On the Spectrum of Graphs with Pockets

Abstract

Let $S_{r,l}$ be a generalized star on $rl+1$ vertices with central vertex $v$. Let $H_v$ be a graph of order $m$ with a specified vertex $v$ of degree $m-1$. For simple connected graphs $G_{r,l,H_v}$, obtained by attaching $v$ of $H_v$ to each vertex of $S_{r,l}$ except the central vertex, we derive the adjacency, Laplacian, and signless Laplacian spectrum of $G_{r,l,H_v}$ in terms of the corresponding spectrum of $S_{r,l}$ and $H_v$. Furthermore, we extend these results to obtain the adjacency, Laplacian, and signless Laplacian characteristic polynomials of general graphs.