On the normalized Laplacian spectra of some subdivision joins of two graphs

TL;DR

This paper derives the normalized Laplacian spectra of subdivision joins of regular graphs, providing spectral characterizations and applications such as constructing cospectral graphs and computing spanning trees and Kirchhoff indices.

Contribution

It determines the normalized Laplacian spectra of subdivision-vertex and subdivision-edge joins of regular graphs, linking spectra to original graphs and enabling new spectral graph constructions.

Findings

Spectra of subdivision-vertex and subdivision-edge joins expressed in terms of original graphs' spectra

Construction of non-regular normalized Laplacian cospectral graphs

Formulas for spanning trees and degree-Kirchhoff index of the joins

Abstract

For two simple graphs $G_1$ and $G_2$, we denote the subdivision-vertex join and subdivision-edge join of $G_1$ and $G_2$ by $G_1\dot{\vee}G_2$ and $G_1\veebar G_2$, respectively. This paper determines the normalized Laplacian spectra of $G_1\dot{\vee}G_2$ and $G_1\veebar G_2$ in terms of these of $G_1$ and $G_2$ whenever $G_1$ and $G_2$ are regular. As applications, we construct some non-regular normalized Laplacian cospectral graphs. Besides we also compute the number of spanning trees and the degree-Kirchhoff index of $G_1\dot{\vee}G_2$ and $G_1\veebar G_2$ for regular graphs $G_1$ and $G_2$.