Proof of conjectures on adjacency eigenvalues of graphs

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