The eigenvalues of $A(G)$ are called the *eigenvalues* of $\Gamma$ and the multiset of such eigenvalues together with their multiplicities is called the *spectrum* of $\Gamma$, denoted by $\mathrm{Spec}(\Gamma)$. The *distance* $d(v_i,v_j)$ (or $d_{ij}$ for short) between two vertices $v_i$ and $v_j$ is the length of a shortest path from $v_i$ to $v_j$. The largest distance in $\Gamma$ is the *diameter* of $\Gamma$, denoted by $d(\Gamma)$, that is, $d(\Gamma)=\max\{d(v_i,v_j)\mid v_i,v_j\in V\}$. For $1\le l\le d(\Gamma)$, let $N^{(l)}(v_i)=\{v_j\mid d(v_i,v_j)=l\}$. Therefore, $V=N^{(1)}(v_i)\cup \cdots\cup N^{(d(\Gamma))}(v_i)$ is a partition of $V$ for any vertex $v_i$. Particularly, we alway write $N(v_i)$ for $N^{(1)}(v_i)$, which is just the neighborhood of $v_i$. Moreover, the size of $N(v_i)$ is the *degree* of $v_i$, denoted by $d(v_i)$. A bijection $\sigma$: $V\rightarrow V$ is an *automorphism* of $\Gamma$ if $v_i^{\sigma}\sim v_j^{\sigma}$ if and only if $v_i\sim v_j$. The set of all automorphisms forms the *automorphism group* of $\Gamma$, denoted by $Aut(\Gamma)$. The graph $\Gamma$ is *vertex transitive* if for any vertex $v_i$ there exists $\sigma\in Aut(\Gamma)$ such that $v_1^{\sigma}=v_i$.

In 1970, Doob [@Doob] started the investigation of graphs with few distinct eigenvalues. Such graphs has been proved to possess nice combinatorial properties and rich structures [@Dam]. It is clear that 