At the end of the spectrum: Chromatic bounds for the largest eigenvalue of the normalized Laplacian
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911945014116352 |
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| author | Beers, Lies Mulas, Raffaella |
| author_facet | Beers, Lies Mulas, Raffaella |
| contents | For a graph with largest normalized Laplacian eigenvalue $λ_N$ and (vertex) coloring number $χ$, it is known that $λ_N\geq χ/(χ-1)$. Here we prove properties of graphs for which this bound is sharp, and we study the multiplicity of $χ/(χ-1)$. We then describe a family of graphs with largest eigenvalue $χ/(χ-1)$. We also study the spectrum of the $1$-sum of two graphs (also known as graph joining or coalescing), with a focus on the maximal eigenvalue. Finally, we give upper bounds on $λ_N$ in terms of $χ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_09160 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | At the end of the spectrum: Chromatic bounds for the largest eigenvalue of the normalized Laplacian Beers, Lies Mulas, Raffaella Combinatorics Spectral Theory For a graph with largest normalized Laplacian eigenvalue $λ_N$ and (vertex) coloring number $χ$, it is known that $λ_N\geq χ/(χ-1)$. Here we prove properties of graphs for which this bound is sharp, and we study the multiplicity of $χ/(χ-1)$. We then describe a family of graphs with largest eigenvalue $χ/(χ-1)$. We also study the spectrum of the $1$-sum of two graphs (also known as graph joining or coalescing), with a focus on the maximal eigenvalue. Finally, we give upper bounds on $λ_N$ in terms of $χ$. |
| title | At the end of the spectrum: Chromatic bounds for the largest eigenvalue of the normalized Laplacian |
| topic | Combinatorics Spectral Theory |
| url | https://arxiv.org/abs/2402.09160 |