Spectral properties of edge Laplacian matrix

TL;DR

This paper investigates the spectral properties of the edge Laplacian matrix derived from directed graphs, focusing on bipartiteness, spectrum computation for specific graph classes, and polynomial divisibility related to graph covers.

Contribution

It introduces spectral analysis of the edge Laplacian matrix for various graph classes and proves polynomial divisibility involving Kronecker double covers.

Findings

Spectrum computed for regular, complete bipartite graphs, and trees.

Bipartiteness characterized via spectral properties.

Characteristic polynomial divisibility established for graph covers.

Abstract

Let $N(X)$ be the Laplacian matrix of a directed graph obtained from the edge adjacency matrix of a graph $X.$ In this work, we study the bipartiteness property of the graph with the help of $N(X).$ We computed the spectrum of the edge Laplacian matrix for the regular graphs, the complete bipartite graphs, and the trees. Further, it is proved that given a graph $X,$ the characteristic polynomial of $N(X)$ divides the characteristic polynomial of $N(X^{\prime\prime}),$ where $X^{\prime\prime}$ denote the Kronecker double cover of $X.$