Upper bounds on the average eccentricity of [formula omitted]-free and [formula omitted]-free graphs

Abstract

Let G be a finite, connected graph. The eccentricity of a vertex v of G is the distance from v to a vertex farthest from v. The average eccentricity of G is the arithmetic mean of the eccentricities of the vertices of G. It was shown that the average eccentricity of a graph of order n and minimum degree δ cannot exceed avec(G)≤9n4(δ+1)+154, and that this bound is best possible apart from a small additive constant. In this paper we show that for triangle-free graphs this bound can be improved to avec(G)≤3⌈n2δ⌉+5. We also show that for graphs not containing a 4-cycle as a subgraph, the bound can be improved to avec(G)≤154⌈nδ2−2⌊δ2⌋+1⌉+112. We construct graphs to show that the bound for triangle-free graphs is sharp apart from a small additive constant. We also show that the bound for C4-free graphs is close to being best possible by constructing for every value of δ for which δ+1 is a prime power, an infinite sequence of C4-free graphs with average eccentricity at least 154nδ2+3δ+2−52.