is a critical transition.

    On the other hand, we have for a given subspace $U \subseteq \mathbb{R}^n$ a permutation group $\mathrm{Aut}\hspace{0.5mm} U$ which is defined as $\{ \sigma \in S_n : \sigma \hspace{0.5mm} U = U \}$. As a matter of fact, $\mathrm{Aut}\hspace{0.5mm} G = \cap_{\lambda \in \mathrm{spec} \hspace{0.3mm} \mathbf{A}(G) } \mathrm{Aut}\hspace{0.5mm} V_{\lambda}$, where $V_{\lambda}$ is the eigenspace of the adjacency matrix $\mathbf{A}(G)$ corresponding to $\lambda$, and moreover we can obtain a good approximation $\Pi[ \oplus V_{\lambda} ; v ]$ to $\Pi_v^*$ by analyzing a decomposition of $V_{\lambda}$ resulted from the division of $V_{\lambda}$ by subspaces $\{ {\mathrm{proj}\big[  V_{\lambda}  \big]}( \pmb{e}_v )^{\perp} : v \in V(G) \}$, where ${\mathrm{proj}\big[  V_{\lambda}  \big]}( \pmb{e}_v )$ denotes the orthogonal projection of the vector $\pmb{e}_v$ onto $V_{\lambda}$ and ${\mathrm{proj}\big[  V_{\lambda}  \big]}( \pmb{e}_v )^{\perp}$ stands for the orthogonal complement of the subspace spanned by ${\mathrm{proj}\big[  V_{\lambda}  \big]}( \pmb{e}_v )$ in $V_{\lambda}$. In fact, there is a close relation among subspaces spanned by cells of the equitable partition $\Pi[ \oplus V_{\lambda} ; v ]$ of $G$, which enables us to determine $\Pi_v^*$ more efficiently. In virtue of that, we devise a deterministic algorithm solving GI in time $n^{ O( \log n ) }$, which is equal 