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In this paper, we revise Conway’s law from a mathematical point of view. We model it more mathematically by introducing several mathematical tools. By introducing a task graph, we first rigorously state Conway’s law based on the homomorphisms in graph theory for the software system and the organizations that created it. Although Conway did not mention it, the task graph shows the geometric structure of tasks, which plays a crucial role. However, the model in terms of homomorphism is not sufficient for recent requirements for high-level treatment of communication (from security, knowledge hiding, etc.) in organizations and hierarchical treatment of organizations. In order to reformulate it for the requirement, we introduce mathematical tools, weakened homomorphisms and graph topology. Then we provide a novel mathematical model of Conway’s law with hierarchical and complicated structures along the line of the recent requirement. [ABSTRACT FROM AUTHOR]

Heuristics aim to provide approximate solutions for NP-hard combinatorial optimization (CO) problems in a reasonable time. They are faster than deterministic methods, where the solving process may take days. This paper presents an attempt to combine a heuristic algorithm and some machine learning (ML) methods for solving the Multidimensional Knapsack Problem (MKP), which is a well-known CO case. The approach named Genetic Algorithm and Machine Learning (GAML) is a two-step algorithm that first uses four machine methods to produce four solutions and then integrates the best one within the GA. For this purpose, (1) This approach builds models by training four machine learning (ML) methods using the decision variables obtained from the optimal solutions of simple MKP instances. The decision variable of an item is equal to one if it is selected in the knapsack and zero otherwise. (2) The four models are applied to predict the decision variables for complex MKP instances. (3) A dummy algorithm constructs feasible solutions from the predictions by adding the selected items randomly while the constraints are not violated. (4) For each MKP instance, the best feasible solution among the four built is integrated with the initialization and the crossing operators of a genetic algorithm. An experiment has been conducted on well-known complex MKP benchmarks. The results proved that the models were able to provide high-quality solutions. Also, the proposed algorithm was faster than other approaches in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we will introduce a hypergraph associated with a given module. The module, denoted as B, is defined over a ring S. The hypergraph consists of vertices that represent nontrivial submodules of B, which we denote as 풟S(B). A set Ei of vertices forms a hyperedge if the intersection of any two distinct elements in Ei is a δ-small submodule of B and Ei is maximal with respect to this property. The connectivity of 풟S(B) is discussed, as well as the diameter and girth for a connected hypergraph. Various examples are provided to illustrate the concepts presented. [ABSTRACT FROM AUTHOR]

Given an undirected graph G and a real edge-weight vector, the connected subgraph problem consists of finding a maximum-weight subset of edges which induces a connected subgraph of G. In this paper, we establish a link between the complexity of the connected subgraph problem and the matching number. We study the separation problem associated with the so-called matching-partition inequalities, which were introduced by Didi Biha, Kerivin and Ng [Polyhedral study of the connected subgraph problem, Discr. Math. 338 (2015) 80–92] for the connected subgraph polytope. [ABSTRACT FROM AUTHOR]

In this paper, the concepts of nilpotent and Engel orthomodular lattices, similar with the solvable orthomodular lattices are defined and their properties are investigated. The notion of n-Engel orthomodular lattices as a natural generalization of distributive orthomodular lattices is introduced, and we discuss Engel orthomodular lattices, which is defined by left and right normed commutators. [ABSTRACT FROM AUTHOR]

The edge version of the Nirmala index is defined in terms of edge degrees as EN(G) =∑e∼fd(e) + d(f), where d(x) denotes the degree of the edge x in graph G and e ∼ f means that the edges e and f are adjacent. In this paper, we obtain some sharp bounds for the edge version of the Nirmala index in terms of the edge version of topological indices. Also, we give the extremal unicyclic graphs of order n with the minimum edge version of the Nirmala index. [ABSTRACT FROM AUTHOR]

