Normalized Laplacians for Gain Graphs

M. Rajesh Kannan; Navish Kumar; Shivaramakrishna Pragada

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arXiv · 2009.13788

Normalized Laplacians for Gain Graphs

M. Rajesh Kannan·Navish Kumar·Shivaramakrishna Pragada

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Abstract

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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BibTeX RIS

M. Rajesh Kannan, Navish Kumar, Shivaramakrishna Pragada. 2022-03-10. Normalized Laplacians for Gain Graphs. https://arxiv.org/abs/2009.13788

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