On bipartite graphs with complete bipartite star complements

Let G be a bipartite graph with μ   as an eigenvalue of multiplicity k>1k>1. We show that if G   has Kr,sKr,s(1≤r≤s)(1≤r≤s) as a star complement for μ   then k≤s−1k≤s−1; moreover if μ   is non-main then k≤s−2k≤s−2 for large enough s. We provide examples of graphs in which various bounds on k or s   are attained. We also describe the bipartite graphs with K1,sK1,s as a star complement for a non-main eigenvalue of multiplicity s−1>1s−1>1.