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The rings considered in this paper are commutative with identity which are not integral domains. Let R be a ring. Let us denote the set of all annihilating ideals of R by (R) and (R) ∖ { (0) } by (R) ∗ . With R , we associate an undirected graph, denoted by (R) , whose vertex set is (R) ∗ and distinct vertices I and J are adjacent in this graph if and only if I J = (0) and I + J ∈ (R). The aim of this paper is to study the interplay between the graph-theoretic properties of (R) and the ring-theoretic properties of  R. [ABSTRACT FROM AUTHOR]

Given a connected graph G , the distance Laplacian matrix D L (G) is defined as D L (G) = Tr (G) − D (G) , and the distance signless Laplacian matrix D Q (G) is defined as D Q (G) = Tr (G) + D (G) , where Tr (G) is the transmission matrix of G and D (G) is the distance matrix of G. The Indu–Bala product of two graphs G 1 and G 2 , denoted by G 1 ▾ G 2 , was introduced in (G. Indulal and R. Balakrishnan, Distance spectrum of Indu–Bala product of graphs,  AKCE Int. J. Graph Comb. 13(3) (2016) 230–234). In this paper, we first obtain the distance Laplacian spectrum of G 1 ▾ G 2 in terms of Laplacian spectra of G 1 and G 2 . We then obtain the distance signless Laplacian spectrum of G 1 ▾ G 2 in terms of signless Laplacian spectra of G 1 and G 2 . We construct pair of graphs which are distance Laplacian co-spectral as well as pair of graphs which are distance signless Laplacian co-spectral. We further find the metric dimension of G 1 ▾ G 2 in terms of metric dimensions of G 1 and G 2 . Finally, we provide a problem for future research. [ABSTRACT FROM AUTHOR]

The partially balanced incomplete block (PBIB)-designs and association schemes arising from different graph parameters is a well-studied concept. In this paper, we define and construct PBIB-designs with association schemes in strongly regular graphs through the nodes belonging to the diametral paths as blocks. A (v , b , r , k , λ , μ) -design, called a diametral-design (in short) over a strongly regular graph G = (V , E) of degree d , diameter diam (G) , is an ordered pair D = (V , B) , where V = V (G) and B is the set of all diametral paths of G , called blocks, containing the nodes belonging to the diametral paths, satisfying the following conditions: (i) If x , y ∈ V and { x , y } ∈ E , then there are exactly λ blocks containing { x , y } ; (ii) If x , y ∈ V , x ≠ y and { x , y } ∉ E , then there are exactly μ -blocks containing { x , y }. We construct PBIB-designs with two-association schemes through diametral paths corresponding to the strongly regular graphs such as the square lattice graphs L 2 (n) , the triangular graphs T (n) , the three Chang graphs, the Petersen graph, the Shrikhande graph, the Clebsch graph, the complement of Clebsch graph, the complement of Schläfli graph, complete bipartite graphs, the Hoffman–Singleton graph, etc. and in each case we give the composition of the diametral paths. These serve as extensions for the collection of near impossible class of strongly regular graphs with given parameters having two-association schemes. [ABSTRACT FROM AUTHOR]

In this paper, we established a precedent to provide urban planning alternatives, given that in modern mobility problems, it is difficult to provide a corrective solution. The proposal of this work focuses on finding preventive rather than remedial solutions, although in particular cases, it may be applicable. We simulated town design situations with optimal ramified transport and metaheuristic optimization methods. For this purpose, it was necessary to analyze the essential aspects of road transportation and traffic problems in the world's leading cities, considering the hypodamic planes proposed in antiquity and contrasting their properties with optimal ramified transportation. [ABSTRACT FROM AUTHOR]

In this paper, we develop a theory for constructing cyclic codes of odd length over the ring R = ℤ 4 + u ℤ 4 + v ℤ 4 , where u 2 = v 2 = u v = v u = 0 , which plays an important role in DNA computing. A direct link between the elements of R and the 6 4 codons used in the amino acids of the living organisms is established. Then, we investigate reversible cyclic codes and reversible complement cyclic codes of odd length over R. Moreover, we give some properties of binary images of the codons under the Gray map. Finally, two examples of cyclic codes over R with their minimum Hamming distance will be studied as well. [ABSTRACT FROM AUTHOR]

