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A mixed graph is a graph with edges and arcs, which can be considered as a hybrid of an undirected graph and a directed graph. In this paper we define the mixed adjacency matrix and the mixed energy of a mixed graph. The mixed adjacency matrix generalizes both the adjacency matrix of an undirected graph and the skew-adjacency matrix of a digraph. Then we compute the characteristic polynomial of the mixed adjacency matrix of a mixed graph and deduce some basic results from it. Furthermore, we give bounds to the mixed energy of a general mixed graph, and we compute the mixed energy of some special mixed graphs. At the end of the paper, we introduce mixed unitary Cayley graphs and compute their spectra. [ABSTRACT FROM AUTHOR]

An oriented hypergraph is an oriented incidence structure that generalizes and unifies graph and hypergraph theoretic results by examining its locally signed graphic substructure. In this paper we obtain a combinatorial characterization of the coefficients of the characteristic polynomials of oriented hypergraphic Laplacian and adjacency matrices via a signed hypergraphic generalization of basic figures of graphs. Additionally, we provide bounds on the determinant and permanent of the Laplacian matrix, characterize the oriented hypergraphs in which the upper bound is sharp, and demonstrate that the lower bound is never achieved. [ABSTRACT FROM AUTHOR]

Abstract: A threshold graph on n vertices is coded by a binary string of length . We obtain a formula for the inertia of (the adjacency matrix of) a threshold graph in terms of the code of the graph. It is shown that the number of negative eigenvalues of the adjacency matrix of a threshold graph is the number of ones in the code, whereas the nullity is given by the number of zeros in the code that are preceded by either a zero or a blank. A formula for the determinant of the adjacency matrix of a generalized threshold graph and the inverse, when it exists, of the adjacency matrix of a threshold graph are obtained. Results for antiregular graphs follow as special cases. [Copyright &y& Elsevier]

Abstract: Let be an unweighted rooted tree of k levels such that in each level the vertices have equal degree. Let d k−j+1 denotes the degree of the vertices in the level j. We find the eigenvalues of the adjacency matrix and of the Laplacian matrix of . They are the eigenvalues of principal submatrices of two nonnegative symmetric tridiagonal matrices of order k × k. The codiagonal entries for both matrices are , and , while the diagonal entries are zeros, in the case of the adjacency matrix, and d j , 1⩽ j ⩽k, in the case of the Laplacian matrix. Moreover, we give some results concerning to the multiplicity of the above mentioned eigenvalues. [Copyright &y& Elsevier]

Abstract: Let J be the all-ones matrix, and let A denote the adjacency matrix of a graph. An old result of Johnson and Newman states that if two graphs are cospectral with respect to yJ − A for two distinct values of y, then they are cospectral for all y. Here we will focus on graphs cospectral with respect to yJ − A for exactly one value of y. We call such graphs -cospectral. It follows that is a rational number, and we prove existence of a pair of -cospectral graphs for every rational . In addition, we generate by computer all -cospectral pairs on at most nine vertices. Recently, Chesnokov and the second author constructed pairs of -cospectral graphs for all rational , where one graph is regular and the other one is not. This phenomenon is only possible for the mentioned values of , and by computer we find all such pairs of -cospectral graphs on at most eleven vertices. [Copyright &y& Elsevier]

Let G be a graph. An edge orientation of G is called smooth if the in-degree and the out-degree of every vertex differ by at most one. In this paper, we show that if G is a 2-edge-connected non-bipartite graph with δ ( G ) ≥ 3 , then G has a smooth primitive orientation. Among other results, using the spectral radius of digraphs, we show that if D 1 is a primitive regular orientation and D 2 is a non-regular orientation of a given graph, then for sufficiently large t , the number of closed walks of length t in D 1 is more than the number of closed walks of length t in D 2 . [ABSTRACT FROM AUTHOR]

