General eccentric distance sum of graphs with given diameter

Abstract

For [Formula: see text], the general eccentric distance sum of a connected graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the vertex set of [Formula: see text], [Formula: see text] is the eccentricity of [Formula: see text], [Formula: see text] and [Formula: see text] is the distance between vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. For [Formula: see text] and [Formula: see text], we present the graphs having the smallest general eccentric distance sum among graphs with given order and diameter, and among bipartite graphs with given order and odd diameter. The extremal graphs for the classical eccentric distance sum are corollaries of our results on the general eccentric distance sum.