A graph G has a ( k 1 , k 2 ) -partition if V (G) can be partitioned into two nonempty disjoint subsets V 1 and V 2 so that G [ V 1 ] and G [ V 2 ] are graphs whose components are paths of order at most k 1 and k 2 , respectively. In this paper, we proved that every planar graph with girth at least six giving that i -cycle is not intersecting with j -cycle admits a ( 3 , 3) -partition, where i ∈ { 6 , 7 } and j ∈ { 6 , 7 , 8 , 9 }. [ABSTRACT FROM AUTHOR]

This paper focuses on the matrix sequences that represent negative indices Pell and Pell–Lucas number. It will specifically provide the important relationships between negative indices Pell and Pell–Lucas number and Pell and Pell–Lucas matrix sequences. The research of new error correcting codes has received a lot of attention in recent years, specifically due to its potential use in quantum-resistant cryptosystems. These matrices are then applied in cryptology and have a great opportunity to report some cases that might be a mistake between the sender and the receiver by using Lucas blocking algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the configuration space G , b , ℓ associated with the movement of a robotic arm of length ℓ on a grid over an underlying graph G , anchored at a vertex b ∈ G. We study an associated poset with inconsistent pairs (PIP) IP G , b , ℓ consisting of indexed paths on G. This PIP acts as a combinatorial model for the robotic arm, and we use IP G , b , ℓ to show that the space G , b , ℓ is a CAT(0) cubical complex, generalizing work of Ardila, Bastidas, Ceballos, and Guo. This establishes that geodesics exist within the configuration space, and yields explicit algorithms for moving the robotic arm between different configurations in an optimal fashion. We also give a tight bound on the diameter of the robotic arm transition graph — the maximal number of moves necessary to change from one configuration to another — and compute this diameter for a large family of underlying graphs G. [ABSTRACT FROM AUTHOR]

Dominant local metric dimension (DLMD) combines two interesting graph theory concepts: domination and local metric dimension (LMD). Graph invariants are a fantastic tool for analyzing graph abstract structures. Metric dimension and DLMD as graph invariants have a wide range of applications, which includes robot navigation, pharmaceutical chemistry, pattern recognition, and image processing. In this paper, we computed the DLMD of the prism graph Y n , antiprism graph A n , and the sunflower graph S F n . [ABSTRACT FROM AUTHOR]

In this paper, we analyze the intersection graphs of subtrees in a cactus, called cactus subtree graphs. A max-k-cliquesubgraph of a graph is an induced subgraph whose maximum clique has at most k vertices. We describe a polynomial time algorithm to find a maximum weight max- k -clique subgraph in a cactus subtree graph when k is fixed. In perfect graphs this problem is equivalent to the maximum weight k -colorable subgraph problem. In many applications like scheduling production lines and communication networks on imperfect graphs, a maximum max- k -clique subgraph solution is better than a maximum k -colorable subgraph solution. In addition, we describe cactus subtree graphs polynomial time algorithms for recognition, maximum independent sets, maximum cliques, maximum induced bipartite graphs, maximum induced forests and minimum feedback vertex sets. [ABSTRACT FROM AUTHOR]

Let G be a simple graph and connected. A function l : V (G) → { 1 , 2 , ... , k } is called vertex irregular k -labeling and w : V (G) → N where w (u) = ∑ v ∈ N (u) l (v). The function of f is called local irregular vertex coloring if every u v ∈ E (G) , w (u) ≠ w (v) and opt (l) = min { max l i ; l i vertex irregular labeling}. The local irregular chromatic number is denoted by χ lis (G). In this paper, we study local irregular vertex coloring of S m ▹ v 0 S n , S m ▹ v 1 S n , and S m ▹ v 1 P n. [ABSTRACT FROM AUTHOR]

