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Using Minimum Degree to Bound Average Distance (1998)
| Citations: | 9 - 0 self |
BibTeX
@MISC{Beezer98usingminimum,
author = {R.A. Beezer and J. E. Riegsecker and B.A. Smith},
title = {Using Minimum Degree to Bound Average Distance},
year = {1998}
}
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Abstract
We show the average distance of a connected graph with n vertices, e edges and minimum degree d satisfies # j (n+1)n(n-1)-2e d+1 k n (n - 1) # n + 1 d + 1 - 2e (d + 1) n (n - 1) . This improves conjectures of the computer program Gra#ti [8], and He and Li [11] and the results of Kouider and Winkler [13]. The bound is sharp since complete graphs yield equality. 1 Introduction Let G = (V, E) be a connected, simple, undirected graph on n vertices. For each unordered pair of vertices {u, v} in G, select a path of minimal length. Designate the corresponding directed paths from u to v and from v to u as, P (u, v) and P (v, u) , respectively. The number of edges in P (u, v) is the distance # (u, v) from u to v and the average distance of G is = 1 n (n - 1) X u,v #V # (u, v) . 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 edg...
