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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]

Abstract: Let be the smallest number such that the adjacency matrix of any undirected graph with vertices or more has at least nonpositive eigenvalues. We show that is well-defined and prove that the values of for are 1, 3, 6, 10, 16 respectively. In addition, we prove that for all , , in which is the Ramsey number for and , and is the triangular number. This implies new lower bounds for eigenvalues of Laplacian matrices: the largest eigenvalue is bounded from below the largest degree, which generalizes some prior results. [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 problem of identifying those simple, undirected graphs with vertices and edges that have the smallest minimum eigenvalue of the adjacency matrix is considered. Several general properties of the minimizing graphs are described. These strongly suggest bipartition, to the extent possible for the number of edges. In the bipartite case, the precise structure of the minimizing graphs is given for a number of infinite classes. [Copyright &y& Elsevier]

Abstract: In this paper we consider threshold graphs (also called nested split graphs) and investigate some invariants of these graphs which can be of interest in bounding the largest eigenvalue of some graph spectra. [Copyright &y& Elsevier]

Abstract: We consider weighted graphs, where the edge weights are positive definite matrices. The eigenvalues of a graph are the eigenvalues of its adjacency matrix. We obtain an upper bound on the spectral radius of the adjacency matrix and characterize graphs for which the bound is attained. [Copyright &y& Elsevier]

Abstract: In this note we study some properties of the spectrum of a connected graph G with four different eigenvalues, and (spectrally maximum) diameter three. When G is regular, this is the case, for instance, when G is a distance-regular graph. [Copyright &y& Elsevier]

In this paper, all simple connected signed graphs with maximum degree at most 4 and with just two distinct adjacency eigenvalues are completely characterized, there exists an infinite family of 4-regular signed graphs with just two distinct adjacency eigenvalues. [ABSTRACT FROM AUTHOR]