Normalized Laplacians for gain graphs
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...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
American Journal of Combinatorics
2022-01-01
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| Series: | The American Journal of Combinatorics |
| Subjects: | |
| Online Access: | https://ajcombinatorics.org/ojs/index.php/AmJC/article/view/3 |
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| Summary: | 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 \(\mathcal{L}(\Phi)\).
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| ISSN: | 2768-4202 |