In this paper, we use a novel approach to describe generator polynomials of quasi-cyclic (QC) and generalized QC (GQC) codes over finite fields. Our study of QC- and GQC-codes will be general and not only restricted to one-generator codes. We prove that generator polynomials of QC-codes and GQC-codes are unique. Further, we use our results to obtain an expression for the dimensions of QC-codes and GQC-codes. As an application of our construction of these codes, we obtain many optimal linear codes over finite fields GF (2) , GF (3) , GF (4) and GF (5). [ABSTRACT FROM AUTHOR]

This paper shows that the topological structures of particle orbits generated by a generic class of vector fields on spherical surfaces, called the flow of finite type, are in one-to-one correspondence with discrete structures such as trees/graphs and sequences of letters. The flow of finite type is an extension of structurally stable Hamiltonian vector fields, which appear in many theoretical and numerical investigations of two-dimensional (2D) incompressible fluid flows. Moreover, it contains compressible 2D vector fields such as the Morse–Smale vector fields and the projection of 3D vector fields onto 2D sections. The discrete representation is not only a simple symbolic identifier for the topological structure of complex flows, but it also gives rise to a new methodology of topological data analysis for flows when applied to data brought by measurements, experiments, and numerical simulations of complex flows. As a proof of concept, we provide some applications of the representation theory to 2D compressible vector fields and a 3D vector field arising in an industrial problem. [ABSTRACT FROM AUTHOR]

Difference sets have wide applications in constructing sequences and codes in engineering and cryptography, and in imaging with coded masks in astronomical events and in medical events. In this paper, the relation between difference sets and binary sequences is presented and a coded mask as an application of difference sets is given. First of all, an algorithm for an appropriate difference set is written in C++ by using twin primes and then a coded mask for imaging astronomical events is designed with the aid of this algorithm. [ABSTRACT FROM AUTHOR]

Given an ordered k -decomposition D = { G 1 , G 2 , ... , G k } of a connected graph G = (V , E) , the representation of an edge e ∈ E with respect to the decomposition D is the k -tuple γ (e / D) = (d (e , G 1) , d (e , G 2) , ... , d (e , G k)) , where d (e , G i) represents the distance from e to G i . A decomposition D of E is a resolving decomposition if for every pair of edges e , f ∈ E , γ (e / D) ≠ γ (f / D). The minimum k for which G has a resolving k -decomposition is its decomposition dimension dec (G). In this paper, upper bounds for the decomposition dimension of some class of graphs are determined. [ABSTRACT FROM AUTHOR]

In this paper, we construct an infinite family of weighted growing complex networks, namely, weighted neighborhood networks (WNN) which are constructed in an iterative way by using a base network and a sequence of growing weighted networks. We determine the weighted Laplacian spectra of WNN which is expressed in terms of the spectra of base network and the sequence of weighted regular networks. Using the weighted Laplacian spectra, we obtain the Kirchhoff index, the entire mean weighted first-passage time and the number of spanning trees of WNN. Also, we compute the weighted normalized Laplacian spectra of these networks which is expressed in terms of the spectra of regular base network and the sequence of weighted regular networks and from that, we derive the multiplicative Kirchhoff index, Kemeny's constant and the number of spanning trees in terms of the weighted normalized Laplacian spectra. [ABSTRACT FROM AUTHOR]

The order two element graph of a group G is the simple undirected graph whose vertex set consists of all elements of G and two distinct vertices u , v are adjacent if and only if either u v o r v u ∈ { t 2 : t ∈ G } ∪ { a ∈ G : a 2 = e } \ { e } , where e is the identity element of G. We denote this graph by 2 (G). In this paper, we identify those commutative groups G for which the graph 2 (G) is connected. We also characterize the structures of the graph 2 (ℤ n) for any positive integer n. Moreover, we consider the symmetric group S n , the dihedral group D n for any positive integer n and for these groups, we look into the structure and other graph-theoretic properties of the corresponding graphs. Finally, we give a list of finite groups of order ≤ 1 6 with their corresponding order two element graph structures. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a study of the Moore–Penrose inverse L † of the Laplacian matrix of a simple and connected graph, particularly, for some families of graphs such as path, cycle, ladder, fan and wheel graphs. For this purpose, it is used diverse approaches and MP inverse of the Cartesian product of graphs, and are obtained new closed-form formulas of the L † of these families. A comparison of the computational efficiency of the new formulas versus traditional mathematical software is presented, showing the advantage of new formulas. [ABSTRACT FROM AUTHOR]

