On Wiener index and average eccentricity of graphs of girth at least 6 and (C₄,C₅)-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 avec(G) of G is the arithmetic mean of the eccentricities of the vertices of G. The Wiener index W(G) of G is the sum of the distances between all unordered pairs of vertices of G. For these two distance measures we give upper bounds in terms of order n, minimum degree δ and maximum degree Δ for graphs of girth at least six, and for graphs containing neither 4-cycles nor 5-cycles as subgraphs. For graphs of girth at least six we show that the average eccentricity is bounded above by [Formula presented], where Δ^∗=Δδ+(δ−1)Δ(δ−2)+[Formula presented]. We construct graphs that show that for δ−1 a prime power this bound is sharp apart from a term O(Δ). We further show that if the girth condition on G is relaxed to G having neither a 4-cycle nor a 5-cycle as a subgraph, then similar and only slightly weaker bounds hold. For such graphs we also show that the average eccentricity is bounded from above by [Formula presented]⌈[Formula presented]⌉+8, which in some sense is close to being optimal. We obtain similar upper bounds for the Wiener index of graphs of girth at least 6 and for graphs containing neither a 4-cycle nor a 5-cycle as a subgraph.