The beta invariant is closely related to the Chromatic and Tutte Polynomials and has been extensively studied, see Brylawski [A combinatorial model for series-parallel networks, Trans. Amer. Math. Soc. 154 (1971) 1–22], Crapo [A higher invariant for matroids, J. Combin. Theory 2 (1967) 406–417], Lee and Wu [Bounding the beta invariant of 3-connected matroids, Discrete Math. 354 (2022) 1–11], Oxley [On Crapo’s beta invariant for matroids, Stud. Appl. Math. 66(3) (1982) 267–277], and others. Lee and Wu [Bounding the beta invariant of 3-connected matroids, Discrete Math. 354 (2022) 1–11] established that a 3-connected graph with n vertices possesses a beta invariant of at least n − 2, reaching equality only when the graph is a wheel or the Prism. Additionally, Oxley [On Crapo’s beta invariant for matroids, Stud. Appl. Math. 66(3) (1982) 267–277] provided characterizations for matroids with beta invariant values of two, three, and four.This paper extends the findings of Lee and Wu by offering a comprehensive characterization of all 3-connected graphs with n vertices that have a beta invariant of either n or n − 1. As a consequence, we prove that any 3-connected graph with at least 8 vertices other than a Wheel has a beta invariant of at least n + 1. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the pair mean cordial labeling behavior of udukkai graph, octopus graph, double wheel graph, barbell graph, drum graph, vanessa graph and fire cracker graph. [ABSTRACT FROM AUTHOR]

This paper considers the notion of nonbinary (linear) codes and the notion of hypergraphs and makes a relation between two mathematical tools for real-world applications. We consider a nonbinary (linear) code as an underline set and construct a type of hypergraphs called N.C-hypergraph (N.L.C-hypergraph) and establish any given hypergraph is isomorphic to an N.C-hypergraph (N.L.C-hypergraph). This study introduces an equivalence relation on any given code as a positive equivalence relation based on the weight of codewords, generates quotient N.C-hypergraph, and computes the cardinal of its hyperedges and Hamming distance of any given two code words of quotient N.C-hypergraphs. Also, we introduce a binary relation on quotient N.C-hypergraphs, convert them to simple graphs as code-based graphs, and prove that all graphs are positive code-based graphs. We characterize the minimum distance and maximum distance of codewords in hyperedges of code-based (hyper)graphs concerning linear codes. Finally, we design real problems (such as wireless sensor (hyper)networks and American standard code for information interchange) as complex hypernetworks (such that are based on the code words) as a quotient N.C-hypergraph (N.L.C-hypergraph) and convert them to the code-based graphs (such as wireless sensor networks). [ABSTRACT FROM AUTHOR]

Topological indices are numeric quantities that describes the topology of molecular structure in mathematical chemistry. One of these indices is the neighborhood version of the hyper Zagreb index, HMN. The neighborhood version of the hyper Zagreb index is expressed as the sum over all pairs of adjacent vertices a and b of the terms [δG(a) + δG(b)]2, where δG(a) is the neighborhood degree sum of the vertex a which is the sum of degrees of all of its neighboring vertices. In this paper, we show that over all simple connected graphs, the path graph Pn has the minimum value of HMN and the complete graph Kn has the maximum value of HMN. Also, we give the minimum value of this index within the set of trees with given vertices number and maximum vertex degree and specify the minimal trees. [ABSTRACT FROM AUTHOR]

This paper presents a formula for the a-number of certain maximal curves characterized by the equation yq+1 2 = xm + x over the finite field 픽q2. a-number serves as an invariant for the isomorphism class of the p-torsion group scheme. By utilizing the action of the Cartier operator on H0(풳, Ω1), we establish a closed formula for the a-number of 풳. [ABSTRACT FROM AUTHOR]

Let s be a positive integer. A graph is s-transitive if its automorphism group is transitive on s-arcs but not on (s + 1)-arcs. In this paper, we study all tetravalent s-transitive graphs of order 6p2. [ABSTRACT FROM AUTHOR]

Let R be a finite commutative ring with identity. In this paper, formal expressions of the number of m × n matrices over R of rank r and the number of invertible matrices over R are presented. The number of matrices over R with a given rank and a given number of single unit-entry rows, rows in which a single entry is a unit and all other entries are zero, is finally determined. [ABSTRACT FROM AUTHOR]

