On bipartite graphs with complete bipartite star complements

Citation

Rowlinson P (2014) On bipartite graphs with complete bipartite star complements. Linear Algebra and Its Applications, 458, pp. 149-160. https://doi.org/10.1016/j.laa.2014.06.011

Abstract
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.

Keywords
Bipartite graph; Eigenvalue; Star complement; Symmetric design

Journal
Linear Algebra and Its Applications: Volume 458