Normalized Laplacians for gain graphs

Abstract

We propose the notion of normalized Laplacian matrix \(\mathcal{L}(\Phi)\) for a gain graph \(\Phi\) 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 \(\mathcal{L}(\Phi)\).