Proof of conjectures on adjacency eigenvalues of graphs

AbstractLet G be a simple graph of order n with t triangle(s). Also let λ1(G),λ2(G),…,λn(G) be the eigenvalues of the adjacency matrix of graph G. X. Yong [X. Yong, On the distribution of eigenvalues of a simple undirected graph, Linear Algebra Appl. 295 (1999) 73–80] conjectured that (i) G is complete if and only if det(A(G))=(−1)n−1(n−1) and also (ii) G is complete if and only if |det(A(G))|=n−1. 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 Ft 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 ecc(G) of a graph G is the mean value of eccentricities of all vertices of G. Moreover, we mention three conjectures, obtained by the system AutoGraphiX, about the average eccentricity (ecc(G)), girth (g(G)) and the spectral radius (λ1(G)) 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