on \( n \) vertices is \( \frac{2}{3} n + O(1) \), and there is a constant \( \varepsilon > 0 \) such that for every caterpillar on \( n \) vertices, the value is at most \( (1 – \varepsilon) n + 1 \).

Let \(G\) be a group, and let \(c\in\mathbb{Z}^+\cup\{\infty\}\). We let \(\sigma_c(G)\) be the maximal size of a subset \(X\) of \(G\) such that, for any distinct \(x_1,x_2\in X\), the group \(\langle x_1,x_2\rangle\) is not \(c\)-nilpotent; similarly we let \(\Sigma_c(G)\) be the smallest number of \(c\)-nilpotent subgroups of \(G\) whose union is equal to \(G\). In this note we study \(D_{2k}\), the dihedral group of order \(2k\). We calculate \(\sigma_c(D_{2k})\) and \(\Sigma_c(D_{2k})\), and we show that these two numbers coincide for any given \(c\) and \(k\).

Let \(p > 2\) be prime and \(r \in \{1,2, \ldots, p-1\}\). Denote by \(c_{p}(n)\) the number of \(p\)-regular partitions of \(n\) in which parts can occur not more than three times. We prove the following: If \(8r + 1\) is a quadratic non-residue modulo \(p\), \(c_{p}(pn + r) \equiv 0 \pmod{2}\) for all nonnegative integers \(n\).

Let \( G=(V,E) \) be a simple connected graph with vertex set \( G \) and edge set \( E \). The harmonic index of graph \( G \) is the value \( H(G)=\sum_{uv\in E(G)} \frac{2}{d_u+d_v} \), where \( d_x \) refers to the degree of \( x \). We obtain an upper bound for the harmonic index of trees in terms of order and the total domination number, and we characterize the extremal trees for this bound.

One of the fundamental properties of the hypercube \( Q_n \) is that it is bipancyclic as \( Q_n \) has a cycle of length \( l \) for every even integer \( l \) with \( 4 \leq l \leq 2^n \). We consider the following problem of generalizing this property: For a given integer \( k \) with \( 3 \leq k \leq n \), determine all integers \( l \) for which there exists an \( l \)-vertex, \( k \)-regular subgraph of \( Q_n \) that is both \( k \)-connected and bipancyclic. The solution to this problem is known for \( k = 3 \) and \( k = 4 \). In this paper, we solve the problem for \( k = 5 \). We prove that \( Q_n \) contains a \( 5 \)-regular subgraph on \( l \) vertices that is both \( 5 \)-connected and bipancyclic if and only if \( l \in \{32, 48\} \) or \( l \) is an even integer satisfying \( 52 \leq l \leq 2^n \). For general \( k \), we establish that every \( k \)-regular subgraph of \( Q_n \) has \( 2^k, 2^k + 2^{k-1} \) or at least \( 2^k + 2^{k-1} + 2^{k-3} \) vertices.

Coded caching technology can better alleviate network traffic congestion. Since many of the centralized coded caching schemes now in use have high subpacketization, which makes scheme implementation more challenging, coded caching schemes with low subpacketization offer a wider range of practical applications. It has been demonstrated that the coded caching scheme can be achieved by creating a combinatorial structure named placement delivery array (PDA). In this work, we employ vector space over a finite field to obtain a class of PDA, calculate its parameters, and consequently achieve a coded caching scheme with low subpacketization. Subsequently, we acquire a new MN scheme and compare it with the new scheme developed in this study. The subpacketization \(F\) of the new scheme has significant advantages. Lastly, the number of users \(K\), cache fraction \(\frac{M}{N}\), and subpacketization \(F\) have advantages to some extent at the expense of partial transmission rate \(R\) when compared to the coded caching scheme in other articles.

