Average distance, minimum degree, and irregularity index

Let G = (V, E) be a connected graph of order n. The distance, dG(x, y), between vertices x and y in G is defined as the length of a shortest x -y path in G. The average distance, mu(G), of G is defined as mu(G) = (n )-1 E{x,y}subset of V dG(x, y). We give an upper bound on the average 2 distance of a connected graph of given order and minimum degree where irregularity index is prescribed. Our results are a strengthening of the classical theorems by Kouider and Winkler (1997) [9] and by Dankelmann and Entringer (2000) [5] on average distance and minimum degree if the number of distinct terms in the degree sequence of the graph is prescribed.(c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons .org /licenses /by-nc -nd /4 .0/).