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Recently there is huge interest in graph theory and intensive study on computing integer powers of matrices. In this paper, we consider one type of directed graph. Then we obtain a general form of the adjacency matrices of the graph. By using the well-known property which states the (i, j) entry of Am (A is adjacency matrix) is equal to the number of walks of length m from vertex i to vertex j, we show that elements of mth positive integer power of the adjacency matrix correspond to well-known Jacobsthal numbers. As a consequence, we give a Cassini-like formula for Jacobsthal numbers. We also give a matrix whose permanents are Jacobsthal numbers. [ABSTRACT FROM AUTHOR]

Detecting communities in real world networks is an important problem for data analysis in science and engineering. By clustering nodes intelligently, a recursive algorithm is designed to detect community. Since the relabeling of nodes does not alter the topology of the network, the problem of community detection corresponds to the finding of a good labeling of nodes so that the adjacency matrix form blocks. By putting a fictitious interaction between nodes, the relabeling problem becomes one of energy minimization, where the total energy of the network is defined by putting interaction between the labels of nodes so that clustering nodes that are in the same community will decrease the total energy. A greedy method is used for the computation of minimum energy. The method shows efficient detection of community in artificial as well as real world network. The result is illustrated in a tree showing hierarchical structure of communities on the basis of sub-matrix density. Applications of the method to weighted and directed networks are discussed. [ABSTRACT FROM AUTHOR]

Abstract: We define cut-and-paste, a construction which, given a quadriculated disk obtains a disjoint union of quadriculated disks of smaller total area. We provide two examples of the use of this procedure as a recursive step. Tilings of a disk receive a parity: we construct a perfect or near-perfect matching of tilings of opposite parities. Let be the black-to-white adjacency matrix: we factor , where and are lower and upper triangular matrices, is obtained from a larger identity matrix by removing rows and columns and all entries of , and are equal to , or . [Copyright &y& Elsevier]

Abstract: The Randić index R of a graph G is defined as the sum of over all edges of G, where denotes the degree of a vertex in G. is the largest eigenvalue of the signless Laplacian matrix of G, where D is the diagonal matrix with degrees of the vertices on the main diagonal and A is the adjacency matrix of G. Hansen and Lucas [18] conjectured (1) and equality holds for and (2) with equality if and only if for and for , respectively. In this paper, we prove the conjecture (1) and obtain a result very close to the conjecture (2). Moreover, we give some results relating harmonic index and the largest eigenvalue of the adjacency matrix. [Copyright &y& Elsevier]

Abstract: We calculate the Smith normal form of the adjacency matrix of each of the following graphs or their complements (or both): complete graph, cycle graph, square of the cycle, power graph of the cycle, distance matrix graph of cycle, Andrásfai graph, Doob graph, cocktail party graph, crown graph, prism graph, Möbius ladder. The proofs operate by finding the abelianization of a cyclically presented group whose relation matrix is column equivalent to the required adjacency matrix. [Copyright &y& Elsevier]

The energy of a simple graph G , E ( G ) , is the sum of the absolute values of the eigenvalues of its adjacency matrix. The energy of line graph and the signless Laplacian energy of graph G are denoted by E ( L G ) ( L G is the line graph of G ) and LE + ( G ) , respectively. In this paper we obtain a relation between E ( L G ) and LE + ( G ) of graph G . From this relation we characterize all the graphs satisfying E ( L G ) = LE + ( G ) + 4 m n − 4 . We also present a relation between E ( G ) and E ( L G ) . Moreover, we give an upper bound on E ( L G ) of graph G and characterize the extremal graphs. [ABSTRACT FROM AUTHOR]

In this paper the unique graph with maximal index (i.e. the largest eigenvalue of the adjacency matrix) is identified among all connected bipartite graphs of order n and size n + k , under the assumption that k ≥ 0 and n ≥ k + 5 . [ABSTRACT FROM AUTHOR]

Abstract: The rank of a graph G is defined to be the rank of its adjacency matrix. In this paper, we consider the following problem: what is the structure of a connected graph G with rank 5? or equivalently, what is the structure of a connected n-vertex graph G whose adjacency matrix has nullity ? In this paper, we completely characterize connected graphs G whose adjacency matrix has rank 5. [Copyright &y& Elsevier]

