The maximal geometric-arithmetic energy of trees with at most two branched vertices

Abstract

Let G be a graph of order n with vertex set V(G)={v1,v2,…,vn} and edge set E(G), and d(vi) be the degree of the vertex vi. The geometric-arithmetic matrix of G, recently introduced by Rodríguez and Sigarreta, is the square matrix of order n whose (i, j)-entry is equal to 2d(vi)d(vj)d(vi)+d(vj) if vivj ∈ E(G), and 0 otherwise. The geometric-arithmetic energy of G is the sum of the absolute values of the eigenvalues of geometric-arithmetic matrix of G. In this paper, we characterize the tree of order n which has the maximal geometric-arithmetic energy among all trees of order n with at most two branched vertices.