Regular sets of circulant quartic graphs

For a graph $Γ=(V,E)$ and nonnegative integers $a$ and $b$, a nonempty proper subset $C \subset V$ is called an $(a,b)$-regular set if every vertex in $C$ has exactly $a$ neighbors in $C$, and every vertex in $V\setminus C$ has exactly $b$ neighbors in $C$. In this paper, we study the existence of s