On bounds of eigenvalues of Randić vertex-degree-based adjacency matrix

Let G = (V,E), V = {1,2, …, n}, be a simple graph of order n and size m, without isolated vertices. Denote by d1 ≥ d2 … ≥ dn > 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) = Σi~j F(di,dj), where F is an appropriately chosen function with the property F(x,y)=F(y,x). Randic vertex-degree-based adjacency matrix RA=(ri j) is defined as rij = Fp(di,dj) √didj, if i ~ j, and 0 otherwise. Denote by f1 ≥ f2 ≥ … ≥ fn the eigenvalues of RA. Upper and lower bounds for fi, i = 1,2, … n are obtained.