Abstract: Let G =(V, E) be a simple graph and {λ 1(G),…, λ n (G)} be its adjacency spectrum. It is easy to see that if an edge is added between two isolated vertices, then one zero eigenvalue increases by 1, and another zero eigenvalue decreases by 1. Let G + be a connected graph obtained from G by adding an edge e ∉ E(G). In this paper, it will be proved that the spectrum of G + is different from that of G only in two places with one eigenvalue increases by m and another eigenvalue decreases by m, where m >0 is a rational number, if and only if G is an empty graph with order 2. It will also be proved that one cannot construct a new adjacency integral connected graph with order n ⩾3 from a known one by adding an edge. [Copyright &y& Elsevier]

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]

Let J(λ;n1,…,nk) be the set of matrices A such that λ is an eigenvalue of A and n1⩽⋯⩽nk are the sizes of the Jordan blocks associated with λ. For a given index v of A, denote by A−v the principal submatrix of co-order one obtained from A by deleting the vth row and column. In the present paper, all possible changes of the part of the Jordan form corresponding to λ under the transition from A to A−v are determined for matrices A∈J(λ;n1,…,nk) such that for the eigenvalue λ of both A and A⊤, there exists a Jordan chain of the largest length nk whose eigenvector has nonzero vth entry. In particular, it is shown that for almost every matrix A∈J(λ;n1,…,nk), n1,…,nk−1 are the sizes of Jordan blocks for λ considered as an eigenvalue of A−v. Moreover, it is also proved that if A is the adjacency matrix of a vertex-transitive digraph and k⩾2, then the change n1,…,nk→n1,…,nk−2,2nk−1−1 holds for the eigenvalue λ under the transition from A to A−v. In the case of k=1, λ is a simple eigenvalue of A and does not belong to the spectrum of A−v. [Copyright &y& Elsevier]

The rank of a digraph is defined as the rank of its adjacency matrix. In this paper we show how to find all digraphs of rank k from the reduced bipartite graphs of rank 2 k. In particular, since the bipartite graphs of rank 2 , 4 and 6 are known, then we characterize digraphs of rank 1 , 2 and 3. The results are based on rank-preserving operations on digraphs such as splitting and fusion of vertices or twin-deletion and vertex-multiplication. [ABSTRACT FROM AUTHOR]

In an earlier work, the author together with Guo [7] introduced the Hermitian adjacency matrix of directed (and partially directed) graphs. However, it appears that a more natural Hermitian matrix exists, and it is the purpose of this note to bring this new Hermitian matrix to the attention of researchers in algebraic graph theory. [ABSTRACT FROM AUTHOR]

Let G be a simple graph with n vertices v 1 , ... , v n and m edges. Let A (G) = [ a i j ] n × n be the adjacency matrix of G , that is a i j = 1 if v i and v j are adjacent, and a i j = 0 , otherwise. Let λ 1 , ... , λ n be the eigenvalues of A (G). The energy of G , denoted by E (G) , is defined as | λ 1 | + ⋯ + | λ n |. It is well known that 2 m ≤ E (G) ≤ 2 m n. In this paper we improve the latter lower bound and prove that if all eigenvalues of G are non-zero, then E (G) ≥ 2 | λ 1 λ n | | λ 1 | + | λ n | 2 m n , where | λ 1 | ≥ ⋯ ≥ | λ n |. We show that the equality holds if and only if G = n 2 K 2 or G = p K β + 1 ⋃ q T β + 1 , where p and q are some non-negative integers and β ≥ 2 and T β + 1 is the graph K β + 1 , β + 1 ∖ M , where M is a perfect matching of K β + 1 , β + 1. Finally by studying the | λ 1 λ n | | λ 1 | + | λ n | of graphs, we find some lower bounds for the energy of graphs. In particular, we obtain that if G has no eigenvalue in the interval (− 1 , 1) , then E (G) ≥ 2 m 2 n − 2 n , and the equality holds if and only if G is the complete graph K n. [ABSTRACT FROM AUTHOR]