In this paper, we construct a weighted 3 -simplicial complex S w = (V , E , F , T , w V , w E , w F , w T) on a connected oriented locally finite graph (V , E) by the introduction of the notion of oriented tetrahedrons T , the notion of oriented triangular faces F , a weight on V , a weight on E , a weight on F and a weight on T. Next, we create the weighted Gauss–Bonnet operator of S w and we use it to construct the weighted Laplacian associated to V , the weighted Laplacian associated to E , the weighted Laplacian associated to F , the weighted Laplacian associated to T and the weighted Laplacian associated to S w . After that, we introduce the notion of the χ -completeness of S w and we give necessary conditions for S w to be χ -complete. Finally, we prove that the weighted Gauss–Bonnet operator and the weighted Laplacians are essentially self-adjoint based on the χ -completeness. [ABSTRACT FROM AUTHOR]

The n -cube graph or hypercube Q n is the graph whose vertex set is the set of all n -dimensional Boolean vectors, two vertices being joined if and only if they differ in exactly one co-ordinate. The purpose of the paper is to investigate the signed domination number of this hypercube graphs. In this paper, signed domination number n -cube graph for odd n is found and the lower and upper bounds of hypercube for even n are found. [ABSTRACT FROM AUTHOR]

The online social platforms, like Twitter, Facebook, LinkedIn and WeChat, have grown really fast in last decade and have been one of the most effective platforms for people to communicate and share information with each other. Due to the word-of-mouth effects, information usually can spread rapidly on these social media platforms. Therefore, it is important to study the mechanisms driving the information diffusion and quantify the consequence of information spread. A lot of efforts have been focused on this problem to help us better understand and achieve higher performance in viral marketing and advertising. On the other hand, the development of neural networks has blossomed in the last few years, leading to a large number of graph representation learning (GRL) models. Compared with traditional models, GRL methods are often shown to be more effective. In this paper, we present a comprehensive review for recent works leveraging GRL methods for popularity prediction problem, and categorize related literatures into two big classes, according to their mainly used model and techniques: embedding-based methods and deep learning methods. Deep learning method is further classified into convolutional neural networks, graph convolutional networks, graph attention networks, graph neural networks, recurrent neural networks, and reinforcement learning. We compare the performance of these different models and discuss their strengths and limitations. Finally, we outline the challenges and future chances for popularity prediction problem. [ABSTRACT FROM AUTHOR]

Super strongly perfect graphs and their association with certain other classes of graphs are discussed in this paper. It is observed that every split graph is super strongly perfect. An existing result on super strongly perfect graphs is disproved providing a counter example. It is also established that if a graph G contains a cycle of odd length, then its line graph L(G) is not always super strongly perfect. Complements of cycles of length six or above are proved to be non-super strongly perfect. If a graph is strongly perfect, then it is shown that they are super strongly perfect and hence all P 4 -free graphs are super strongly perfect. [ABSTRACT FROM AUTHOR]

A signed graph Σ is a simple undirected graph in which each edge is either positive or negative. Restrained dominating set D in Σ is a restrained dominating set of the underlying graph | Σ | where the subgraph induced by the edges across Σ [ D : V ∖ D ] and within V ∖ D is balanced. The minimum cardinality of a restrained dominating set of Σ is called the restrained domination number, denoted by γ r (Σ). In this paper, we initiate the study on various critical concepts to investigate the effect of edge removal or edge addition on restrained domination number in signed graphs. [ABSTRACT FROM AUTHOR]

