On bipartite graphs with complete bipartite star complements

Let G be a bipartite graph with mu as an eigenvalue of multiplicity k > 1. We show that if G has K-r,K-s (1 <= r <= s) as a star complement for mu then k <= s - 1; moreover if mu is non-main then k <= 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 K-1,K-s as a star complement for a non-main eigenvalue of multiplicity s - 1 > 1. (C) 2014 Elsevier Inc. All rights reserved.