is called an [*automorphism*]{} of $G$. Clearly, all automorphisms of $G$ form a group under composition of maps, which is denoted by ${\mathrm{Aut} \hspace{0.4mm} G}$. Suppose the vertex set $V(G)$ is $\{1,2,\ldots,n\}$ abbreviated to $[n]$. Then a bijective map $\phi$ on $V(G)$ is a permutation of $[n]$, and thus the automorphism group ${\mathrm{Aut} \hspace{0.4mm} G}$ is a permutation group of $[n]$.

There is a natural action of ${\mathrm{Aut} \hspace{0.4mm} G}$ on the vertex set $[n]$: $I v = v$, where $I$ is the identity of ${\mathrm{Aut} \hspace{0.4mm} G}$, and $\gamma (\sigma v) = (\gamma \sigma) v$ for any two permutations $\gamma$ and $\sigma$ in ${\mathrm{Aut} \hspace{0.4mm} G}$. Accordingly, we can obtain a subset $\{ \sigma v : \sigma \in {\mathrm{Aut} \hspace{0.4mm} G} \}$ of $[n]$, which is called an [*orbit*]{} of ${\mathrm{Aut} \hspace{0.4mm} G}$. Obviously, the orbits of ${\mathrm{Aut} \hspace{0.4mm} G}$ constitute a partition of $[n]$, which is denoted by $\Pi_G^*$, and each orbit is called a cell of $\Pi_G^*$. One can readily see that for any subgroup $\mathfrak{S}$ of ${\mathrm{Aut} \hspace{0.4mm} G}$ we have a partition of $[n]$ consisting of orbits of $\mathfrak{S}$.

Suppose $G$ and $H$ are isomorphic and $\phi$ is an isomorphism between $G$ and $H$. It 