Randić degree-based energy of graphs

Abstract

Let G = (V, E), V = {1, 2,. .. , n}, be a simple graph of order n and size m, without isolated vertices. Denote by ∆ = d 1 ≥ d 2 ≥ • • • ≥ d n = δ > 0, d i = d(i), a sequence of its vertex degrees. If vertices i and j are adjacent, we write i ∼ j. With T I we denote a topological index that can be represented as T I = T I(G) = ∑ i∼ j F(d i , d j), where F is an appropriately chosen function with the property F(x, y) = F(y, x). Randić degree-based adjacency matrix RA = (r i j) is defined as r i j = F(d i ,d j) √ d i d j if i ∼ j, and 0 otherwise. Denote by f i , i = 1, 2,. .. , n, the eigenvalues of RA. The Randić degree-based energy of graph could be defined as RE T I = RE T I (G) = ∑ n i=1 | f i |. Upper and lower bounds for RE T I are obtained.