  Let $G$ be a graph of order $n$ and $\mu$ be an adjacency eigenvalue of $G$
with multiplicity $k\geq 1$. 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 $K_{t,s}\
(s\geq t\geq 2)$ 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.
