On the spectral characterization of pineapple graphs

Abstract

The pineapple graph Kpq is obtained by appending q pendant edges to a vertex of a complete graph Kp (q≥1, p≥3). Zhang and Zhang (2009) [7] claim that the pineapple graphs are determined by their adjacency spectrum. We show that their claim is false by constructing graphs which are cospectral and non-isomorphic with Kpq for every p≥4 and various values of q. In addition we prove that the claim is true if q=2, and refer to the literature for q=1, p=3, and (p,q)=(4,3).