of partitions of $[n]$ each of which consists of orbits of a stabilizer that fixes a sequence of vertices of $G$, and thus the determination of the partition $\Pi^*_v$ is a critical transition.

The [*adjacency matrix*]{} of $G$, denoted by $\mathbf{A}(G)$, is a $n\times n$ (0,1)-matrix where each entry $a_{ij}$ of the matrix is equal to 1 if and only if the two vertices $v_i$ and $v_j$ are adjacent in $G$. In the second part, we will reveal some of geometric features of ${\mathrm{Aut} \hspace{0.4mm} G}$ by means of the decomposition $\oplus V_{\lambda} = \operatorname{\mathbb{R}}^n$, where $V_{\lambda}$ is the eigenspace of ${\mathbf{A}( G )}$ corresponding to the eigenvalue $\lambda$. In virtue of that, we could build a partition $\Pi[ \oplus V_{\lambda};v ]$, which is a good approximation to $\Pi^*_{v}$.

Geometric Features of $\Pi_{G}^*$
---------------------------------

Let $\Pi$ be a partition of $[n]$ with cells $C_1,\ldots,C_t$, which is said to be [*equitable*]{} if for any vertex $v$ in $C_i$, the number of neighbors of $v$ in $C_j$ is a constant $b_{ij}$ $(1\leq i,j \leq t)$, [*i.e.,*]{} the number of neighbors in every cell is independent of the vertex $v$. Clearly, if $\mathfrak{S}$ is a subgroup of ${\mathrm{Aut} \hspace{0.4mm} G}$ then the partition 