Laplacian Spectra of Semigraphs

TL;DR

This paper investigates the eigenvalues of the Laplacian matrix of semigraphs, establishing fundamental properties, bounds, and eigenvalue enumeration for specific types of semigraphs.

Contribution

It introduces the spectral properties of Laplacian matrices for semigraphs, including positivity, connectivity characterization, and bounds on eigenvalues, extending graph theory concepts.

Findings

Laplacian of a semigraph is positive semi-definite.

Semigraph connectivity is characterized by the second smallest eigenvalue.

Derived bounds for the largest Laplacian eigenvalue of semigraphs.

Abstract

Consider a semigraph $G=(V,\,E)$; in this paper, we study the eigenvalues of the Laplacian matrix of $G$. We show that the Laplacian of $G$ is positive semi-definite, and $G$ is connected if and only if $\lambda_2 >0.$ Along the similar lines of graph theory bounds on the largest eigenvalue, we obtain upper and lower bounds on the largest Laplacian eigenvalue of G and enumerate the Laplacian eigenvalues of some special semigraphs such as star semigraph, rooted 3-uniform semigraph tree.