Randić degree-based energy of graphs

Let G = (V;E), V = f1;2;:::;ng, be a simple graph of order n and size m, without isolated vertices. Denote by ∆ = d1 ≥ d2 ≥ ··· ≥ dn = d > 0, di = d(i), a sequence of its vertex degrees. If vertices i and j are adjacent, we write i ∼ j. With TI we denote a topological index that can be represented as TI = TI(G) = ∑i∼ j F(di;dj), where F is an appropriately chosen function with the property F(x;y) = F(y;x). Randic degree-based adjacency matrix ' RA = (ri j) is defined as ri j = Fp(ddi;iddjj) if i ∼ j, and 0 otherwise. Denote by fi, i = 1;2;:::;n, the eigenvalues of RA. The Randic degree-based energy of graph could be defined as ' RETI = RETI(G) = ∑n i=1 j fij. Upper and lower bounds for RETI are obtained. .