Albertson energy and Albertson-Estrada index of graphs

Let G be a graph of order n with vertices labeled as v1; v2; : : : ; vn. Let di be the degree of the vertex vi for i = 1; 2; ; n. The Albertson matrix of G is the square matrix of order n whose (i; j)-entry is equal to jdi dj j if vi is adjacent to vj and zero, otherwise. The main purposes of this paper is to introduce the Albertson energy and Albertson-Estrada index of a graph, both base on the eigenvalues of the Albertson matrix. Moreover, we establish upper and lower bounds for these new graph invariants and relations between them.