Proof of conjectures on adjacency eigenvalues of graphs

Let G be a simple graph of order n with t triangle(s). Also let λ 1 ( G ) , λ 2 ( G ) , … , λ n ( 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 F t is determined by its adjacency spectrum. Here we prove this conjecture. centricity 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 ( λ 1 ( 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.