On the Gutman index and minimum degree

The Gutman index Gut(G) of a graph G is defined as ∑ {x,y}⊆V(G) deg(x)deg(y)d(x,y), where V(G) is the vertex set of G, deg(x),deg(y) are the degrees of vertices x and y in G, and d(x,y) is the distance between vertices x and y in G. We show that for finite connected graphs of order n and minimum degree δ, where δ is a constant, Gut(G) ≤ 24·3/55(δ+1)n5+ O(n4). Our bound is asymptotically sharp for every δ≥2 and it extends results of Dankelmann, Gutman, Mukwembi and Swart (2009) and Mukwembi (2012), whose bound is sharp only for graphs of minimum degree 2. © 2014 Elsevier B.V. All rights reserved.