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Let G be a graph with n vertices and m edges. The minimum edge dominating energy is defined as the sum of the absolute values of eigenvalues of the minimum edge dominating matrix of the graph G. In this paper, some lower and upper bounds for the minimum edge dominating energy of graph G are established. [ABSTRACT FROM AUTHOR]

Abstract A (1 , ≤ ℓ) -identifying code in digraph D is a dominating subset C of vertices of D , such that all distinct subsets of vertices of D with cardinality at most ℓ have distinct closed in-neighborhoods within C. As far as we know, it is the very first time that the spectral graph theory has been applied to the identifying codes. We give a new method to obtain an upper bound on ℓ for digraphs. The results obtained here can also be applied to graphs. [ABSTRACT FROM AUTHOR]

Abstract Motivated by the study of genomes evolving by reversals, we consider pseudograph transformations known as "pressing sequences". In particular, we address the question of when a graph has precisely one pressing sequence resulting in the empty graph, thus answering an question from Cooper and Davis (2015) [13]. We characterize such "uniquely pressable" graphs, count the number of them on a given number of vertices, and provide a polynomial-time recognition algorithm. We conclude with several open questions. [ABSTRACT FROM AUTHOR]

Abstract Finding a diagonal matrix congruent to A − c I for constants c , where A is the adjacency matrix of a graph G allows us to quickly tell the number of eigenvalues in a given interval. If G has clique-width k and a corresponding k -expression is known, then diagonalization can be done in time O (poly (k) n) where n is the order of G. [ABSTRACT FROM AUTHOR]

Let G 1 , G 2 and H be three graphs on disjoint sets of vertices and G 1 has m 1 edges. Let S (G 1 , H) be the graph obtained from G 1 and H in the following way: (1) Delete all the edges of G 1 and consider m 1 disjoint copies of H. (2) Join each vertex of the i th copy of H to the end vertices of the i th edge of G 1 . Let G 1 (∨ H) G 2 be the graph obtained from S (G 1 , H) by joining each vertex of G 1 with each vertex of G 2. In this paper, we determine the adjacency (respectively, Laplacian, signless Laplacian) spectrum of G 1 (∨ H) G 2 in terms of those of G 1 , G 2 and H. As an application, we construct infinite pairs of cospectral graphs. [ABSTRACT FROM AUTHOR]

Representing a subset of vertices in a graph by means of a matrix was introduced by E. Sampath Kumar. Let G(V, E) be a graph and S ⊆ V be a set of vertices. We can represent the set S by means of a matrix as follows, in the adjacency matrix A(G) of G replace the aii element by 1 if and only if, vi ∊ S. In this paper we study the set S being dominating set and corresponding domination energy of some class of graphs. [ABSTRACT FROM AUTHOR]

Let G be a graph with adjacency matrix A ( G ) and let D ( G ) be the diagonal matrix of the degrees of G . For any real α ∈ [ 0 , 1 ] , Nikiforov [3] defined the matrix A α ( G ) as A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . The largest eigenvalue of A α ( G ) is called the A α -spectral radius of G . In this paper, we give three edge graft transformations on A α -spectral radius. As applications, we determine the unique graph with maximum A α -spectral radius among all connected graphs with diameter d , and determine the unique graph with minimum A α -spectral radius among all connected graphs with given clique number. In addition, some bounds on the A α -spectral radius are obtained. [ABSTRACT FROM AUTHOR]

We give counterexamples to a result in F. Esser and F. Harary [2, Theorem 3] asserting that two nonisomorphic complete r -partite graphs with the same number of vertices have different spectral radii. We then derive some results on invariant factors and apply them to obtain relationship between the parameters of integral complete multipartite graphs and their integer eigenvalues. Necessary conditions for complete multipartite graphs to be integral are obtained. [ABSTRACT FROM AUTHOR]

In this work, we introduce balance-binomial graphs, whose entries of adjacency matrix are balance-binomial coefficients. Then we obtain some characteristic properties of the graph. [ABSTRACT FROM AUTHOR]

For an undirected simple graph G , the line graph L ( G ) is the graph whose vertex set is in one-to-one correspondence with the edge set of G where two vertices are adjacent if their corresponding edges in G have a common vertex. The energy E ( G ) is the sum of the absolute values of the eigenvalues of G . The vertex connectivity κ ( G ) is referred as the minimum number of vertices whose deletion disconnects G . The positive inertia ν + ( G ) corresponds to the number of positive eigenvalues of G . Finally, the matching number β ( G ) is the maximum number of independent edges of G . In this paper, we derive a sharp upper bound for the energy of the line graph of a graph G on n vertices having a vertex connectivity less than or equal to k . In addition, we obtain upper bounds on E ( G ) in terms of the edge connectivity, the inertia and the matching number of G . Moreover, a new family of hyperenergetic graphs is obtained. [ABSTRACT FROM AUTHOR]