For given two graphs G 1 and G 2 , and integer j ≥ 2 , the size multipartite Ramsey number m j (G 1 , G 2) = t is the smallest integer such that every factorization of graph K j × t : = F 1 ⊕ F 2 satisfies the following condition: either F 1 contains G 1 as a subgraph or F 2 contains G 2 as a subgraph. In this paper, we determine that m 4 (P n , W 4) for n ≥ 2 where P n is path on n ≥ 2 vertices and W 4 is a wheel on five vertices. [ABSTRACT FROM AUTHOR]

The notion of the metric dimension of a graph is well-known and its study is well entrenched in the literature. In this paper, we introduce new classes of graphs that exhibit remarkable characteristic: the metric dimension and the diameter of the unit graph are equal. Additionally, we provide a characterization of finite commutative rings R, wherein the metric dimension assumes a value k, where 1 ≤ k ≤ 3. Further, we also demonstrate an exhaustive examination that ascertains the precise finite commutative rings R, in which domination number is equal to metric dimension of unit graph. [ABSTRACT FROM AUTHOR]

Codes over integer rings equipped with the Lee metric are gaining attention among researchers recently. In this paper, we investigate some specific linear codes over integer rings and characterize their dimensions and Lee weights using greatest common divisors (GCD). Furthermore, we construct equidistant Lee codes over ℤ2b, ℤ2pandℤp which have applications in error detection and error correction. [ABSTRACT FROM AUTHOR]

Cayley graph is a graph that encodes the abstract structure of a group. It gives a way of encoding information about a group in a graph. On the other hand, hypergroup is a generalization of group in which the composition of any two elements is a non-empty set. The purpose of this paper is to find a suitable generalization of Cayley graphs to cover hypergroups. More precisely, we introduce generalized Cayley graphs over hypergroups, study their properties and find a simple tool to construct large connected GCH-graphs from smaller GCH-graphs by using graph product. [ABSTRACT FROM AUTHOR]

Let n ≥ 1 and Un be the multiplicative group of the additive group ℤn of integers modulo n. The unitary Haar graph Hn = Haar(ℤn,Un) is a graph with vertex set ℤn ×{0, 1} and the edge set {{(a, 0), (b, 1)}|b − a ∈ Un}. In this paper, we study some structural properties of Hn such as total chromatic number, diameter, planarity, the girth, vertex-connectivity and edge-connectivity. Also, some spectral properties of the graph Hn are determined. [ABSTRACT FROM AUTHOR]

Luo, Tian and Wu [Discrete Math. 345(4) (2022) 112788] conjectured that for any tree T with bipartition (X,Y ), every k-connected bipartite graph G with minimum degree at least k + w, where w =max{|X|,|Y |}, contains a tree T′≅T such that κ(G − V (T′)) ≥ k. In the paper, we confirm the conjecture when T is an odd path on m vertices. We remind that Yang and Tian [Proof of a conjecture on connectivity keeping odd paths in k-connected bipartite graphs, arXiv:2209.08373v2] also prove the same result by a different way. [ABSTRACT FROM AUTHOR]

The traveling salesman problem (TSP) involves determining the shortest length (optimal tour) for a complete circuit through an arbitrary number of points. In the special case that the points to be visited are randomly distributed in the plane within a unit square, and travel costs correspond to the Euclidean distance between points, it is known that the length of the optimal tour divided by the square root of the number of points (n) asymptotically approaches a constant (βTSP) as n becomes large. If individual points are randomly drawn from a uniform distribution, the optimal lengths for a sufficiently large number of TSP instances of the same size (n) comprise a normal distribution whose mean approaches βTSP as n increases. Moreover, as n approaches infinity, this distribution gradually narrows such that its standard deviation asymptotically approaches zero. Although the precise value of βTSP is unknown, published studies indicate its magnitude is approximately 0.712. In this paper, possibly for the first time, precise formulae are proposed for the parameters of the underlying normal distribution and βTSP, based on intuition, mathematical reasoning, and empirical fits to both published and experimentally derived data. [ABSTRACT FROM AUTHOR]

