Spectra of the blow-up graphs

TL;DR

This paper derives the eigenvalues of the Laplacian and signless Laplacian matrices for blow-up graphs and their complements, providing a comprehensive spectral analysis of these graph transformations.

Contribution

It explicitly determines all eigenvalues of Laplacian and signless Laplacian matrices for blow-up graphs and their complements, advancing spectral graph theory.

Findings

Eigenvalues of Laplacian matrices for blow-up graphs are explicitly characterized.

Eigenvalues of signless Laplacian matrices for blow-up graphs are explicitly characterized.

Spectral properties of the complements of blow-up graphs are also determined.

Abstract

Let $G$ be graph on $n$ vertices and $G^{(t)}$ its blow-up graph of order $t.$ In this paper, we determine all eigenvalues of the Laplacian and the signless Laplacian matrix of $G^{(t)}$ and its complement $\bar{G^{(t)}}.$