International Journal of Mathematics and Mathematical Sciences
Volume 16 (1993), Issue 3, Pages 579-586

Antipodal graphs and digraphs

Garry Johns and Karen Sleno

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.


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.