The Gershgorin Circle Theorem is a well-known and efficient method for bounding the eigenvalues of a matrix in terms of its entries. If A is a symmetric matrix, by writing A = B + x 1 , where 1 is the matrix with unit entries, we consider the problem of choosing x to give the optimal Gershgorin bound on the eigenvalues of B , which then leads to one-sided bounds on the eigenvalues of A. We show that this x can be found by an efficient linear program (whose solution can in may cases be written in closed form), and we show that for large classes of matrices, this shifting method beats all existing piecewise linear or quadratic bounds on the eigenvalues. We also apply this shifting paradigm to some nonlinear estimators and show that under certain symmetries this also gives rise to a tractable linear program. As an application, we give a novel bound on the lowest eigenvalue of a adjacency matrix in terms of the "top two" or "bottom two" degrees of the corresponding graph, and study the efficacy of this method in obtaining sharp eigenvalue estimates for certain classes of matrices. [ABSTRACT FROM AUTHOR]

To track the gradual change of the adjacency matrix of a simple graph G into the signless Laplacian matrix, V. Nikiforov in [35] suggested the study of the convex linear combination A α ( α-adjacency matrix ), A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) , for α ∈ [ 0 , 1 ] , where A ( G ) and D ( G ) are the adjacency and the diagonal vertex degrees matrices of G , respectively. Taking this definition as an idea the next matrix was considered for a , b ∈ R . The matrix A a , b defined by A a , b ( G ) = a D ( G ) + b A ( G ) , extends the previous α -adjacency matrix. This matrix is designated the ( a , b ) -adjacency matrix of G . Both adjacency matrices are examples of universal matrices already studied by W. Haemers. In this paper, we study the ( a , b ) -adjacency spectra for a family of compound graphs formed by disjoint balanced trees whose roots are identified to the vertices of a given graph. In consequence, new families of cospectral (adjacency, Laplacian and signless Laplacian) graphs, new hypoenergetic graphs (graphs whose energy is less than its vertex number) and new explicit formulae for Estrada, signless Laplacian Estrada and Laplacian Estrada indices of graphs were obtained. Moreover, sharp upper bounds of the above indices for caterpillars, in terms of length of the path and of the maximum number of its pendant vertices, are given. [ABSTRACT FROM AUTHOR]

Abstract The spectral radius of a graph is the largest eigenvalue of the adjacency matrix corresponding to this graph. Let G be a graph with long pendant paths. In this note, we estimate the spectral radius of G , by obtaining appropriate upper and lower bounds. As an application, the spectral radius of some k-ary trees are estimated. [ABSTRACT FROM AUTHOR]

Abstract It is proved that for any finite connected graph G , there exists an orientation of G such that the spectral radius of the corresponding Hermitian adjacency matrix is smaller or equal to the spectral radius of the universal cover of G (with equality if and only if G is a tree). This provides a suitable answer to a problem proposed by Mohar. The proof uses the method of interlacing families of polynomials that was developed by Marcus, Spielman, and Srivastava in their seminal work on the existence of infinite families of Ramanujan graphs. [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 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]

Abstract: For a simple graph G, let rank(G) and dnzr(G) denote respectively the rank and the number of distinct nonzero rows of the adjacency matrix A(G) of G. Equivalent conditions are given for the join of two vertex-disjoint graphs to satisfy . A new proof is provided for the known relation rank(G)=dnzr(G) for cographs G. Our approach relies on the concepts of neighborhood equivalence classes, reduced graph and reduced adjacency matrix, and also on a known result that relates the spectrum of the adjacency matrix of a graph with that of its reduced adjacency matrix as well as a new characterization of the nonsingularity of a real symmetric matrix in a special 2×2 block form. Our treatment provides ways to construct graphs G, other than cographs, that satisfy rank(G)=dnzr(G). As a side result we also show that every rational number is equal to the sum of the entries of the inverse of the adjacency matrix of a connected nonsingular graph. [Copyright &y& Elsevier]

Abstract: Consider a graph on vertices with adjacency matrix and degree sequence . A universal adjacency matrix of is any matrix in Span with a nonzero coefficient for , where and and are the identity and all-ones matrix, respectively. Thus a universal adjacency matrix is a common generalization of the adjacency, the Laplacian, the signless Laplacian and the Seidel matrix. We investigate graphs for which some universal adjacency matrix has just two eigenvalues. The regular ones are strongly regular, complete or empty, but several other interesting classes occur. [Copyright &y& Elsevier]