We continue the study of Token Sliding (reconfiguration) graphs of independent sets initiated by the authors in an earlier paper [Graphs Comb. 39.3, 59, 2023]. Two of the topics in that paper were to study which graphs \(G\) are Token Sliding graphs and which properties of a graph are inherited by a Token Sliding graph. In this paper, we continue this study specializing in the case of when \(G\) and/or its Token Sliding graph \(\mathsf{TS}_k(G)\) is a tree or forest, where \(k\) is the size of the independent sets considered. We consider two problems. The first is to find necessary and sufficient conditions on \(G\) for \(\mathsf{TS}_k(G)\) to be a forest. The second is to find necessary and sufficient conditions for a tree or forest to be a Token Sliding graph. For the first problem, we give a forbidden subgraph characterization for the cases of \(k=2,3\). For the second problem, we show that for every \(k\)-ary tree \(T\) there is a graph \(G\) for which \(\mathsf{TS}_{k+1}(G)\) is isomorphic to \(T\). A number of other results are given along with a join operation that aids in the construction of \(\mathsf{TS}_k\)-graphs.

In this paper, we introduce graceful and near graceful labellings of several families of windmills. In particular, we use Skolem-type sequences to prove (near) graceful labellings exist for windmills with \(C_{3}\) and \(C_{4}\) vanes, and infinite families of \(3,5\)-windmills and \(3,6\)-windmills. Furthermore, we offer a new solution showing that the graph obtained from the union of \(t\) 5-cycles with one vertex in common (\(C_{5}^{t}\)) is graceful if and only if \(t \equiv 0, 3 \pmod{4}\) and near graceful when \(t \equiv 1, 2 \pmod{4}\).

We study groups generated by sets of pattern avoiding permutations. In the first part of the paper, we prove some general results concerning the structure of such groups. In particular, we consider the sequence \((G_n)_{n \geq 0}\), where \(G_n\) is the group generated by a subset of the symmetric group \(S_n\) consisting of permutations that avoid a given set of patterns. We analyze under which conditions the sequence \((G_n)_{n \geq 0}\) is eventually constant. Moreover, we find a set of patterns such that \((G_n)_{n \geq 0}\) is eventually equal to an assigned symmetric group. Furthermore, we show that any non-trivial simple group cannot be obtained in this way and describe all the non-trivial abelian groups that arise in this way. In the second part of the paper, we carry out a case-by-case analysis of groups generated by permutations avoiding a few short patterns.

We consider the eccentric graph of a graph \(G\), denoted by \(\mathrm{ecc}(G)\), which has the same vertex set as \(G\), and two vertices in the eccentric graph are adjacent if and only if their distance in \(G\) is equal to the eccentricity of one of them. In this paper, we present a fundamental requirement for the isomorphism between \(\mathrm{ecc}(G)\) and the complement of \(G\), and show that the previous necessary condition given in the literature is inadequate. Also, we obtain that the diameter of \(\mathrm{ecc}(T)\) is at most 3 for any tree and get some characterizations of the eccentric graph of trees.

Let \(G\) be a finite simple undirected \((p, q)\)-graph, with vertex set \(V(G)\) and edge set \(E(G)\) such that \(p = |V(G)|\) and \(q = |E(G)|\). A super edge-magic total labeling \(f\) of \(G\) is a bijection \(f \colon V(G) \cup E(G) \longrightarrow \{1, 2, \dots, p+q\}\) such that for all edges \(uv \in E(G)\), \(f(u) + f(v) + f(uv) = c(f)\), where \(c(f)\) is called a magic constant, and \(f(V(G)) = \{1, \dots, p\}\). The minimum of all \(c(f)\), where the minimum is taken over all the super edge-magic total labelings \(f\) of \(G\), is defined to be the super edge-magic total strength of the graph \(G\). In this article, we work on certain classes of unicyclic graphs and provide evidence to conjecture that the super edge-magic total strength of a certain family of unicyclic \((p, q)\)-graphs is equal to \(2q + \frac{n+3}{2}\).

For a poset \(P = C_a \times C_b\), a subset \(A \subseteq P\) is called a chain blocker for \(P\) if \(A\) is inclusion-wise minimal with the property that every maximal chain in \(P\) contains at least one element of \(A\), where \(C_i\) is the chain \(1 < \cdots < i\). In this article, we define the shelter of the poset \(P\) to give a complete description of all chain blockers of \(C_5 \times C_b\) for \(b \geq 1\).

