Coronae graphs and their $\alpha$-eigenvalues

TL;DR

This paper investigates the $eta$-eigenvalues of various graph constructions involving coronae graphs and introduces methods to determine these eigenvalues using the graph invariant called coronal, leading to the creation of infinitely many non-isomorphic $eta$-isospectral graphs.

Contribution

It introduces new formulas for $eta$-eigenvalues of complex graph operations using the coronal invariant, expanding spectral graph theory techniques.

Findings

Derived explicit $eta$-eigenvalue formulas for multiple graph operations.

Constructed infinitely many pairs of non-isomorphic $eta$-isospectral graphs.

Extended spectral analysis to new classes of graph equations.

Abstract

Let $G_1$ and $G_2$ be two simple connected graphs. The invariant \textit{coronal} of graph is used in order to determine the $\alpha$-eigenvalues of four different types of graph equations that are $G_1 \circ G_2, G_1\lozenge G_1$ and the other two`s are $G_1 \odot G_2$ and $G_1 \circleddash G_2$ which are obtained using the $R$-graph of $G_1$. As an application we construct infinitely many pairs of non-isomorphic $\alpha$-Isospectral graph.