Abstract: A weighted directed graph is a directed graph G whose underlying undirected graph is simple and whose edges have nonzero (directional) complex weights, that is, the presence of an edge of weight is as good as the presence of the edge with weight , the complex conjugate of . Let G be a weighted directed graph on vertices . Denote by the weight of an edge . The adjacency matrix of G is an matrix with entries or or 0, depending on whether or or otherwise, respectively. We supply a characterization of those unicyclic weighted directed graphs G whose edges have weights from the set and whose adjacency matrix satisfies the following property: ‘λ is an eigenvalue of with multiplicity k if and only if is an eigenvalue of with the same multiplicity’. [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]

A spectral version of extremal graph theory problem is as follows: what is the maximum spectral radius of an H -free graph of order n ? Nikiforov posed a conjecture on spectral extremal value of cycles. In the past decade, much attention has been paid to this conjecture and other questions on spectral extremal graphs. In order to approach to Nikiforov's conjecture, we provide a new conjecture on spectral extremal value involving fans and wheels. Moreover, we show it is true for k = 2. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to highlight the role played by the anti-regular graph within the class of threshold graphs. Using the fact that every threshold graph contains a maximal anti-regular graph, we show that some known results, and new ones, on the spectral properties of threshold graphs can be deduced from (i) the known results on the eigenvalues of anti-regular graphs, (ii) the subgraph structure of threshold graphs, and (iii) eigenvalue interlacing. In particular, we prove a strengthened version of the recently proved fact that no threshold graph contains an eigenvalue in the interval Ω = [ − 1 − 2 2 , − 1 + 2 2 ] , except possibly the trivial eigenvalues −1 and/or 0, determine the inertia of a threshold graph, and give partial results on a conjecture regarding the optimality of the non-trivial eigenvalues of an anti-regular graph within the class of threshold graphs. [ABSTRACT FROM AUTHOR]

Given a graph G , the mixed graph D G is obtained from G by orienting some of its edges, where G is called the underlying graph of D G. Let p (D G) , n (D G) (resp. p (G) , n (G)) be the positive inertia index and negative inertia index of D G (resp. G). In this paper, we first establish the inequalities − d (G) ⩽ p (D G) − p (G) ⩽ d (G) and − d (G) ⩽ n (D G) − n (G) ⩽ d (G) , where d (G) is the dimension of cycle space of G. Furthermore, all the corresponding extremal graphs are characterized. [ABSTRACT FROM AUTHOR]

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]

The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. P. Hansen and D. Stevanović (2008) [9] determined the graphs with maximum spectral radius among all connected graphs of order n with diameter D . In this paper, we generalize this result to k -connected graphs of order n with diameter D . [ABSTRACT FROM AUTHOR]

Let G be a graph on n vertices and λ 1 , λ 2 , ... , λ n their eigenvalues. The Estrada index of G is defined as E E (G) = ∑ i = 1 n e λ i . In this paper, we obtain new upper and lower bounds on the Estrada Index of a given graph, considering numerical inequalities present in the literature. [ABSTRACT FROM AUTHOR]

Let G be a graph on n vertices and λ 1 ≥ λ 2 ≥ ... ≥ λ n its eigenvalues. The Estrada index of G is defined as E E (G) = ∑ i = 1 n e λ i . In this work, we use an increasing sequence converging to the λ 1 to obtain an increasing sequence of lower bounds for E E (G). In addition, we generalize this succession for the Estrada index of an arbitrary symmetric nonnegative matrix. [ABSTRACT FROM AUTHOR]

