Spectral characterization of the complete graph removing a cycle

Abstract

A graph is determined by its spectrum if there is not another graph with the same spectrum. Cámara and Haemers proved that the graph $K_n\setminus C_k$, obtained from the complete graph $K_n$ with n vertices by deleting all edges of a cycle $C_k$ with k vertices, is determined by its spectrum for $k\in \{3,4,5\}$, but not for $k=6$. In this paper, we show that $k=6$ is the unique exception for the spectral determination of $K_n\setminus C_k$.