Normalized Laplacians for Gain Graphs

We propose the notion of normalized Laplacian matrix $\mathcal{L}(Φ)$ for a gain graphs and study its properties in detail, providing insights and counterexamples along the way. We establish bounds for the eigenvalues of $\mathcal{L}(Φ)$ and characterize the classes of graphs for which equality holds. The relationships between the balancedness, bipartiteness, and their connection to the spectrum of $\mathcal{L}(Φ)$ 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 $\mathcal{L}(Φ)$.

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