In this paper, we study near approximations in Cayley graphs, which are an extended study of rough approximations in Cayley graphs by using more than one equivalence relations. Furthermore, we introduce the notion of a near edge Cayley graphs, expanded to the near vertex pseudo-Cayley graphs and near pseudo-Cayley graphs. Some theorems dealing with the properties discussed are derived. We illustrate some examples especially in cyclic groups for simplification in graphs visualization. [ABSTRACT FROM AUTHOR]

A star coloring of a graph G is a proper vertex coloring in which every path on four vertices uses at least three distinct colors. Equivalently, in a star coloring, the induced subgraphs formed by the vertices of any two colors have connected components that are star graphs. A graph G is k - star-colorable if there exists a star coloring of G from a set of k colors. The minimum positive integer k for which G is k -star-colorable is the star chromatic number of G and is denoted by χ s (G). In this paper, upper and lower bounds are presented for the star chromatic number of the rooted product, hierarchical product, and lexicographic product. [ABSTRACT FROM AUTHOR]

A nonempty set S ⊆ V (G) of a graph G = (V , E) is an open packing set of G if no two vertices of S have a common neighbor in G. The maximum cardinality of an open packing set is called the open packing number of G and is denoted by ρ o (G). The open packing subdivision number  sd ρ o (G) is the minimum number of edges in G that must be subdivided (each edge in G can be subdivided at most once) in order to increase the open packing number. In this paper, we initiate a study on this parameter. [ABSTRACT FROM AUTHOR]

For a simple connected graph G , let D (G) and T (G) be the distance matrix and the diagonal matrix of the vertex transmissions, respectively. The convex linear combination D α (G) of D (G) and T (G) is defined as, D α (G) = α T (G) + (1 − α) D (G) , 0 ≤ α ≤ 1. The matrix D α (G) , known as generalized distance matrix, is effective in merging the distance spectral and distance signless Laplacian spectral theories. In this paper, we study the spectral radius and energy of the generalized distance matrix D α (G) of a graph G. We obtain bounds for the generalized distance spectral radius and generalized distance energy of connected graphs in terms of various parameters associated with the structure of graph. [ABSTRACT FROM AUTHOR]

Let G be a graph. A vertex-cut S of G is said to be k-restricted if every component of G − S has at least k vertices, and cyclic if G − S has at least two components which contain a cycle. The minimum cardinality over all k -restricted vertex-cuts of G is called the k-restricted connectivity of G and is denoted by κ k (G). Also the minimum cardinality over all cyclic vertex-cuts of G is called the cyclic connectivity of graph G and is denoted by κ c (G). In this paper, the k -restricted connectivity and cyclic connectivity of the direct product of two graphs G 1 and G 2 is obtained for some k ≥ 2 , where G 1 is a complete graph, and G 2 is a complete graph, a complete bipartite graph or a cycle. [ABSTRACT FROM AUTHOR]

In this paper, we prove that every simple connected graph with α > n 2 has 1 as a normalized Laplacian eigenvalue with multiplicity at least 2 α − n , where n and α are the order and the independence number of the graph, respectively. Then we investigate graphs with α ≤ n 2 and having or not having 1 as a normalized Laplacian eigenvalue. [ABSTRACT FROM AUTHOR]

The graph obtained by joining cycle C m to a path P n with a bridge is called Tadpole graph denoted by T m , n . A subset D of V (G) is said to be a tadpole dominating set of G if D is a dominating set and the set of vertices of D spans a tadpole graph T m , n where m ≥ 3 , n ≥ 1. In this paper, we find the tadpole domination number of cartesian product of certain graphs, namely, paths, cycles and complete graphs. Also the tadpole domination number for the Mycielskian of cycles and paths is obtained. [ABSTRACT FROM AUTHOR]