An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of either +1 or −1. This generalizes signed graphs to a hypergraph setting and simultaneously provides a natural definition for a signed adjacency which is used to define the adjacency and Laplacian matrices. Many properties of these matrices are known, but there are no nontrivial families of oriented hypergraphs with their spectrum determined. In this paper we define and study hypergraph families that are analogous to cycles and paths in graphs and signed graphs. The adjacency and Laplacian eigenvalues for some of these families is determined. [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: In this paper we introduce a new parameter for a graph called the minimum universal rank. This parameter is similar to the minimum rank of a graph. For a graph G the minimum universal rank of G is the minimum rank over all matrices of the form where A is the adjacency matrix of G, J is the all ones matrix and D is the matrix with the degrees of the vertices in the main diagonal, and are scalars. Bounds for general graphs based on known graph parameters are given, as is a formula for the minimum universal rank for regular graphs based on the multiplicity of the eigenvalues of A. The exact value of the minimum universal rank of some families of graphs are determined, including complete graphs, complete bipartite graph, paths and cycles. Bounds on the minimum universal rank of a graph obtained by deleting a single vertex are established. It is shown that the minimum universal rank is not monotone on induced subgraphs, but bounds based on certain induced subgraphs, including bounds on the union of two graphs, are given. [Copyright &y& Elsevier]

Abstract: Some graphs Γ have the following property : the configuration graph (i.e. the non–collinearity graph) of the neighbourhood geometry of Γis isomorphic to Γ. For instance, the ubiquitous Petersen graph satisfies . The purpose of this paper is to reveal repercussions of property on adjacency matrices A for Γ. This will be achieved in terms of invariance under (powers of) the following mapping Θ: denote by I and J the identity matrix and the all one matrix, respectively, both of order n = k 2+1, and define Θ(A) := (k−1)I + J−A 2. If k =3 and A is an adjacency matrix for the Petersen graph, it is well known that Θ(A)= A. In 1960, Hoffman and Singleton showed that for arbitrary k the matrix equation Θ(A)= A has only very few solutions, namely for k =2,3,7 and possibly for k =57. We prove that the property implies the existence of an integer m ⩾1 such that Θ m (A)= A. We determine a standard form for such matrices which is invariant under the action of Θ. In particular, we characterize all solutions, on A, to Θ m (A)= A for k =3 and k =4, where A is an adjacency matrix of some k–regular graph, solving a conjecture posed by the authors. [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 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: Let be a simple graph with vertex set and edge set . The adjacency matrix of a graph is denoted by and defined as the matrix , where for and 0 otherwise. The largest eigenvalue () of is called the spectral radius or the index of . The Laplacian matrix of is , where is the diagonal matrix of its vertex degrees and is the adjacency matrix. Among all eigenvalues of the Laplacian matrix of a graph, the most studied is the second smallest, called the algebraic connectivity () of a graph [12]. In [1,2], Aouchiche et al. have given a series of conjectures on index () and algebraic connectivity () of (see also [3]). Here we prove two conjectures and disprove one by a counter example. [Copyright &y& Elsevier]

Abstract: We present a class of graphs whose adjacency matrices are nonsingular with integral inverses, denoted h-graphs. For the h-graph with adjacency matrix , we consider the problem of identifying exactly when is signature-similar to the adjacency matrix of another h-graph, . When this holds, is a type of graph-inverse of , which is known as the dual of . We present necessary and sufficient conditions for the existence of . As an application, we provide a characterization of all dual pairs and where both graphs are unicyclic. This characterization allows us to identify those h-graphs that are unicyclic and self-dual. [Copyright &y& Elsevier]

Abstract: A Bethe tree is a rooted unweighted of k levels in which the root vertex has degree equal to d, the vertices at level j have degree equal to and the vertices at level k are the pendant vertices. In this paper, we first derive an explicit formula for the eigenvalues of the adjacency matrix of . Moreover, we give the corresponding multiplicities. Next, we derive an explicit formula for the simple nonzero eigenvalues, among them the largest eigenvalue, of the Laplacian matrix of . Finally, we obtain upper bounds on the largest eigenvalue of the adjacency matrix and of the Laplacian matrix of any tree . These upper bounds are given in terms of the largest vertex degree and the radius of , and they are attained if and only if is a Bethe tree. [Copyright &y& Elsevier]