A signed graph G σ consists of an underlying graph G and a sign function σ , which assigns each edge uv of G a sign σ ( u v ) , either positive or negative. The adjacency matrix of G σ is defined as A ( G σ ) = ( a u , v σ ) with a u , v σ = σ ( u v ) a u , v , where a u , v = 1 if u , v ∈ V ( G ) are adjacent, and a u , v = 0 otherwise. The positive inertia index of G σ , written as p ( G σ ) , is defined to be the number of positive eigenvalues of A ( G σ ) . Recently, Yu et al. (2016) [12] characterized the signed graphs G σ with pendant vertices such that p ( G σ ) = 2 . In this paper, we extend the above work to a more general case, characterizing the signed graphs G σ with cut points whose positive inertia index is 2. [ABSTRACT FROM AUTHOR]

Given a graph G with n vertices and m edges, the term energy of graph E(G) was introduced by Gutman in chemistry, due to its relevance to the total π ― electron energy of carbon compounds. In 1971 McClelland obtained both lower and upper bounds for π 8213; electron energy. An improved upper bound was obtained by Koolen-Moulton in 2001. The lower and upper bounds for E(G) obtained in this paper are better than McClelland and Koolen-Moulton bounds. Also we obtained an upper bound for graph energy in terms of n as E(G) ≤ n/2[ 1 + √ n-2/2]. [ABSTRACT FROM AUTHOR]

In a certain class of graphs, a graph is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum. A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to its number of vertices plus one. In this paper, we characterize the minimizing graph among all the connected graphs that belong to a class of graphs whose complements are bicyclic with two cycles. [ABSTRACT FROM AUTHOR]

For a (simple) graph H and non-negative integers c0, c1, . . ., cd (cd ≠ 0), p(H) = Pd ...k=0 ck · Hk is the lexicographic polynomial in H of degree d, where the sum of two graphs is their join and ck · Hk is the join of ck copies of Hk. The graph Hk is the kth power of H with respect to the lexicographic product (H0 = K1). The spectrum (if H is connected and regular) and the Laplacian spectrum (in general case) of p(H) are determined in terms of the spectrum of H and ck's. Constructions of infinite families of cospectral or integral graphs are announced. [ABSTRACT FROM AUTHOR]

In this paper, bipartite graphs and their adjacency matrices are applied to equivalently represent covering-based rough sets through three sides, which are approximation operators, properties and reducible elements. Firstly, a bipartite graph is constructed through a covering. According to the constructed bipartite graph, two equivalent representations of a pair of covering upper and lower approximation operators are presented. And an algorithm is designed for computing the pair of covering approximation operators from the viewpoint of these equivalent representations. Some properties and reducible elements of covering-based rough sets are also investigated through the constructed bipartite graph. Finally, an adjacency matrix of the constructed bipartite graph is proposed, and reducible elements in the covering are obtained through the proposed adjacency matrix. Moreover, an equivalent representation of the covering upper approximation operator is presented through the proposed adjacency matrix. In a word, these results show two interesting views, which are graphs and matrices, to investigate covering-based rough sets. [ABSTRACT FROM AUTHOR]

A well known upper bound for the spectral radius of a graph, due to Hong, is that μ 1 2 ≤ 2 m − n + 1 if δ ≥ 1 . It is conjectured that for connected graphs n − 1 ≤ s + ≤ 2 m − n + 1 , where s + denotes the sum of the squares of the positive eigenvalues. The conjecture is proved for various classes of graphs, including bipartite, regular, complete q -partite, hyper-energetic, and barbell graphs. Various searches have found no counter-examples. The paper concludes with a brief discussion of the apparent difficulties of proving the conjecture in general. [ABSTRACT FROM AUTHOR]

The k-th power of a graph G, denoted by [ABSTRACT FROM AUTHOR]

Let λ2(G) and τ (G) denote the second largest eigenvalue and themaximum number of edge-disjoint spanning trees of a graph G, respectively. Motivated by a question of Seymour on the relationship between eigenvalues of a graph G and bounds of τ (G), Cioaba andWong conjectured that for any integers d, k ≥ 2 and a d-regular graph G, if λ2(G) < d - 2k-1/d+1, then τ (G) ≥ k. They proved the conjecture for k = 2, 3, and presented evidence for the cases when k ≥ 4. Thus the conjecture remains open for k ≥ 4. We propose a more general conjecture that for a graph G with minimum degree δ ≥ 2k ≥ 4, if λ2(G) < δ - 2k-1 δ+1, then τ (G) ≥ k. In this article, we prove that for a graph G with minimum degree δ, each of the following holds. (i) For k {2, 3}, if δ ≥ 2k and λ2(G) < δ - 2k-1/δ+1, then τ (G) ≥ k. (ii) For k ≥ 4, if δ ≥ 2k and λ2(G) < δ - 3k-1/δ+1, then τ (G) ≥ k. Our results sharpen theorems of Cioaba and Wong and give a partial solution to Cioaba and Wong's conjecture and Seymour's problem. We also prove that for a graph G with minimum degree δ ≥ k ≥ 2, if λ2(G) < δ - 2(k-1) δ+1, then the edge connectivity is at least k, which generalizes a former result of Cioab? a. As corollaries, we investigate the Laplacian and signless Laplacian eigenvalue conditions on τ (G) and edge connectivity. [ABSTRACT FROM AUTHOR]