This project aims at investigating properties of channel detecting codes on specific domains \(1^+0^+\). We focus on the transmission channel with deletion errors. Firstly, we discuss properties of channels with deletion errors. We propose a certain kind of code that is a channel detecting (abbr. \(\gamma\)-detecting) code for the channel \(\gamma = \delta(m, N)\) where \(m < N\). The characteristic of this \(\gamma\)-detecting code is considered. One method is provided to construct \(\gamma\)-detecting code. Finally, we also study a kind of special channel code named \(\tau(m, N)\)-srp code.

A chemical structure specifies the molecular geometry of a given molecule or solid in the form of atom arrangements. One way to analyze its properties is to simulate its formation as a product of two or more simpler graphs. In this article, we take this idea to find upper and lower bounds for the generalized Randić index \(\mathcal{R}_{\alpha}\) of four types of graph products, using combinatorial inequalities. We finish this paper by providing the bounds for \(\mathcal{R}_{\alpha}\) of a line graph and rooted product of graphs.

Let \(G\) be a \((p, q)\) graph. Let \(f: V(G) \to \{1, 2, \ldots, k\}\) be a map where \(k \in \mathbb{N}\) is a variable and \(k > 1\). For each edge \(uv\), assign the label \(\gcd(f(u), f(v))\). \(f\) is called \(k\)-Total prime cordial labeling of \(G\) if \(\left|t_{f}(i) – t_{f}(j)\right| \leq 1\), \(i, j \in \{1, 2, \ldots, k\}\) where \(t_{f}(x)\) denotes the total number of vertices and edges labeled with \(x\). A graph with a \(k\)-total prime cordial labeling is called \(k\)-total prime cordial graph. In this paper, we investigate the 4-total prime cordial labeling of some graphs like dragon, Möbius ladder, and corona of some graphs.

Let \(G = (V, E)\) be a graph with vertex set \(V\) and edge set \(E\). An edge labeling \(f: E \to Z_{2}\) induces a vertex labeling \(f^{+} : V \to Z_{2}\) defined by \( f^{+}(v) \equiv \sum_{uv \in E} f(uv) \pmod 2 \), for each vertex \(v \in V\). For \(i \in Z_{2}\), let \( v_{f}(i) = |\{v \in V : f^+(v) = i\}| \) and \( e_{f}(i) = |\{e \in E : f(e) = i\}| \). An edge labeling \(f\) of a graph \(G\) is said to be edge-friendly if \( |e_{f}(1) – e_{f}(0)| \le 1 \). The set \(\{v_f(1) – v_f(0) : f \text{ is an edge-friendly labeling of } G\}\) is called the full edge-friendly index set of \(G\). In this paper, we shall determine the full edge-friendly index sets of one point union of cycles.

After the Chartrand definition of graph labeling, since 1988 many graph families have been labeled through mathematical techniques. A basic approach in those labelings was to find a pattern among the labels and then prove them using sequences and series formulae. In 2018, Asim applied computer-based algorithms to overcome this limitation and label such families where mathematical solutions were either not available or the solution was not optimum. Asim et al. in 2018 introduced the algorithmic solution in the area of edge irregular labeling for computing a better upper-bound of the complete graph \(es(K_n)\) and a tight upper-bound for the complete \(m\)-ary tree \({es(T}_{m,h})\) using computer-based experiments. Later on, more problems like complete bipartite and circulant graphs were solved using the same technique. Algorithmic solutions opened a new horizon for researchers to customize these algorithms for other types of labeling and for more complex graphs. In this article, to compute edge irregular \(k\)-labeling of star \(S_{m,n}\) and banana tree \({BT}_{m,n}\), new algorithms are designed, and results are obtained by executing them on computers. To validate the results of computer-based experiments with mathematical theorems, inductive reasoning is adopted. Tabulated results are analyzed using the law of double inequality and it is concluded that both families of trees observe the property of edge irregularity strength and are tight for \(\left\lceil \frac{|V|}{2} \right\rceil\)-labeling.

