A new counting methods, including the issue of counting labelled self-complementary graphs

Abstract

Harary and Palmer announced an enumeration problem of labelled self-complementary graphs at the end of their book (Graphical Enumeration, Academic Press, New York and London, 1973). This paper resolves this problem. A method for solving this problem leads to the derivation of following formulas: (a) A formula on the number of labelled graphs with the given order of automorphism groups of those graphs. (b) A formula on the number of unlabelled graphs with the given order of automorphism groups of those graphs. (c) A formula on the number of labelled self-complementary graphs with the given order of automorphism groups of those graphs. (d) A formula on the number of unlabelled self-complementary graphs with the given order of automorphism groups of those graphs.