Using minimum degree to bound average distance

Total and average distance are not only interesting invariants of graphs in their own right but are also used for studying properties or classifying graphical systems that depend on the number of edges traversed. Recent examples include studies of computer networks [3] and the use of graphical invariants to partially classify the structure of molecules [1]. There have been a number of conjectures involving average distance in the literature including a conjecture of the computer program G. Galatea Graffiti [8], proven by Chung [7], that the independence number of a connected graph is at least as large as the average distance. Kouider and Winkler [13] provide an extensive list of references for other problems and results involving average distance: see [4, 5, 6, 9, 10, 12, 14, 15, 17, 18, 19]. The main result of this paper was motivated by Conjecture 62 of Graffiti [8]: If G is a regular graph of degree r and order n, then the average distance of G is at most n r . If we denote the minimum degree of a, possibly non-regular, graph by d, then Conjecture 127 of