Abstract: The rank of a graph G is defined to be the rank of its adjacency matrix. In this paper, we consider the following problem: What is the structure of a connected graph with rank 4? This question has not yet been fully answered in the literature, and only some partial results are known. In this paper we resolve this question by completely characterizing graphs G whose adjacency matrix has rank 4. [Copyright &y& Elsevier]

Abstract: In this paper we obtain several characterizations of the adjacency matrix of a probe interval graph. In course of this study we describe an easy method of obtaining interval representation of an interval bigraph from its adjacency matrix. Finally, we note that if we add a loop at every probe vertex of a probe interval graph, then the Ferrers dimension of the corresponding symmetric bipartite graph is at most 3. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we give a criterion for the adjacency matrix of a Cayley digraph to be normal in terms of the Cayley subset . It is shown with the use of this result that the adjacency matrix of every Cayley digraph on a finite group is normal iff is either abelian or has the form for some non-negative integer , where is the quaternion group and is the abelian group of order and exponent 2. [Copyright &y& Elsevier]

In this paper we determine the unique graph with minimal least eigenvalue (of the adjacency matrix) within the set of connected graphs of fixed order n and size m , whenever m = ⌈ n 2 ⌉ ⌊ n 2 ⌋ + a , where a is a fixed integral constant in [ 1 , ⌈ n 2 ⌉ − 1 ] . [ABSTRACT FROM AUTHOR]

Let G be a simple graph with the adjacency matrix A ( G ) . A well-known result of Cvetković and Gutman states that removing a pendant vertex and its neighbour, does not change the nullity of the graph. We generalize this theorem and some other theorems similar to it for an arbitrary eigenvalue of a graph. Also, for an arbitrary eigenvalue λ , we use a corresponding star set to delete some subgraphs and to determine the multiplicity of λ . We use star sets to find some Parter–Wiener vertices, in trees. We state our results for real symmetric matrices (and so for weighted graphs) and as a corollary we use them for simple graphs. Furthermore, we use these methods to obtain some results for λ = 0 , ± 1 . [ABSTRACT FROM AUTHOR]

A graph in a certain graph class is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum among all graphs in that class. Bell et al. have identified a subclass within the connected graphs of order n and size m in which minimizing graphs belong (the complements of such graphs are either disconnected or contain a clique of size n/2 ). In this paper we discuss the minimizing graphs of a special class of graphs of order n whose complements are connected and contains exactly one cycle (namely the class Ucn of graphs whose complements are unicyclic), and characterize the unique minimizing graph in Ucn when n ≥ 20. [ABSTRACT FROM AUTHOR]

Let G be a graph with adjacency matrix A, and let H(t)=exp(itA). For an eigenvalue μ of A with multiplicity k, a star set for μ in G is a vertex set X of G such that |X|=k and the induced subgraph G-X does not have μ as an eigenvalue. G is said to have perfect state transfer from the vertex u to the vertex v if there is a time τ such that |H(τ)u,v|=1. The unitary operator H(t) has important applications in the transfer of quantum information. In this paper, we give an expression of H(t). For a star set X of graph G, perfect state transfer does not occur between any two vertices in X. We also give some results for the existence of perfect state transfer in a graph. [ABSTRACT FROM AUTHOR]

Let G = G1 ∪ G2 be the sum of two simple graphs G1, G2 having a common edge or G = G1 ∪ e1 ∪ e2 ∪ G2 be the sum of two simple disjoint graphs G1,G2 connected by two edges e1 and e2 which form a cycle C4 inside G. We give a method of computing the determinant detA(G) of the adjacency matrix of G by reducing the calculation of the determinant to certain subgraphs of G1 and G2. To show the scope and effectiveness of our method we give some examples [ABSTRACT FROM AUTHOR]

Abstract: Let be the skew adjacency matrix of the oriented graph of order n and be all eigenvalues of . The skew spectral radius of is defined as . In this paper, we investigate oriented graphs whose skew spectral radii do not exceed 2. [Copyright &y& Elsevier]

