Spectral Invariants and Some Stable Properties of a Graph

Abstract

For an integer $k\ge 0$, a graphical property $P$ is said to be $k$-stable if whenever $G+uv$ has property $P$ and $d_G(u)+d_G(v)\ge k$, where $uv\notin E(G)$, then $G$ itself has property $P$. In this note, we present spectral sufficient conditions for several stable properties of a graph.