In a simple connected graph, the average 2-degree of a vertex is the average degree of its neighbors. With the average 2-degree sequence and the maximum degree ratio of adjacent vertices, we present a sharp upper bound of the spectral radius of the adjacency matrix of a graph, which improves a result in [Y. H. Chen, R. Y. Pan and X. D. Zhang, Two sharp upper bounds for the signless Laplacian spectral radius of graphs, Discrete Math. Algorithms Appl. 3(2) (2011) 185-191]. [ABSTRACT FROM AUTHOR]

In this paper we present an O( nlog n) time algorithm for finding a maximum flow in a directed planar graph, where the vertices are subject to capacity constraints, in addition to the arcs. If the source and the sink are on the same face, then our algorithm can be implemented in O( n) time. For general (not planar) graphs, vertex capacities do not make the maximum flow problem more difficult, as there is a simple reduction that eliminates vertex capacities. However, this reduction does not preserve the planarity of the graph. The essence of our algorithm is a different reduction that does preserve the planarity and can be implemented in linear time. For the special case of undirected planar graphs, an algorithm with the same time complexity was recently claimed, but we show that it has a flaw. We also apply our technique to obtain a linear-time algorithm to convert a flow to an acyclic flow, and a linear-time algorithm to find a largest set of vertex-disjoint s− t paths, in a directed planar graph. [ABSTRACT FROM AUTHOR]

We consider the multivariate interlace polynomial introduced by Courcelle (Electron. J. Comb. 15(1), ), which generalizes several interlace polynomials defined by Arratia, Bollobás, and Sorkin (J. Comb. Theory Ser. B 92(2):199-233, ) and by Aigner and van der Holst (Linear Algebra Appl., ). We present an algorithm to evaluate the multivariate interlace polynomial of a graph with n vertices given a tree decomposition of the graph of width k. The best previously known result (Courcelle, Electron. J. Comb. 15(1), ) employs a general logical framework and leads to an algorithm with running time f( k)⋅ n, where f( k) is doubly exponential in k. Analyzing the GF(2)-rank of adjacency matrices in the context of tree decompositions, we give a faster and more direct algorithm. Our algorithm uses $2^{3k^{2}+O(k)}\cdot n$ arithmetic operations and can be efficiently implemented in parallel. [ABSTRACT FROM AUTHOR]

RC-graphs are special type of graphs. This paper deals with the Eigen values of adjacency matrices and other related matrices of RC-graphs. Many interesting results on eigen values apart from eigen values of general graphs are obtained for RC-graphs. Several directions are initiated towards possible future study. [ABSTRACT FROM AUTHOR]

Given a locally finite simple graph so that its degree is not bounded, every self-adjoint realization of the adjacency matrix is unbounded from above. In this note, we give an optimal condition to ensure it is also unbounded from below. We also consider the case of weighted graphs. We discuss the question of self-adjoint extensions and prove an optimal criterium. [ABSTRACT FROM AUTHOR]

Several matrices can be associated to a graph such as the adjacency matrix or the Laplacian matrix. The spectrum of these matrices gives some informations about the structure of the graph and the question "Which graphs are determined by their spectrum?" remains a difficult problem in algebraic graph theory. In this article we enlarge the known families of graphs determined by their spectrum by considering some unicyclic graphs. An odd (resp. even) sun is a graph obtained by appending a pendant vertex to each vertex of an odd (resp. even) cycle. A broken sun is a graph obtained by deleting pendant vertices of a sun. In this paper we prove that a sun is determined by its Laplacian spectrum, an odd sun is determined by its adjacency spectrum (counter-examples are given for even suns) and we give some spectral characterizations of broken suns. [ABSTRACT FROM AUTHOR]

In this article we follow the study of the hierarchical product of graphs, an operation recently introduced in the context of networks. A well-known example of such a product is the binomial tree which is the (hierarchical) power of the complete graph on two vertices. An appealing property of this structure is that all the eigenvalues are distinct. Here we show how to obtain a graph with this property by applying the hierarchical product. In particular, we propose a generalization of the binomial tree and study some of its main properties. [ABSTRACT FROM AUTHOR]

