of $[n]$ consisting of orbits of $\mathfrak{S}$ is an equitable one. On the other hand, we can construct a direct graph $G / \Pi$ from $G$ and its equitable partition $\Pi$, which is called the [*quotient graph*]{} of $G$ over $\Pi$. The vertex set of $G / \Pi$ is composed of cells of $\Pi$ and there are $b_{ij}$ arcs $(1\leq i,j \leq t)$ from the $i$th vertex to the $j$th vertex of $V(G  / \Pi)$.

For each cell $C_i$ ($i=1,\ldots,t$) of the partition $\Pi$, one can build a vector $\pmb{R}_{C_i}$, or abbreviated to $\pmb{R}_i$, to indicate $C_i$, that is called the [*characteristic vector*]{} of $C_i$, such that the $k$th coordinate ($1\leq k\leq n$) of the vector is 1 if $k$ belongs to $C_i$ otherwise it is 0. By means of characteristic vectors, we can define the [*characteristic matrix*]{} $\mathbf{R}_{\Pi}$ of $\Pi$ as $(\pmb{R}_1 \pmb{R}_2 \cdots \pmb{R}_t)$. It is not difficult to verify that a partition $\Pi$ of $[n]$ is equitable if and only if the column space of $\mathbf{R}_{\Pi}$ is ${\mathbf{A}( G )}$-invariant (see [@GodRoy] for details).

As well-known, if the partition $\Pi$ involved is equitable, there is a close relation between eigenvalues and eigenvectors of $\mathbf{A}(G)$ and that of 