Spectral characterization of the complete graph removing a path

Abstract

A graph G is said to be A−DS if every graph having the same adjacency spectrum is isomorphic to G. Let Kn∖Pk be the graph obtained from the complete graph Kn with n vertices by removing all edges of a path Pk with k vertices. It was shown by Doob and Haemers that Kn∖Pn is A−DS. In 2014, Cámara and Haemers conjectured that Kn∖Pk is A−DS for every 2≤k≤n, and they succeeded in proving it for 2≤k≤6. Recently, Mao, Cioabă and Wang verified the conjecture for 7≤k≤9. In this paper, we show that the conjecture is true for all k≥20.