Abstract: The positive index of inertia of a graph G, denoted by , is the number of the positive eigenvalues of the adjacency matrix of G. In this paper, we investigate the minimal positive index of inertia among all bicyclic graphs of order n with pendant vertices, and characterize the bicyclic graphs with positive index 1 or 2 among all bicyclic graphs of order n. [Copyright &y& Elsevier]

Abstract: Let be a simple graph of order with triangle(s). Also let be the eigenvalues of the adjacency matrix of graph . X. Yong [X. Yong, On the distribution of eigenvalues of a simple undirected graph, Linear Algebra Appl. 295 (1999) 73–80] conjectured that (i) is complete if and only if and also (ii) is complete if and only if . Here we disprove this conjecture by a counter example. Wang et al. [J.F. Wang, F. Belardo, Q.X. Huang, B. Borovićanin, On the two largest Q-eigenvalues of graphs, Discrete Math. 310 (2010) 2858–2866] conjectured that friendship graph is determined by its adjacency spectrum. Here we prove this conjecture. The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity of a graph is the mean value of eccentricities of all vertices of . Moreover, we mention three conjectures, obtained by the system AutoGraphiX, about the average eccentricity , girth and the spectral radius of graphs (see Aouchiche (2006) [1], available online at http://www.gerad.ca/~agx/). We give a proof of one conjecture and disprove two conjectures by counter examples. [Copyright &y& Elsevier]

Abstract: The rank of a graph is that of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. We determine the maximum order of reduced trees as well as bipartite graphs with a given rank and characterize those graphs achieving the maximum order. [Copyright &y& Elsevier]

Abstract: Suppose that is the graph obtained by taking the box product of a path of length and a path of length . Let be the adjacency matrix of . In 1996, Rara showed that, if , then . We extend this result to allow and to be any positive integers, and show that [Copyright &y& Elsevier]

Abstract: Let be a function. Let be the matrix defined by if for some i and j and otherwise. We describe the Jordan canonical form of the matrix in terms of the directed graph for which is the adjacency matrix. We discuss several examples including a connection with the Collatz conjecture. [Copyright &y& Elsevier]

Abstract: The nullity of a graph is defined as the multiplicity of the eigenvalue zero in the spectrum of the adjacency matrix of the graph. We investigate a class of graphs with pendant trees, and express the nullity of such graph in terms of that of its subgraphs. As an application of our results, we characterize unicyclic graphs with a given nullity. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we focus on some operations of graphs and give a kind of eigenvalue interlacing in terms of the adjacency matrix, standard Laplacian, and normalized Laplacian. Also, we explore some applications of this interlacing. [Copyright &y& Elsevier]

Abstract: The spread of a graph is defined to be the difference between the largest eigenvalue and the least eigenvalue of the adjacency matrix of the graph. Let denote the set of connected unicyclic graphs of order n and girth k, and let denote the set of connected unicyclic graphs of order n. In this paper, we determine the unique graph with minimum least eigenvalue (respectively, the unique graph with maximum spread) among all graphs in . We, finally, characterize the unique graph with minimum least eigenvalue (respectively, the unique graph with maximum spread) among all graphs in . [Copyright &y& Elsevier]

Abstract: Let G be a graph of order n and rank(G) denotes the rank of its adjacency matrix. Clearly, . In this paper we characterize all graphs G such that or n +2. Also for every integer n ⩾5 and any k, 0⩽ k ⩽ n, we construct a graph G of order n, such that . [Copyright &y& Elsevier]

Abstract: Let G be a simple graph. Let and denote the largest eigenvalue of the adjacency matrix and the Laplacian matrix of G, respectively. Let denote the largest vertex degree. If G has just one cycle, thenThe equality holds if and only if . AndThe equality holds if and only if , n is even. [Copyright &y& Elsevier]

Abstract: The skew energy of a digraph D is defined as the sum of the singular values of its skew adjacency matrix . In this paper, we first interpret the entries of the power of the skew adjacency matrix of a digraph in terms of the number of its walks and then focus on the question posed by Adiga et al. [C. Adiga, R. Balakrishnan, Wasin So, The skew energy of a graph, Linear Algebra Appl. 432 (2010) 1825–1835] of determining all -regular connected digraphs with optimum skew energy. [Copyright &y& Elsevier]

