Graphs with Prescribed Star Complement for \(1\) as the Second Largest Eigenvalue

Abstract

Let \(G\) be a graph of order \(n\) and let \(\mu\) be an eigenvalue of multiplicity \(m\). A star complement for \(\mu\) in \(G\) is an induced subgraph of \(G\) of order \(n-m\) with no eigenvalue \(\mu\). In this paper, we investigate maximal and regular graphs that have \(K_{r,s} + t{K_{1}}\) as a star complement for \(\mu\) as the second largest eigenvalue. Interestingly, it turns out that some well-known strongly regular graphs are uniquely determined by such a star complement.