Indicated coloring is a graph coloring game in which two players collectively color the vertices of a graph in the following way. In each round, the first player (Ann) selects a vertex and then the second player (Ben) colors it properly, using a fixed set of colors. The goal of Ann is to achieve a proper coloring of the whole graph, while Ben is trying to prevent the realization of this project. The smallest number of colors necessary for Ann to win the game on a graph G (regardless of Ben’s strategy) is called the indicated chromatic number of G, denoted by χi(G).In this paper, we observe that the Nordhaus–Gaddum inequalities for the chromatic number are also satisfied by the indicated chromatic number. Further, we prove that outerplanar graphs, 3-colorable maximal planar graphs with col(G)≠5, uniquely 4-colorable planar graphs and Mn (obtained from Cn by joining diagonally opposite vertices) are k-indicated colorable for all k greater than or equal to its chromatic number. Also, we show that Ben wins the game with 4 colors when Ann uses any connected strategy on M4t+1 for any t ≥ 2. This turns out to be a generalization of the result due to Grzesik in [Indicated coloring of graphs, Discrete Math. 312(23) (2012) 3467–3472]. Finally, we prove that Mycielskian of G is (k + 1)-indicated colorable when G is k-indicated colorable for some k ≥ χi(G). These partially answer one of the questions which was raised by Grzesik in [Indicated coloring of graphs, Discrete Math. 312(23) (2012) 3467–3472]. [ABSTRACT FROM AUTHOR]

As we know that some chemically interesting graphs can be obtained by different graph operations on some general or particular graphs, it is important to study such graph operations in order to understand how it is related to the corresponding topological indices of the original graphs. In [A. Miličević, A. Nikolić and S. Trinajstić, On reformulated Zagreb indices, Mol. Divers. 8 (2004) 393–399], authors defined two new variants of Corona product and investigated some topological indices. In this paper, we extended the work and found the formulas of the F-index, hyper-Zagreb index, and Sigma index for double join of graphs based on the total graph and double corona of graphs based on the total graph. Moreover, some applications of the derived results are also discussed. [ABSTRACT FROM AUTHOR]

Let 픽ql be a finite field of cardinality ql, where q is a power of a prime and l is a prime integer. An 픽q-linear code over 픽ql of length n is an 픽q-subspace of 픽qln. If an 픽q-linear code over 픽ql with parameters (n,qk,d H) meets the Singleton bound, i.e., dH = n −⌈k l ⌉ + 1, then it is called an MDS code. In this paper, by use of a Fourier matrix and a canonical form decomposition of constacyclic 픽q-linear codes over 픽ql, we construct some optimal 픽q-linear codes over 픽ql which have the same parameters as the MDS codes over 픽ql. [ABSTRACT FROM AUTHOR]

Inspired by the studies on conditional connectivity by Harary [1], we worked on k-restricted edge connectivity of a graph G [4]. The k-restricted edge connectivity of a graph G is the cardinality of a minimum edge-cut whose deletion disconnects the graph G into components where each component has order at least k. In this work, we take into account some results on k-restricted edge connectivity and the novel parameter exact k-restricted edge connectivity. In the definition of k-restricted edge connectivity, we impose some extra constraints to achieve this parameter. In this paper, we consider some fundamental aspects of exact k-restricted edge connectivity and how they relate to the k-restricted edge connectivity. [ABSTRACT FROM AUTHOR]

The nonzero component graph of a finite-dimensional vector space over a finite field is a graph whose vertices are the nonzero vectors in the vector space, and any two vertices are adjacent if the corresponding linear combination contains a common basis vector. In this paper, we discuss the centrality measures and entropy of the nonzero component graph and also analyze the sensitivity of the graph using the centrality measures. [ABSTRACT FROM AUTHOR]

