On graphs with prescribed star complements

Let μ be an eigenvalue of a simple graph G with multiplicity k≥1. A star complement for μ in G is an induced subgraph of G of order n−k with no eigenvalue μ. In this paper, we study the maximal graphs with the star Sm as a star complement for −2. The maximal graphs with S3, S4, S13 and S21 as a star complement for −2 are described. We also describe the regular graphs with K2,s(s≥2) as a star complement for an eigenvalue μ.