On the spectra of graphs with edge-pockets

Abstract

Let and be two vertex disjoint graphs of orders and respectively, where and Let be a specified edge of such that is isomorphic to Let be a subset of the edge set of and let denote the subgraph of induced by Let be the graph obtained by taking one copy of and vertex disjoint copies of and then pasting the edge in the th copy of with the edge where Then the copies of the graph that are pasted to the edges are called as edge-pockets, and we say is a graph with edge-pockets. In this article, we prove some results describing the Laplacian (resp. adjacency) spectrum of using the Laplacian (resp. adjacency) spectra of and The complete Laplacian (resp. adjacency) spectrum of is also described in some particular cases. As an application, we show that these results enable us to construct infinitely many pairs of Laplacian (resp. adjacency) cospectral graphs.