Graphs of order n with exactly \(n+1\) complementarity eigenvalues

Abstract

This work explores the complementarity spectrum \(\Pi (G)\) of a graph G. Previous studies have established that for a graph of order n, the complementarity spectrum satisfies \(|\Pi (G)| \ge n\), with equality holding only for elementary graphs. This implies that any non-elementary graph of order n has at least \(n+1\) distinct complementarity eigenvalues. In this article, we focus on connected graphs of order n that admit exactly \(n+1\) complementarity eigenvalues. We characterize this class by examining the structural features responsible for such spectral behavior. For instance, we show that any graph of sufficiently large order in this family induces exactly one subgraph of order \(n-1\) that also belongs to the same family. Furthermore, we demonstrate that graphs in this class are uniquely determined by their complementarity spectra, emphasizing the potential of complementarity eigenvalues in graph identification where traditional spectral methods may fall short.