Abstract: Let be an unweighted tree of k levels such that in each level the vertices have equal degree. Let n k−j+1 and d k−j+1 be the number of vertices and the degree of them in the level j. We find the eigenvalues of the adjacency matrix and Laplacian matrix of for the case of two vertices in level 1 (n k =2), including results concerning to their multiplicity. They are the eigenvalues of leading principal submatrices of nonnegative symmetric tridiagonal matrices of order k × k. The codiagonal entries for these matrices are , 2⩽ j ⩽ k, while the diagonal entries are 0,…,0,±1, in the case of the adjacency matrix, and d 1, d 2,…, d k−1, d k ±1, in the case of the Laplacian matrix. Finally, we use these results to find improved upper bounds for the largest eigenvalue of the adjacency matrix and of the Laplacian matrix of any given tree. [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]

Abstract In this study we derive certain upper bounds for the largest eigenvalue (called the index and denoted λ 1) of a signed graph. In particular, we prove the following upper bound: λ 1 2 ≤ max ⁡ { d i m i − n i : 1 ≤ i ≤ n } , where d i is the vertex degree of i , m i = 1 d i ∑ j ∼ i d j and n i = ∑ j ∼ i (| N i σ (i j) ∩ N j | − | | N i σ (i j) ∩ N j σ (i j) | − | N i σ (i j) ∩ N j − σ (i j) | |) , with N i , N i + and N i − denoting the neighbourhood, the positive neighbourhood and the negative neighbourhood of a vertex i. In our proofs we use standard techniques transferred from the field of (unsigned, simple) graphs. Using the fact that all switching equivalent signed graphs share the same spectrum, we derive some more sophisticated bounds. [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 Let μ be an eigenvalue of a simple graph G with multiplicity k ≥ 1. A star complement for μ in G is an induced subgraph of G of order n − k with no eigenvalue μ. In this paper, we study the maximal graphs with the star S m as a star complement for −2. The maximal graphs with S 3 , S 4 , S 13 and S 21 as a star complement for −2 are described. We also describe the regular graphs with K 2 , s (s ≥ 2) as a star complement for an eigenvalue μ. [ABSTRACT FROM AUTHOR]

In this paper we develop the concept of energy of a vertex introduced in Arizmendi and Juarez-Romero (2018). We derive basic inequalities, continuity properties, give examples and extend the definition to locally finite graphs. [ABSTRACT FROM AUTHOR]

Abstract: We give sharp upper and lower bounds for the spectral radius of a nonnegative matrix with all row sums positive using its average 2-row sums, and characterize the equality cases if the matrix is irreducible. We compare these bounds with the known bounds using the row sums by examples. We also apply these bounds to various matrices associated with a graph, including the adjacency matrix, the signless Laplacian matrix and some distance-based matrices. Some known results are generalized and improved. [Copyright &y& Elsevier]

Abstract: Let be a weighted graph. The inertia of is the triple , where , , are the numbers of the positive, negative and zero eigenvalues of the adjacency matrix of including their multiplicities, respectively. , are called the positive, negative indices of inertia of , respectively. In this paper we present a lower bound for the positive, negative indices of weighted unicyclic graphs of order n with fixed girth and characterize all weighted unicyclic graphs attaining this lower bound. Moreover, we characterize the weighted unicyclic graphs of order n with two positive, two negative and at least zero eigenvalues, respectively. [Copyright &y& Elsevier]

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: We give upper and lower bounds for the spectral radius of a nonnegative matrix using its row sums and characterize the equality cases if the matrix is irreducible. Then we apply these bounds to various matrices associated with a graph, including the adjacency matrix, the signless Laplacian matrix, the distance matrix, the distance signless Laplacian matrix, and the reciprocal distance matrix. Some known results in the literature are generalized and improved. [Copyright &y& Elsevier]

Abstract: Let G be a threshold graph of order n with adjacency matrix A. We present an algorithm for constructing a diagonal matrix congruent to for any x. An application, using Sylvesterʼs Law of Inertia, can determine how many eigenvalues lie in an interval, allowing efficient approximation. We prove that eigenvalues of threshold graphs, other than 0 or −1, are simple. We give the spectrum for threshold graphs , represented by . When , has the minimum eigenvalue among threshold graphs of order n, and a formula for is given. There is one more graph if . [Copyright &y& Elsevier]