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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: We consider simple graphs and their adjacency matrices. In , Rara (1996) gives methods of reducing graphs which simplify the procedure of computing the determinant of their adjacency matrices. We continue this subject matter and give a general method of reducing graphs. By the use of this method we define a formula for computing the determinant of any planar grid and in particular settle the problem of their singularity. [Copyright &y& Elsevier]

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]

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]

The early research on regular graphs of the French mathematician Georges Brunel (1856–1900) is discussed. Brunel developed early graph terminology and started the French “applied” approach towards graph theory. [Copyright &y& Elsevier]