On the Automorphism Group of a Graph

TL;DR

This paper explores the relationship between a graph's automorphism group and its representations, introducing an algorithm to compute all block systems and generators efficiently.

Contribution

It establishes connections between block systems and irreducible representations of automorphism groups, and presents a polynomial-time algorithm for their computation.

Findings

Algorithm outputs all block systems within time $n^{C \, \log n}$

Provides a method to generate the automorphism group of a graph

Links group representations with graph symmetry structures

Abstract

An automorphism of a graph $G$ with $n$ vertices is a bijective map $\phi$ from $V(G)$ to itself such that $\phi(v_i)\phi(v_j)\in E(G)$ $\Leftrightarrow$ $v_i v_j\in E(G)$ for any two vertices $v_i$ and $v_j$ of $G$. Denote by $\mathfrak{G}$ the group consisting of all automorphisms of $G$. As well-known, the structure of the action of $\mathfrak{G}$ on $V(G)$ is represented definitely by its block systems. On the other hand for each permutation $\sigma$ on $[n]$, there is a natural action on any vector $\pmb{v}=(v_1,v_2,\ldots,v_n)^t\in \mathbb{R}^n$ such that $\sigma\pmb{v}=(v_{\sigma^{-1}1},v_{\sigma^{-1}2},\ldots,v_{\sigma^{-1} n})^t$. Accordingly, we actually have a permutation representation of $\mathfrak{G}$ in $\mathbb{R}^n$. In this paper, we establish the some connections between block systems of $\mathfrak{G}$ and its irreducible representations, and by virtue of that we finally devise an algorithm outputting a generating set and all block systems of $\mathfrak{G}$ within time $n^{C \log n}$ for some constant $C$.