Relation between degree distance and gutman index of graphs

Let G = (V; E) be a simple connected graph with n vertices and m edges. The degree distance of a graph G is D′(G) = ∑ {v_i; v_j}⊆V (G) (dG(v_i) + dG(v_j)) d_G(v_i; v_j): where d_G(v_i; v_j) is the shortest distance between vertices v_i and v_j, and d_G(v_i) is the degree of the vertex v_i in G. The Gutman index (also known as Schultz index of the second kind) of a graph G is defined as Gut(G) = ∑ {v_i; v}⊆V (G) d_G(v_i) d_G(v_j) d_G(v_i; v_j): We obtain some lower and upper bounds on D′ (G) and Gut(G) of a graph G in terms of n, m,Δ and δ and characterize the extremal graphs. Moreover, we present some relations between D′ (G) and Gut(G) of graph G.