Upper bounds on the average eccentricity of K₃-free and C₄-free graphs

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 [Formula presented], 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 [Formula presented]. We also show that for graphs not containing a 4-cycle as a subgraph, the bound can be improved to [Formula presented]. 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 [Formula presented].