Maximal graphs with a prescribed complete bipartite graph as a star complement

Abstract

Let $ G $ be a graph of order $ n $ and $ \mu $ be an adjacency eigenvalue of $ G $ with multiplicity $ k\geq 1 $. A star complement for $ \mu $ in $ G $ is an induced subgraph of $ G $ of order $ n-k $ with no eigenvalue $ \mu $. In this paper, we characterize the maximal graphs with the bipartite graph $ K_{2, s} $ as a star complement for eigenvalues $ \mu = -2, 1 $ and study the cases of other eigenvalues for further research.