N(v_2)|=b_{32}$, where $y^{\tau}=v_2\in N^2(v)$ for some $\tau\in Aut(\Gamma)$ and $x^{\tau}=v$. It means that $\Gamma$ is a $(n,r,b_{22},b_{32})$-strongly regular graph.

For a subset $S$ of a group $G$, let $\overline{S}=G\setminus(S\cup\{1\})$. Denote by $C(v,S)=\{(x,y)\mid x,y\in S \textrm{ and } v=xy\}$. By applying Lemma \[lem-z-8\], we get a criterion for a Cayley graph to be strongly regular.

\[cor-z-3\] Let $S$ be a subset of a group $G$ such that the identity element $1\not\in S$ and $S=S^{-1}$. Then the Cayley graph $Cay(G,S)$ is $(|G|,|S|,\lambda,\mu)$-strongly regular if and only if $|C(v,S)|=\lambda$ for any $v\in S$ and $|C(v',S)|=\mu$ for any $v'\in \overline{S}$.

At first, we will verify that $|N(v)\cap S|=|C(v,S)|$ for any $v\in G$. On the one hand, for any $x\in N(v)\cap S$, we have $x^{-1}v=y\in S$, that is, $v=xy$. It means that $(x,y)\in C(v,S)$ and thus $|N(v)\cap S|\le |C(v,S)|$. On the other hand, for any $(x,y)\in C(v,S)$, we have $x,y\in S$ and $v=xy$, that is, $x^{-1}v=y\in S$. It means that $x\in N(v)\cap S$ and thus $|C(v,S)|\le |N(v)\cap S|$.

Assume that $X=Cay(G,S)$ is $(|G|,|S|,\lambda,\mu)$-strongly regular. For any $v\in S$, we have $v\sim 1$ and thus $|N(v)\cap N(1)|=\lambda$. Therefore, by noticing $N(1)=S$, we have $|C(v,S)|=|N(v)\cap S|=|N(v)\cap N(1)|=\lambda$. For any $v'\in \overline{S}$, we have $v'\not\sim 1$ and thus 