On Wiener index and average eccentricity of graphs of girth at least 6 and [formula omitted]-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 n − Δ ∗ n 9 n + 3 Δ ∗ 4 (δ 2 − δ + 1) + 21 , where Δ ∗ = Δ δ + (δ − 1) Δ (δ − 2) + 3 2 . 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 9 2 ⌈ n 4 δ 2 − 10 δ + 14 ⌉ + 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. [ABSTRACT FROM AUTHOR]