On the Gutman index and minimum degree

Abstract 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 ) ≤ 2 4 ⋅ 3 5 5 ( δ + 1 ) n 5 + O ( n 4 ) . 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 .