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

TL;DR

This paper addresses an enumeration problem for labelled self-complementary graphs, providing formulas for counting graphs with specified automorphism group orders, advancing graph enumeration theory.

Contribution

It introduces a new counting method that resolves a longstanding problem and derives formulas for enumerating graphs and self-complementary graphs with given automorphism group orders.

Findings

Formulas for counting labelled graphs with specified automorphism groups

Formulas for counting unlabelled graphs with specified automorphism groups

Formulas for counting labelled and unlabelled 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.