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.