On the Gutman index and minimum degree

The Gutman index Gut(G) of a graph GG is defined as ∑{x,y}⊆V(G)deg(x)deg(y)d(x,y), where V(G)V(G) is the vertex set of GG, deg(x),deg(y) are the degrees of vertices xx and yy in GG, and d(x,y)d(x,y) is the distance between vertices xx and yy in GG. We show that for finite connected graphs of order nn and minimum degree δδ, where δδ is a constant, Gut(G)≤24⋅355(δ+1)n5+O(n4). Our bound is asymptotically sharp for every δ≥2δ≥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 22.