The Spectrum of the Corona of Two Graphs

Source abstract

We consider only simple graphs. Given two graphs G with vertices $1,\ldots,n$ and H, the corona $G\circ H$ is defined as the graph obtained by taking n copies of H and for each i inserting edges between the ith vertex of G and each vertex of the ith copy of H. For a connected graph G and any r‐regular graph H we provide complete information about the spectrum of $G\circ H$ using the spectrum of G and spectrum of H. Complete information about the Laplacian spectrum of $G\circ H$ is also provided even when H is not regular. A graph G is said to have the property (R) if $\frac{1}{\lambda}$ is an eigenvalue of G whenever $\lambda$ is an eigenvalue of G. Further, if $\lambda$ and $\frac{1}{\lambda}$ have the same multiplicity, for each eigenvalue $\lambda$, then it is said to have the property (SR). We characterize all trees with property (SR) and show that such a tree is the corona product of some tree and an isolated vertex. We supply a family of bipartite graphs with property (R). As an application we construct infinitely many pairs of nonisomorphic graphs with the same spectrum and the same Laplacian spectrum. We prove some results about the eigenvector related to the second smallest eigenvalue of the Laplacian matrix of $G\circ H$ and give an application.