Identifying code number of some Cayley graphs

Let $\Gamma=(V, E)$ be a simple graph. A set $C$ of vertices $\Gamma$ is an identifying set of $\Gamma$ if for every two vertices $x$ and $y$ the sets $N_{\Gamma}[x] \cap C$ and $N_{\Gamma}[y] \cap C$ are non-empty and different. Given a graph $\Gamma,$ the smallest size of an identifying set of $\Gamma$ is called the identifying code number of $\Gamma$ and is denoted by $\gamma^{ID}(\Gamma).$ Two vertices $x$ and $y$ are twins when $N_{\Gamma}[x]=N_{\Gamma}[y].$ Graphs with at least two twin vertices are not identifiable graph. In this paper, we study identifying code number of some Cayley graphs.