On a conjecture by tomescu

Abstract

Let G be a connected graph of order n with vertex set V. The degree distance of G, D'(G), is defined as σ ⊆v(deg(u) + deg(v))d(u,v), where deg(w) is the degree of a vertex w and d(u,v) is the distance between u and v. We give an asymptotically sharp upper bound on the degree distance in terms of order and diameter. Our result, apart from refining a bound given by Dankelmann, Gutman, Mukwembi and Swart [2], corrects an error in the proof of their theorem. As a corollary, we prove that D'(G) ≤ 1/27 n4 + O(n3), thus completely settling, in the affirmative, a conjecture proposed by Tomescu [13] in 1999.