Graph Automorphisms from the Geometric Viewpoint

TL;DR

This paper explores graph automorphisms through a geometric lens, linking automorphism groups to eigenspaces of the adjacency matrix and characterizing extremal vectors to understand symmetry properties.

Contribution

It provides a novel geometric characterization of graph automorphisms using eigenspaces and linear representations, including extremal vectors and their span dimensions.

Findings

Automorphisms correspond to invariant eigenspaces of the adjacency matrix.

Characterization of extremal vectors with maximal span under automorphism group action.

Exact dimensions of span for eigenvectors are determined.

Abstract

An automorphism of a graph $G=(V,E)$ is a bijective map $\phi$ from $V$ to itself such that $\phi(v_i)\phi(v_j)\in E$ $\Leftrightarrow$ $v_i v_j\in E$ for any two vertices $v_i$ and $v_j$. Denote by $\mathfrak{G}$ the group consisting of all automorphisms of $G$. Apparently, an automorphism of $G$ can be regarded as a permutation on $[n]=\{1,\ldots,n\}$, provided that $G$ has $n$ vertices. For each permutation $\sigma$ on $[n]$, there is a natural action on any given vector $\boldsymbol{u}=(u_1,\ldots,u_n)^t\in \mathbb{C}^n$ such that $\sigma\boldsymbol{u}=(u_{\sigma^{-1}1},u_{\sigma^{-1}2},\ldots,u_{\sigma^{-1} n})^t$, so $\sigma$ can be viewed as a linear operator on $\mathbb{C}^n$. Accordingly, one can formulate a characterization to the automorphisms of $G$, {\it i.e.,} $\sigma$ is an automorphism of $G$ if and only if every eigenspace of $\mathbf{A}(G)$ is $\sigma$-invariant, where $\mathbf{A}(G)$ is the adjacency matrix of $G$. Consequently, every eigenspace of $\mathbf{A}(G)$ is $\mathfrak{G}$-invariant, which is equivalent to that for any eigenvector $\boldsymbol{v}$ of $\mathbf{A}(G)$ corresponding to the eigenvalue $\lambda$, $\mathrm{span}(\mathfrak{G}\boldsymbol{v})$ is a subspace of the eigenspace $V_{\lambda}$. By virtue of the linear representation of the automorphism group $\mathfrak{G}$, we characterize those extremal vectors $\boldsymbol{v}$ in an eigenspace of $\mathbf{A}(G)$ so that $\mathrm{dim}~\mathrm{span}(\mathfrak{G}\boldsymbol{v})$ can attain extremal values, and furthermore, we determine the exact value of $\mathrm{dim}~\mathrm{span}(\mathfrak{G}\boldsymbol{v})$ for any eigenvector $\boldsymbol{v}$ of $\mathbf{A}(G)$.