International Journal of Mathematics and Mathematical Sciences
Volume 16 (1993), Issue 3, Pages 579-586
doi:10.1155/S0161171293000717
Antipodal graphs and digraphs
Department of Mathematical Sciences, Saginaw Valley State University, University Center 48710, Michigan, USA
Received 7 May 1991; Revised 5 November 1991
Copyright © 1993 Garry Johns and Karen Sleno. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The antipodal graph of a graph G, denoted by A(G), has the same vertex set
as G with an edge joining vertices u and v if d(u,v) is equal to the diameter of G. (If G is
disconnected, then diam G=∞.) This definition is extended to a digraph D where the arc
(u,v) is included in A(D) if d(u,v) is the diameter of D. It is shown that a digraph D is an
antipodal digraph if and only if D is the antipodal digraph of its complement. This generalizes
a known characterization for antipodal graphs and provides an improved proof. Examples
and properties of antipodal digraphs are given. A digraph D is self-antipodal if A(D) is
isomorphic to D. Several characteristics of a self-antipodal digraph D are given including
sharp upper and lower bounds on the size of D. Similar results are given for self-antipodal
graphs.