On the Aα-characteristic polynomial of a graph

Let G be a graph with n vertices, and let A(G) and D(G) denote respectively the adjacency matrix and the degree matrix of G. Define A alpha(G) = alpha D(G)H- (1-alpha)A(G) for any real a E [0, 11] The A(alpha)-characteristic polynomial of G is defined to be det(xI(n) A(alpha)(G)) = Sigma(j) C alpha j (G)x(n-j) where det(*) denotes the determinant of *, and I,, is the identity matrix of size n. The An -spectrum of G consists of all roots of the Aa-characteristic polynomial of G. A graph G is said to be determined by its A(alpha) -spectrum if all graphs having the same A(alpha)-spectrum as G are isomorphic to G. In this paper, we first formulate the first four coefficients c(alpha 0)(G), c(alpha 1)(G), c(alpha 2)(G) and c(alpha 3)(G) of the A(alpha)-characteristic polynomial of G. And then, we observe that Aa-spectra are much efficient for us to distinguish graphs, by enumerating the A(alpha)-characteristic polynomials for all graphs on at most 10 vertices. To verify this observation, we characterize some graphs determined by their Aa-spectra. (C) 2018 Elsevier Inc. All rights reserved.