Per-spectral and adjacency spectral characterizations of a complete graph removing six edges

Abstract

Cámara and Haemers (2014) investigated when a complete graph with some edges deleted is determined by its adjacency spectrum (DAS for short). They claimed: for any m≥6 and every large enough n one can obtain graphs which are not DAS by removing m edges from a complete graph Kn. Let Gn denote the set of all graphs obtained from a complete graph Kn by deleting six edges. In this paper, we show that all graphs in Gn are uniquely determined by their permanental spectra. However, we show that for each n≥7 or n=5 there is just one pair of nonisomorphic cospectral graphs in Gn, and for n=4 or 6 all graphs in Gn are DAS.