A graph \(G\) is called a fractional ID-\((g,f)\)-factor-critical covered graph if for any independent set \(I\) of \(G\) and for every edge \(e \in E(G-I)\), \(G-I\) has a fractional \((g,f)\)-factor \(h\) such that \(h(e) = 1\). We give a sufficient condition using degree condition for a graph to be a fractional ID-\((g,f)\)-factor-critical covered graph. Our main result is an extension of Zhou, Bian, and Wu’s previous result [S. Zhou, Q. Bian, J. Wu, A result on fractional ID-\(k\)-factor-critical graphs, Journal of Combinatorial Mathematics and Combinatorial Computing 87(2013) 229–236] and Yashima’s previous result [T. Yashima, A degree condition for graphs to be fractional ID-\([a,b]\)-factor-critical, Australasian Journal of Combinatorics 65(2016) 191–199].

In this article, we define \(q\)-generalized Fibonacci polynomials and \(q\)-generalized Lucas polynomials using \(q\)-binomial coefficient and obtain their recursive properties. In addition, we introduce generalized \(q\)-Fibonacci matrix and generalized \(q\)-Lucas matrix, then we derive their basic identities. We define \((k,q,t)\)-symmetric generalized Fibonacci matrix and \((k,q,t)\)-symmetric generalized Lucas matrix, then we give the Cholesky factorization of these matrices. Finally, we give determinantal and permanental representations of these new polynomial sequences.

Stanley considered Dyck paths where each maximal run of down-steps to the \(x\)-axis has odd length; they are also enumerated by (shifted) Catalan numbers. Prefixes of these combinatorial objects are enumerated using the kernel method. A more challenging version of skew Dyck paths combined with Stanley’s restriction is also considered.

For \(r=1,2,…, 6\), we obtain generating functions \(F^{(r)}_{k}(y)\) for words over the alphabet \([k]\), where \(y\) tracks the number of parts and \([y^n]\) is the total number of distinct adjacent \(r\)-tuples in words with \(n\) parts. In order to develop these generating functions for \(1\le r\le 3\), we make use of intuitive decompositions but for larger values of \(r\), we switch to the cluster analysis method for decorated texts that was introduced by Bassino et al. Finally, we account for the coefficients of these generating functions in terms of Stirling set numbers. This is done by putting forward the full triangle of coefficients for all the sub-cases where \(r=5\) and 6. This latter is shown to depend on both periodicity and number of letters used in the \(r\)-tuples.

We consider the following variant of the round-robin scheduling problem: \(2n\) people play a number of rounds in which two opposing teams of \(n\) players are reassembled in each round. Each two players should play at least once in the same team, and each two players should play at least once in opposing teams. We provide an explicit formula for calculating the minimal numbers of rounds needed to satisfy both conditions. Moreover, we also show how one can construct the corresponding playing schedules.

Two binary structures \(\mathfrak{R}\) and \(\mathfrak{R’}\) on the same vertex set \(V\) are \((\leq k)\)-hypomorphic for a positive integer \(k\) if, for every set \(K\) of at most \(k\) vertices, the two binary structures induced by \(\mathfrak{R}\) and \(\mathfrak{R’}\) on \(K\) are isomorphic. A binary structure \(\mathfrak{R}\) is \((\leq k)\)-reconstructible if every binary
structure \(\mathfrak{R’}\) that is \((\leq k)\)-hypomorphic to \(\mathfrak{R}\) is isomorphic to \(\mathfrak{R}\). In this paper, we describe the pairs of \((\leq 3)\)-hypomorphic posets and the pairs of \((\leq 3)\)-hypomorphic bichains. As a consequence, we characterize the \((\leq 3)\)-reconstructible posets and the \((\leq 3)\)-reconstructible bichains. This answers a question suggested by Y. Boudabbous and C. Delhommé during a personal communication.

A tremendous amount of drug experiments revealed that there exists a strong inherent relation between the molecular structures of drugs and their biomedical and pharmacology characteristics. Due to the effectiveness for pharmaceutical and medical scientists of their ability to grasp the biological and chemical characteristics of new drugs, analysis of the bond incident degree (BID) indices is significant of testing the chemical and pharmacological characteristics of drug molecular structures that can make up the defects of chemical and medicine experiments and can provide the theoretical basis for the manufacturing of drugs in pharmaceutical engineering. Such tricks are widely welcomed in developing areas where enough money is lacked to afford sufficient equipment, relevant chemical reagents, and human resources which are required to investigate the performance and the side effects of existing new drugs. This work is devoted to establishing a general expression for calculating the bond incident degree (BID) indices of the line graphs of various well-known chemical structures in drugs, based on the drug molecular structure analysis and edge dividing technique, which is quite common in drug molecular graphs.