A spectral graph theory is a theory in which graphs are studied by means of eigenvalues of a matrix M which is in a prescribed way defined for any graph. This theory is called M-theory. We outline a spectral theory of graphs based on the signless Laplacians Q and compare it with other spectral theories, in particular with those based on the adjacency matrix A and the Laplacian L. The Q-theory can be composed using various connections to other theories: equivalency with A-theory and L-theory for regular graphs, or with L-theory for bipartite graphs, general analogies with A-theory and analogies with A-theory via line graphs and subdivision graphs. We present results on graph operations, inequalities for eigenvalues and reconstruction problems. [ABSTRACT FROM AUTHOR]

We propose a simple data structure for an efficient implementation of the Italiano algorithms for the dynamic updating of the transitive closure of a directed graph represented as adjacency matrix on a model of associative (or content addressable) parallel processors with vertical processing (the STAR–machine). Associative versions of the Italiano algorithms are represented as procedures DeleteArc1 and InsertArc1. We prove the correctness of these procedures and evaluate their time and space complexity. We also present the main advantages of associative versions of the Italiano algorithms. [ABSTRACT FROM AUTHOR]

Abstract: We characterize the graphs which achieve the maximum value of the spectral radius of the adjacency matrix in the sets of all graphs with a given domination number and graphs with no isolated vertices and a given domination number. [Copyright &y& Elsevier]

Abstract: For acyclic and unicyclic graphs we determine a necessary and sufficient condition for a graph G to be singular. Further, it is shown that this characterization can be used to construct a basis for the null-space of G. [Copyright &y& Elsevier]

Abstract: We generalize the concept of a Bethe tree as follows: we say that an unweighted rooted tree is a generalized Bethe tree if in each level the vertices have equal degree. If is a generalized Bethe tree of k levels then we characterize completely the eigenvalues of the adjacency matrix and Laplacian matrix of a graph obtained from the union of r copies of and the cycle connecting the r vertex roots. Moreover, we give results on the multiplicity of the eigenvalues, on the spectral radii and on the algebraic conectivity. [Copyright &y& Elsevier]

Abstract: Recently, Chang and Fu [Y.C. Chang, H.L. Fu, The number of 6-cycles in a graph, Bull. Inst. Combin. Appl. 39 (2003) 27–30] derived an exact expression, based on powers of the adjacency matrix, for the number of 6-cycles in a graph. Here, I demonstrate a method for obtaining the number of n-cycles in a graph from the immanants of the adjacency matrix. The method is applicable to cycles of all sizes and to sets of disjoint cycles of any sizes, and the cycles in the set need not be the same size. [Copyright &y& Elsevier]

We consider free bosons hopping on a network (infinite graph). The condition for Bose–Einstein condensation is given in terms of the random walk on a graph. In case of periodic lattices, we also consider boson moving in an external periodic potential and obatin the criterion for Bose–Einstein condensation. [ABSTRACT FROM AUTHOR]

Abstract: This is a survey of recent results showing that the usual concepts of (local or global) distance-regularity in a graph can be thought of as an extremal (numeric) property of the graph. This is because such structures appear when a certain spectral bound is attained, so yielding striking characterizations of distance-regularity, with the peculiarity of involving only numerical (instead of the usual matricial) identities. Other results providing bounds for specific parameters of the graph, such as its eigenvalue multiplicities, are also derived. [Copyright &y& Elsevier]

Let G be a simple undirected graph. For v∈V(G), the 2-degree of v is the sum of the degrees of the vertices adjacent to v. Denote by ρ(G) and μ(G) the spectral radius of the adjacency matrix and the Laplacian matrix of G, respectively. In this paper, we present two lower bounds of ρ(G) and μ(G) in terms of the degrees and the 2-degrees of vertices. [Copyright &y& Elsevier]

The eigenvalues of a graph are the eigenvalues of its adjacency matrix. This paper presents some upper and lower bounds on the greatest eigenvalue and a lower bound on the smallest eigenvalue. [Copyright &y& Elsevier]

It is proved that any set of finite pairwise nonisomorphic weakly cospectral connected pseudographs, as well as any set of pseudodigraphs satisfying various additional conditions, is finite. [ABSTRACT FROM AUTHOR]

It was proved recently by one of the authors that, if H is a path Pt (t>2 with t≠7 or 8) or an odd cycle Ct (t>3), then there is a unique maximal graph having H as a star complement for −2. The methods employed were analytical in nature, making use of the Reconstruction Theorem for star complements. Here we offer an alternative approach, based on the forbidden subgraph technique. In addition, we resolve the exceptional situations arising when H=P7 or P8. [Copyright &y& Elsevier]