Molodtsov developed soft set theory in 1999 as an approach for modeling vagueness and uncertainty. Many academics currently employ soft set theory to solve decision-making problems. A parameterized point of view for graphs is provided using the idea of soft graphs. Soft graph theory is a rapidly growing field in graph theory because of its adaptability to parameterization techniques. In graph theory, the topic of graph products has attracted much attention. It is a binary operation on graphs that has many combinatorial applications. Analogous to the definitions of graph products, we can define product operations on soft graphs. In this paper, we introduce the tensor product, the restricted tensor product, the strong product, and the restricted strong product of soft graphs. We prove that these products of soft graphs are also soft graphs, and we derive formulas for finding vertex count, edge count, and the sum of part degrees in them. [ABSTRACT FROM AUTHOR]

The vertex induced 2-edge coloring number ψ vi 2 ′ (G) of a graph G is the highest number of colors that can occur in an edge coloring of a graph G such that not more than two colors can be used to color the edges in the induced subgraph 〈 N [ v ] 〉 generated by the closed neighborhood N [ v ] of a vertex v in V (G). The vertex induced 2-edge coloring sum of a graph G denoted as ∑ vi 2 ′ (G) , is the greatest sum among all the vertex induced 2-edge coloring of a graph G which concedes ψ vi 2 ′ (G) colors. The vertex incident 2-edge coloring number of a graph G is the highest number of colors required to color the edges of a graph G such that not more than two colors can be ceded to the edges incident at the vertex v of G. The vertex incident 2-edge coloring sum of a graph G denoted as ∑ vi 2 ′ (G) , is the maximum sum among all the vertex incident 2-edge coloring of graph G which receives maximum ψ vin 2 ′ (G) colors. In this paper, we initiate a study on the vertex induced 2-edge coloring sum and vertex incident 2-edge coloring sum concepts and apply the same to some graph classes. Besides finding the exact values of these parameters, we also obtain some bounds and a few comparative results. [ABSTRACT FROM AUTHOR]

A proper vertex coloring of a graph G is equitable if the sizes of any two color classes differ by at most one. The equitable chromatic number of a graph G , denoted by χ = (G) , is the minimum k such that G is equitably k -colorable. In this paper, we discuss some basic properties of equitable critical graphs as well as equitable k -critical graphs. A graph G is called equitable critical if χ = (H) ≠ χ = (G) for every proper subgraph H of G. G is called equitable k -critical if it is equitable k -chromatic and equitable critical. Furthermore, we discuss that equitable vertex (edge) critical, equitable critical vertex (edge) graphs. [ABSTRACT FROM AUTHOR]

For a simple connected graph G , the generalized distance matrix D α (G) is defined as D α (G) = α Tr (G) + (1 − α) D (G) , 0 ≤ α ≤ 1 , where Tr (G) and D (G) are, respectively, the diagonal matrix of vertex transmission degrees and the distance matrix of G. In this paper, we will study the spectral radius of the generalized distance matrix D α (G) of a connected graph G. We obtain some upper and lower bounds for the generalized distance spectral radius in terms of various parameters, like the vertex transmission degrees, the average transmission 2-degrees, etc., associated with the structure of the graph. We characterize the extremal graphs attaining these bounds. Further, we show that our bounds improve some previously known bounds existing in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we give a class of graphs that do not admit disjoint maximum and maximal independent (MMI) sets. The concept of inverse independence was introduced by Bhat and Bhat in [Inverse independence number of a graph, Int. J. Comput. Appl. 42(5) (2012) 9–13]. Let A be a α -set in G. An independent set D ⊂ V (G) − A is called an inverse independent set with respect to A. The inverse independence number α − 1 (G) is the size of the largest inverse independent set in G. Bhat and Bhat gave few bounds on the independence number of a graph, we continue the study by giving some new bounds and exact value for particular classes of graphs: spider tree, the rooted product and Cartesian product of two particular graphs. [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]

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]

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]

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]

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]

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 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]

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 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 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]

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]

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

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]

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]

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]

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 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]

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]

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]

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]

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]