The graphs which are cospectral with the generalized pineapple graph

Let $p,k,q$ be positive integers with $p-2⩾k$ and let $K_{p,k}^q$ be the generalized pineapple graph which is obtained by joining independent set of $q$ vertices with $k$ vertices of a complete graph $K_p\mathrm{.}$ In \cite{TSH2}, Haemers et al. constructed graphs which cospectral with $K_{p,1}^q\mathrm{.}$ In this paper, we determine all graphs which are cospectral with $K_{p,k}^q$ by considering the eigenvalues of its adjacency matrix. Moreover, We extend the conclusions of Haemers et al. to a broader context.