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The generalized arithmetic–geometric matrix of a simple connected graph G = (V , E) is defined to be the | V | × | V | matrix whose i j -entry is d i α + d j α 2 d i α d j α if v i v j ∈ E , and 0 otherwise, where α is an arbitrary real number and d i is the degree of v i ∈ V. This matrix is a general form of the arithmetic–geometric matrix (α = 1) and the extended adjacency matrix (α = 2), both of which have been well studied in spectral graph theory and chemical graph theory. In this paper, we focus on the spectral radius ρ a g α of the generalized arithmetic–geometric matrix of connected graphs. Some chemical applications of ρ a g α are explored, and some extremal results on ρ a g α are obtained; in particular, the connected graphs (including trees) having the maximum and minimum ρ a g α are determined. [ABSTRACT FROM AUTHOR]

Abstract: Let be one of the -dimensional classical spaces and P be a fixed subspace of type ϑ of . Let be the set of all m-dimensional totally isotropic subspaces Q of satisfying . In this paper, we compute the size of when P is non-isotropic or totally isotropic. As applications, we give lower bounds of ranks of incidence matrices of totally isotropic subspaces of over the real number field , and compute eigenvalues of some graphs. [Copyright &y& Elsevier]

Abstract: In this paper, we analyse a special generalized inverse eigenvalue problem for the pair of matrices each of whose graph is a path on n vertices, by investigating the leading principal minors of the matrix . From the given data consisting of , a matrix ( ) whose graph is a path on k vertices, two column vectors and distinct real numbers λ and μ we construct and two column vectors such that is the leading principal sub-matrix of and , are the eigenpairs of , where and . Further, numerical examples are also given to demonstrate the applicability of the results developed here. [Copyright &y& Elsevier]

Abstract: Ray nonsingular (RNS) matrices are a generalization of sign nonsingular (SNS) matrices from the real field to complex field. The problem of how to recognize ray nonsingular matrices is still open. In this paper, we give an algorithm to recognize the ray nonsingularity for the so called “cycle chain matrices”, whose associated digraphs are of certain simple structure. Furthermore, the cycle chain matrices that are not ray nonsingular are classified into two classes according to whether or not they can be majorized by some RNS cycle chain matrices, and the case when they can is characterized. [Copyright &y& Elsevier]

Abstract: For a graph and a real number , the graph invariant is the sum of the th power of the non-zero Laplacian eigenvalues of . This note presents some bounds for in terms of the vertex degrees of , and a relation between and the first general Zagreb index, which is a useful topological index and has important applications in chemistry. [ABSTRACT FROM AUTHOR]

Abstract: For a given graph , denote by its adjacency matrix and . We say that there exist a perfect state transfer (PST) in if , for some vertices and a positive real number . Such a property is very important for the modeling of quantum spin networks with nearest-neighbor couplings. We consider the existence of the perfect state transfer in integral circulant graphs (circulant graphs with integer eigenvalues). Some results on this topic have already been obtained by Saxena et al. (2007) , Bašić et al. (2009)  and Basić & Petković (2009) . In this paper, we show that there exists an integral circulant graph with vertices having a perfect state transfer if and only if . Several classes of integral circulant graphs have been found that have a perfect state transfer for the values of divisible by . Moreover, we prove the nonexistence of a PST for several other classes of integral circulant graphs whose order is divisible by . These classes cover the class of graphs where the divisor set contains exactly two elements. The obtained results partially answer the main question of which integral circulant graphs have a perfect state transfer. [ABSTRACT FROM AUTHOR]

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]

Abstract: For every rational number x ∈(0,1), we construct a pair of graphs, one regular and one nonregular with adjacency matrices A 1 and A2, having the property that A 1 − xJ and A 2 − xJ have the same spectrum (J is the all-ones matrix). This solves a problem of Van Dam and the second author. For some values of x, we have generated the smallest examples (with respect to the number of vertices) by computer. [Copyright &y& Elsevier]