An incidence of a graph G is a pair (u,e) where u is a vertex of G and e is an edge of G incident with u. Two incidences (u,e) and (v,f) of G are adjacent whenever (i) u = v, or (ii) e = f or (iii) uv = e or f. A strong incidence coloring of a graph G is a mapping from the set of incidences of G to the set of colors {1,…,k}, such that every two incidences that are adjacent, or adjacent to the same incidence receive distinct colors. In this paper, we prove that every connected subcubic graph G except K3,3 has a strong incidence coloring with at most 17 colors. [ABSTRACT FROM AUTHOR]

For a graph G, let ik(G) be the number of independent sets of size k in G. The independence polynomial I(G; x) =∑k≥0ik(G)xk has been the focus of considerable research. In this paper, using the coefficients of independence polynomials, we order graphs with some given parameters. We first determine the extremal graph whose all coefficients of I(G; x) are the largest (respectively, smallest) among all connected graphs (respectively, bipartite graphs) with given vertex cover number. Then we also derive the extremal graph whose all coefficients of I(G; x) are the largest among all connected graphs (respectively, bipartite graphs) with given vertex dominate number. [ABSTRACT FROM AUTHOR]

A 2-distance coloring of a graph is a proper k-coloring in which any two vertices with distance at most two get different colors. The 2-distance number is the smallest number k such that G has a 2-distance k-coloring, denoted as χ2(G). In 1977, Wegner conjectured that for each planar graph G with maximum degree Δ, χ2(G) ≤ 7 if Δ ≤ 3, χ2(G) ≤ Δ + 5 if 4 ≤ Δ ≤ 7, and χ2(G) ≤⌊3Δ 2 ⌋ + 1 if Δ ≥ 8. In 2001, Thomassen supported the conjecture by proving the case Δ = 3. The conjecture is still open even for Δ = 4. In this paper, we show that χ2(G) ≤ 17 for the case Δ ≤ 5 which improves the upper bound 18 recently obtained by Hou et al. [ABSTRACT FROM AUTHOR]

Numerous graph-theoretic methods can be used to investigate the efficiency and reliability of networks. The reliability of a network is determined based on its connectivity. At the very least, the system must continue to function in the event of part of its component (node or link) failures. An ideal system remains reliable even when errors occur. Diameter is a measure of network efficiency used to study the effects of link failures in networks. In this paper, we look into the eliminating edges affect to the hypercube graph’s diameter (call it by d) and calculate the maximum diameter that is denoted by f(t,d) after deleting t edges from hypercube graphs Qn for some n. [ABSTRACT FROM AUTHOR]

The edge incident 2-edge coloring of a graph G is an edge coloring of the graph G such that not more than two colors are assigned to the edges incident to an edge e = uv in G. In other words, for every edge e in G, the edge e and all the edges that are incident to the edge e is in at most two different color classes. The edge incident 2-edge coloring number ψein2′(G) is the maximum number of colors in any edge incident 2-edge coloring of G. The main objective of this paper is to study the edge incident 2-edge coloring concept and apply the same to some graph classes. Besides finding the exact values of these parameters, we also obtain some bounds. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the crosscap of 3-annihilating-ideal hypergraph 3 (R) of a commutative ring R and the topological embedding of 3 (R) to the nonorientable compact surfaces. Furthermore, we determine all Artinian commutative non-local rings R (up to isomorphism) such that 3 (R) is a projective graph. [ABSTRACT FROM AUTHOR]

The sum-connectivity index of a graph G is defined as the sum of weights 1 / d (u) + d (v) over all edges u v of G , where d (u) and d (v) are the degrees of the vertices u and v in G , respectively. A graph G is called quasi-tree, if there exists u ∈ V (G) such that G − u is a tree. In the paper, we give a sharp lower bound on the sum-connectivity index of quasi-tree graphs. [ABSTRACT FROM AUTHOR]

