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 |di‎−‎dj| 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.