# On the Laplacian spectra of some double join operations of graphs

## TL;DR

This paper investigates the Laplacian spectra of four variants of double join graph operations, providing explicit formulas and eigenvector characterizations that generalize existing spectral results for graph joins.

## Contribution

It introduces the concept of double join matrix and derives complete spectral characterizations for four double join graph variants based on Laplacian eigenvalues.

## Key findings

- Explicit eigenvalues and eigenvectors for double join variants
- Generalization of known join spectral results
- Complete characterization in terms of factor graphs' spectra

## Abstract

Many variants of join operations of graphs have been introduced and their spectral properties have been studied extensively by many researchers. This paper mainly focuses on the Laplacian spectra of some double join operations of graphs. We first introduce the conception of double join matrix and provide a complete information about its eigenvalues and the corresponding eigenvectors. Further, we define four variants of double join operations based on subdivision graph, $Q$-graph, $R$-graph and total graph. Applying the result obtained for the double join matrix, we give an explicit complete characterization of the Laplacian eigenvalues and the corresponding eigenvectors of four variants in terms of the Laplacian eigenvalues and the eigenvectors of the factor graphs. These results generalize some well-known results about some join operations of graphs.