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 delta and maximum degree increment 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- increment * 4(delta 2-delta+1) + 21, where increment * = increment delta + (delta - 1)root increment (delta - 2) + 3 9n+3 increment * 2 . We construct n graphs that show that for delta - 1 a prime power this bound is sharp apart from a term O(root increment ). 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 92 left ceiling n 4 delta 2-10 delta+14 right ceiling +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.(c) 2023 Elsevier B.V. All rights reserved.