In this paper, we propose a set cover game and show that any Nash equilibrium is a minimal set cover, and also a Pareto optimal solution. Then we present a distributed algorithm to realize the game. The algorithm is self-organizing in the sense that all players can make decisions by themselves simultaneously. It is local in the sense that each player makes his decision based only on information in his neighborhood. It is efficient in the sense that it converges to a minimal set cover in polynomial time when c max / c min is upper bounded by a polynomial of the input size, where c max and c min are the maximum cost and the minimum cost of sets, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we obtain explicitly the r -dynamic chromatic number of the direct product of a path P m with a k -subdivision S n 1 k of a star graph, for every positive integers r , m , k and n. Particularly, it is obtained that 2 ≤ χ r (P m × S n 1 k ) ≤ 2 n + 1. [ABSTRACT FROM AUTHOR]

In this paper, we want to investigate some theorems about -cubic sets in [Sh. Rashid, M. Gulistan, Y. B. Jun, S. Khan and S. Kadry, -cubic sets and aggregation operators, J. Intell. Fuzzy Syst.37(4) (2019) 5009–5023] and while correcting these theorems, we get new results in this issue and at the end we present an application regarding decision making. First, by using some examples, we show that the conditions of some theorems in above-mentioned paper are not true in general and then we attempt to correct these theorems and proofs. Moreover, we discuss some new related results in -cubic sets. Finally, we discuss the score function and -cubic weighted average operator of -cubic sets for application in physical and different engineering regions. [ABSTRACT FROM AUTHOR]

In this paper, we characterize uniquely colorable graphs with equal chromatic and game chromatic numbers. [ABSTRACT FROM AUTHOR]

We address in this paper the connected p -median problem on complete multi-layered graphs. We first solve the corresponding 1-median problem on the underlying graph in linear time based on the structure of the induced path. For p ≥ 2 , we prove that the connected p -median contains at least one vertex in the layered set corresponding to the 1-median or its adjacent vertices in the induced path. Then, we develop an O (n log n) algorithm that solves the problem on a complete multi-layered graph with n vertices. [ABSTRACT FROM AUTHOR]

For a d -regular graph G , an edge-signing σ : E (G) → { − 1 , 1 } is called a good signing if the absolute eigenvalues of its adjacency matrix are at most 2 d − 1 . Bilu and Linial conjectured that for each regular graph there exists a good signing. In this paper, by using the concept of "Equitable Partition", we prove the conjecture for some cases. We show how to find out a good signing for special complete graphs and lexicographic product of two graphs. In particular, if there exist two good signings for graph G , then we can find a good signing for a 2-lift of G. [ABSTRACT FROM AUTHOR]

In this paper, we construct a weighted geometric realization of the set of 2-simplexes and 3-simplexes. On this weighted geometric realization, we create the Laplacian associated to 2-simplexes L S 2 and the Laplacian associated to 3-simplexes L S 3 . We prove that the nonzero spectrum of L S 2 is the same as the nonzero spectrum of L S 3 . For 0, we show that 0 belongs to the spectrum of L S 2 or to the spectrum of L S 3 . [ABSTRACT FROM AUTHOR]

Let A (G) and D (G) be the adjacency matrix and the diagonal matrix of vertex degrees of a graph G , respectively. For any real α ∈ [ 0 , 1 ] , Nikiforov defined the matrix A α (G) as A α (G) = α D (G) + (1 − α) A (G). Let G ¯ be complement of G. Define S α (G) = A α (G ¯) − A α (G) = (1 − α) J + (α n − 1) I − 2 A α (G) , α ∈ [ 0 , 1 ] , where I and J are the identity matrix and the all-ones matrix, respectively. Since S 0 (G) = A (G ¯) − A (G) = J − I − 2 A (G) is the Seidel matrix of G , this new matrix is the generalization of Seidel matrix. Further, S α (G) = (1 − α) J + (α n − 1) I − 2 α D (G) − 2 (1 − α) A (G) is a subset of universal adjacency matrices defined by Haemers and Omidi. In this paper, S α -eigenvalues and S α -energy of G , respectively, are investigated, which not only extends Seidel matrix but also enriches the theory of universal adjacency matrices. [ABSTRACT FROM AUTHOR]