In this paper, we introduce a graph structure, called component intersection graph, on a finite dimensional vector space \(\mathbb{V}\). The connectivity, diameter, maximal independent sets, clique number, chromatic number of component intersection graph have been studied.

A linear system is a pair \((P,\mathcal{L})\) where \(\mathcal{L}\) is a finite family of subsets on a finite ground set \(P\) such that any two subsets of \(\mathcal{L}\) share at most one element. Furthermore, if for every two subsets of \(\mathcal{L}\) share exactly one element, the linear system is called intersecting. A linear system \((P,\mathcal{L})\) has rank \(r\) if the maximum size of any element of \(\mathcal{L}\) is \(r\). By \(\gamma(P,\mathcal{L})\) and \(\nu_2(P,\mathcal{L})\) we denote the size of the minimum dominating set and the maximum 2-packing of a linear system \((P,\mathcal{L})\), respectively. It is known that any intersecting linear system \((P,\mathcal{L})\) of rank \(r\) is such that \(\gamma(P,\mathcal{L})\leq r-1\). Li et al. in [S. Li, L. Kang, E. Shan and Y. Dong, The finite projective plane and the 5-Uniform linear intersecting hypergraphs with domination number four, Graphs and 34 Combinatorics (2018) , no.~5, 931–945.] proved that every intersecting linear system of rank 5 satisfying \(\gamma(P,\mathcal{L})=4\) can be constructed from a 4-uniform intersecting linear subsystem \((P^\prime,\mathcal{L}^\prime)\) of the projective plane of order 3 satisfying \(\tau(P^\prime,\mathcal{L}^\prime)=\nu_2(P^\prime,\mathcal{L}^\prime)=4\), where \(\tau(P^\prime,\mathcal{L}^\prime)\) is the transversal number of \((P^\prime,\mathcal{L}^\prime)\). In this paper, we give an alternative proof of this result given by Li et al., giving a complete characterization of these 4-uniform intersecting linear subsystems. Moreover, we prove a general case, that is, we prove if $q$ is an odd prime power and \((P,\mathcal{L})\) is an intersecting linear system of rank \((q+2)\) satisfying \(\gamma(P,\mathcal{L})=q+1\), then this linear system can be constructed from a spanning \((q+1)\)-uniform intersecting linear subsystem \((P^\prime,\mathcal{L}^\prime)\) of the projective plane of order \(q\) satisfying \(\tau(P^\prime,\mathcal{L}^\prime)=\nu_2(P^\prime,\mathcal{L}^\prime)=q+1\).

We classify all near hexagons of order \((3,t)\) that contain a big quad. We show that, up to isomorphism, there are ten such near hexagons.

Let \(G=(V,E)\) be a simple graph. A vertex \(v\in V(G)\) ve-dominates every edge \(uv\) incident to \(v\), as well as every edge adjacent to these incident edges. A set
\(D\subseteq V(G)\) is a vertex-edge dominating set if every edge of \(E(G)\) is ve-dominated by a vertex of \(D.\) The MINIMUM VERTEX-EDGE DOMINATION problem is to find a vertex-edge dominating set of minimum cardinality. A linear time algorithm to find the minimum vertex-edge dominating set for proper interval graphs is proposed. The vertex-edge domination problem is proved to be APX-complete for bounded-free graphs and NP-Complete for Chordal bipartite and Undirected Path graphs.

In this paper, we investigate the \((d,1)\)-total labelling of generalized Petersen graphs \(P(n,k)\) for \(d\geq 3\). We find that the \((d,1)\)-total number of \(P(n,k)\) with \(d\geq 3\) is \(d+3\) for even \(n\) and odd \(k\) or even \(n\) and \(k=\frac{n}{2}\), and \(d+4\) for all other cases.

By employing Kummer and Thomae transformations, we examine four classes of nonterminating \(_3F_2\)(1)-series with five integer parameters. Several new summation formulae are established in closed form.