the connected graphs with exactly two distinct eigenvalues are complete graphs. However, it is far from been solving to characterize the graphs with exactly three distinct eigenvalues. The non-regular graphs with three distinct eigenvalues are studied by many mathematicians, see, for example, [@Dam] and [@Koolen]. A connected regular graph with exactly three distinct eigenvalues must be a so called strongly regular graph [@Godsil]. A *strongly regular graph* with parameters $(n,r,\lambda,\mu)$ is an $r$ regular graph on $n$ vertices in which any two adjacent vertices have exactly $\lambda$ common neighbors and any two non-adjacent vertices have exactly $\mu$ common neighbours. The characterization of strongly regular graphs is a classical problem in algebraic graph theory which has caused wide public attention and numerous results are obtained, especially, on the strongly regular Cayley graphs, we refer the reader to [@Calderbank; @Feng; @Ge; @Momihara] and references therein. Let $G$ be a group and $S$ a subset of $G$. The *Cayley graph* $X=Cay(G,S)$ is the graph such that $V(X)=G$ and two vertices $x$ and $y$ are adjacent if $x^{-1}y\in S$. If $S$ contains no identity element of $G$ and $S^{-1}=S$, then $X$ is an undirected simple graph. The *circulant graphs* are the Cayley graphs over 