Graph Automorphisms from the Geometric Viewpoint

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 $\mathbf{A}(G)$.