Regular graphs with a complete bipartite graph as a star complement

Let $G$ be a graph of order $n$ and $\mu$ be an adjacency eigenvalue of $G$ with multiplicity $k\ge 1$. A star complement $H$ for $\mu$ in $G$ is an induced subgraph of $G$ of order $n-k$ with no eigenvalue $\mu$, and the vertex subset $X=V(G-H)$ is called a star set for $\mu$ in $G$. The study of star complements and star sets provides a strong link between graph structure and linear algebra. In this paper, we study the regular graphs with $K_{t,s}(s\ge t\ge 2)$ as a star complement for an eigenvalue $\mu$, especially, characterize the case of $t=3$ completely, obtain some properties when $t=s$, and propose some problems for further study.