Spectral characterization of the complete graph removing a path: Completing the proof of Cámara–Haemers Conjecture

Abstract

A graph G is A−DS if every A-cospectral graph of G is isomorphic to G. Denote by Kn∖Pk the graph obtained from the complete graph Kn with n vertices by deleting all edges of a path Pk with k vertices. In 2014, Cámara and Haemers conjectured that Kn∖Pk is A−DS for every 2≤k≤n. The conjecture has been confirmed for k=n (Doob and Haemers, 2002), 2≤k≤6 (Cámara and Haemers, 2014), 7≤k≤9 (Mao et al., 2019) and k≥20 (Liu et al., 2020). In this paper, we completely settle the conjecture by proving the remaining cases.