The spectra and the signless Laplacian spectra of graphs with pockets

Let $G[F,V_k,H_v]$ be the graph with $k$ pockets, where $F$ is a simple graph of order $n\ge 1$, $V_k=\{v_1,\dots ,v_k\}$ is a subset of the vertex set of $F$ and $H_v$ is a simple graph of order $m\ge 2$, $v$ is a specified vertex of $H_v$. Also let $G[F,E_k,H_{uv}]$ be the graph with $k$ edge-pockets, where $F$ is a simple graph of order $n\ge 2$, $E_k=\{e_1,\dots ,e_k\}$ is a subset of the edge set of $F$ and $H_{uv}$ is a simple graph of order $m\ge 3$, $uv$ is a specified edge of $H_{uv}$ such that $H_{uv}-u$ is isomorphic to $H_{uv}-v$. In this paper, we obtain some results describing the signless Laplacian spectra of $G[F,V_k,H_v]$ and $G[F,E_k,H_{uv}]$ in terms of the signless Laplacian spectra of $F,H_v$ and $F,H_{uv}$, respectively. In addition, we also give some results describing the adjacency spectrum of $G[F,V_k,H_v]$ in terms of the adjacency spectra of $F,H_v$. Finally, as an application of these results, we construct infinitely many pairs of signless Laplacian (resp. adjacency) cospectral graphs.