Graphs G and H are called cospectral if they have the same characteristic polynomial, equivalently, if they have the same eigenvalues considering multiplicities. In this article we introduce a construction to produce pairs of cospectral regular graphs. We also investigate conditions under which the graphs are integral. [ABSTRACT FROM AUTHOR]

Abstract: Richard Brualdi proposed in Stevanivić (2007) [6] the following problem: (Problem AWGS.4) Let and be two nonisomorphic graphs on n vertices with spectra respectively. Define the distance between the spectra of and as Define the cospectrality of by Let In this paper we study Problem A and determine the cospectrality of certain graphs by the Euclidian distance. Let denote the complete graph on n vertices, denote the null graph on n vertices and denote the disjoint union of the with isolated vertices, where . In this paper we find , , ( ) and . [Copyright &y& Elsevier]

Abstract: This paper studies the problem of estimating the spectral radius of trees with the given number of vertices and maximum degree. We obtain the new upper bounds on the spectral radius of the trees, and the results are the best upper bounds expressed by the number of vertices and maximum degree, at present. Let be a tree on n vertices with maximum degree Δ, where . Denote by the spectral radius of T. We prove that [(1)] if , then , and equality holds if and only if T is an almost completely full-degree tree of 3 levels; [(2)] if , then , and equality holds if and only if T is a completely full-degree tree of 3 levels; [(3)] if , then , where . [Copyright &y& Elsevier]

Abstract: We derive some structural and spectral properties of regular bipartite graphs with three distinct non-negative eigenvalues. Next, we consider the relations between these graphs and two-class partially balanced incomplete block designs, and we present a number of situations when the graphs we consider are in fact the incidence graphs of those designs. As a consequence, we give a several constructions of connected regular bipartite graphs with six distinct eigenvalues, and we also determine all such graphs with degree 3, and all such graphs on at most 20 vertices. [Copyright &y& Elsevier]

Abstract: We give an upper bound for the least eigenvalue of a principal submatrix of a real symmetric matrix with zero diagonal, from which we establish an upper bound for the least eigenvalue of a graph when some vertices are removed using the components of the least eigenvector(s). We give lower and upper bounds for the least eigenvalue of a graph when some edges are removed. We also establish bounds for the components of the least eigenvector(s) of a real symmetric matrix and a graph. [Copyright &y& Elsevier]

Abstract: An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of or . We define the adjacency, incidence and Laplacian matrices of an oriented hypergraph and study each of them. We extend several matrix results known for graphs and signed graphs to oriented hypergraphs. New matrix results that are not direct generalizations are also presented. Finally, we study a new family of matrices that contains walk information. [Copyright &y& Elsevier]

Abstract: We determine all trees whose second largest eigenvalue does not exceed . Next, we consider two classes of bipartite graphs, regular and semiregular, with small number of distinct eigenvalues. For all graphs considered we determine those whose second largest eigenvalue is equal to . Some additional results are also given. [Copyright &y& Elsevier]

Abstract: Let be two simple connected graphs. Denote the corona and the edge corona of by and , respectively. In this paper, we first introduce a new invariant, the -coronal of a graph matrix , where the matrix is associated with a graph in a prescribed way. Then it is used to compute the signless Laplacian spectrum of and in terms of the signless Laplacian spectrum of and . In addition, the spectrum of are also given in terms of the spectrum of and . Finally, as an application of these results, we construct many pairs of nonisomorphic signless Laplacian cospectral graphs. [Copyright &y& Elsevier]

Abstract: A -graph is a graph consisting of two cycles and with just a vertex in common. Fan and Luo in Theorem 4.4 alleged that a -graph is determined by its adjacency spectrum if it contains no cycle . However, according to Theorem 6.3 we find that the result in Theorem 4.4 is not completely correct. In this paper, we prove that is DAS if and only if , and has a unique cospectral mate if . [Copyright &y& Elsevier]