${\mathrm{Aut} \hspace{0.4mm} G}$. For convenience, we use $\mathfrak{G}$ in what follows to denote the permutation group ${\mathrm{Aut} \hspace{0.4mm} G}$. $$\label{Def-SubspacePointStabilizer}
V_{\lambda} \langle u \rangle :=
\{ \pmb{v} \in V_{\lambda} \mid \xi \hspace{0.5mm} \pmb{v} = \pmb{v}, ~ \forall \hspace{0.6mm} \xi\in\mathfrak{G}_u  \}, ~~ u\in [n].$$ $$\label{Def-SubspaceEigenvalue1}
V_{1}^{\gamma} := 
\{ \pmb{v} \in \operatorname{\mathbb{R}}^n \mid \gamma \hspace{0.5mm} \pmb{v} = \pmb{v} \}, ~~ \gamma \in \mathfrak{G}.$$ Apparently the partition $\Pi_u^*$ composed of orbits of $\mathfrak{G}_u$ is equitable and $V_{\lambda} \langle u \rangle = \mathbf{R}_{\Pi_u^*} V_{\lambda}^{ G / \Pi_u^* }$.

One can readily see that there are for any vertex $v$ of $G$ two possibilities: $$\label{Equ-ActionGonV}
\mbox{either } \sigma v = v \mbox{ or } \sigma v \neq v, ~ \forall\hspace{0.6mm} \sigma \in \mathfrak{G}.$$ It is interesting that there might be some subsets of $[n]$ possessing that relation (\[Equ-ActionGonV\]). Let $B$ be a non-empty subset contained in some orbit $T$ of $\mathfrak{G}$, which is called a [*block*]{} for $\mathfrak{G}$ if $$\mbox{ either }\sigma B = B \mbox{ or } \sigma B \cap B = \emptyset, ~ \forall\hspace{0.6mm} \sigma \in \mathfrak{G}.$$ Evidently, any element $t$ of $T$ and the orbit $T$ itself are blocks for $\mathfrak{G}$. If the group $\mathfrak{G}$ has only two such kinds of blocks 