Spectra of Coronae

TL;DR

This paper introduces the coronal invariant to determine the spectrum of graph coronas, extending previous results to non-regular graphs and enabling the construction of cospectral graph pairs.

Contribution

The paper defines the coronal invariant and demonstrates its use in computing spectra of graph coronas beyond regular cases, including explicit coronals for various graph families.

Findings

Spectrum of G∘H determined by spectra of G, H, and the coronal of H

Explicit coronals computed for regular, complete n-partite, and path graphs

Generated infinite families of cospectral graphs using corona construction

Abstract

We introduce a new invariant, the coronal of a graph, and use it to compute the spectrum of the corona $G\circ H$ of two graphs $G$ and $H$. In particular, we show that this spectrum is completely determined by the spectra of $G$ and $H$ and the coronal of $H$. Previous work has computed the spectrum of a corona only in the case that $H$ is regular. We then explicitly compute the coronals for several families of graphs, including regular graphs, complete $n$-partite graphs, and paths. Finally, we use the corona construction to generate many infinite families of pairs of cospectral graphs.