Normalized Laplacians for Gain Graphs

TL;DR

This paper introduces a normalized Laplacian matrix for gain graphs, explores its spectral properties, and establishes bounds and relationships with graph properties like balancedness and bipartiteness.

Contribution

It defines the normalized Laplacian for gain graphs and analyzes its eigenvalues, spectrum, and related graph properties, extending classical results to gain graphs.

Findings

Eigenvalue bounds for the normalized Laplacian of gain graphs

Characterization of graphs where bounds are tight

Extension of eigenvalue interlacing to gain graphs

Abstract

We propose the notion of normalized Laplacian matrix $\mathcal{L}(\Phi)$ for a gain graphs and study its properties in detail, providing insights and counterexamples along the way. We establish bounds for the eigenvalues of $\mathcal{L}(\Phi)$ and characterize the classes of graphs for which equality holds. The relationships between the balancedness, bipartiteness, and their connection to the spectrum of $\mathcal{L}(\Phi)$ are also studied. Besides, we extend the edge version of eigenvalue interlacing for the gain graphs. Thereupon, we determine the coefficients for the characteristic polynomial of $\mathcal{L}(\Phi)$.