Average eccentricity, minimum degree and maximum degree in graphs

Let G be a connected finite graph with vertex set V(G). The eccentricity e(v) of a vertex v is the distance from v to a vertex farthest from v. The average eccentricity of G is defined as 1|V(G)|∑v∈V(G)e(v). We show that the average eccentricity of a connected graph of order n, minimum degree δ and maximum degree Δ does not exceed 94n-Δ-1δ+1(1+Δ-δ3n)+7, and this bound is sharp apart from an additive constant. We give improved bounds for triangle-free graphs and for graphs not containing 4-cycles.