Spectral characterizations of lollipop graphs

The lollipop graph, denoted by Hn,p, is obtained by appending a cycle Cp to a pendant vertex of a path Pn-p. We will show that no two non-isomorphic lollipop graphs are cospectral with respect to the adjacency matrix. It is proved that for p odd the lollipop graphs Hn,p and some related graphs are determined by the adjacency spectrum, and that all lollipop graphs are determined by its Laplacian spectrum.