Star Sets in Regular Graphs

Let \(G\) be a finite graph and let \(\mu\) be an eigenvalue of \(G\) of multiplicity \(k\). A star set for \(\mu\) may be characterized as a set \(X\) of \(k\) vertices of \(G\) such that \(\mu\) is not an eigenvalue of \(G – X\). It is shown that if \(G\) is regular then \(G\) is determined by \(\mu\) and \(G – X\) in some cases. The results include characterizations of the Clebsch graph and the Higman-Sims graph.