The normalized Laplacian spectra of the double corona based on $R$-graph

TL;DR

This paper derives the normalized Laplacian spectra for a specific graph operation called the double corona based on the R-graph, linking it to spectra of component graphs and constructing cospectral pairs.

Contribution

It provides a formula for the normalized Laplacian spectrum of the double corona based on R-graphs for regular graphs, extending known spectra of related graph operations.

Findings

Derived the normalized Laplacian spectrum for the double corona based on R-graphs.

Reduced the spectrum to known cases of R-vertex and R-edge coronas.

Constructed infinitely many pairs of normalized Laplacian cospectral graphs.

Abstract

For simple graphs $G$, $G_1$ and $G_2$, we denote their double corona based on $R$-graph by $G^{(R)}\otimes{\{G_1,G_2\}}$. This paper determines the normalized Laplacian spectrum of $G^{(R)}\otimes{\{G_1,G_2\}}$ in terms of these of $G$, $G_1$ and $G_2$ whenever $G$, $G_1$ and $G_2$ are regular. The obtained result reduces to the normalized Laplacian spectra of the $R$-vertex corona $G^{(R)}\odot{G_1}$ and $R$-edge corona $G^{(R)}\circleddash{G_2}$ by choosing $G_2$ or $G_1$ as a null-graph, respectively. Finally, applying the results of the paper, we construct infinitely many pairs of normalized Laplacian cospectral graphs.