Let G be a graph of order n and μ be an adjacency eigenvalue of G
with multiplicity k≥1. A star complement H for μ in G is an
induced subgraph of G of order n−k with no eigenvalue μ, and the vertex
subset X=V(G−H) is called a star set for μ 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 $K_{t,s}\
(s\geq t\geq 2)asastarcomplementforaneigenvalue\mu,especially,characterizethecaseoft=3completely,obtainsomepropertieswhent=s$,
and propose some problems for further study