On the spectral characterization of pineapple graphs

The pineapple graph Kpq is obtained by appending q   pendant edges to a vertex of a complete graph KpKp (q≥1q≥1, p≥3p≥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≥4p≥4 and various values of q  . In addition we prove that the claim is true if q=2q=2, and refer to the literature for q=1q=1, p=3p=3, and (p,q)=(4,3)(p,q)=(4,3).