A simple proof of the characterization of antipodal graphs

✍ Scribed by Garry Johns

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 88 KB
• Volume: 128
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(94)90131-7

✦ Synopsis

The antipodal graph A(G) of a graph G is defined as the graph on the same vertex set as G with two vertices being adjacent in A(G) if the distance between them in G is the diameter of G. (If G is disconnected then we define &am(G) = co.)

Aravamudhan and Rajendran [l, 21 gave the following characterization of antipo-da1 graphs. Theorem 1. A graph G is an antipodal graph ifand only ifit is the antipodal graph of its complement CT.

The authors verified this indirectly by describing those graphs which are not antipodal graphs. In addition, they proved the following lemma which we will also use. The proof is straightforward and thus omitted.

Lemma 2. A(G) = G if and only if diam(G) = 2 or if G is disconnected and the components of G are complete graphs.

In [2] these same authors observed that if H is a connected graph with diam(H) > 2, then A(H) = A(W), where H' is the graph on the same vertex set such that two vertices