Bounding the diameter and eigenvalues of amply regular graphs via Lin-Lu-Yau curvature
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866916282117390336 |
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| author | Huang, Xueping Liu, Shiping Xia, Qing |
| author_facet | Huang, Xueping Liu, Shiping Xia, Qing |
| contents | An amply regular graph is a regular graph such that any two adjacent vertices have $α$ common neighbors and any two vertices with distance $2$ have $β$ common neighbors. We prove a sharp lower bound estimate for the Lin--Lu--Yau curvature of any amply regular graph with girth $3$ and $β>α$. The proof involves new ideas relating discrete Ricci curvature with local matching properties: This includes a novel construction of a regular bipartite graph from the local structure and related distance estimates. As a consequence, we obtain sharp diameter and eigenvalue bounds for amply regular graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_02988 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Bounding the diameter and eigenvalues of amply regular graphs via Lin-Lu-Yau curvature Huang, Xueping Liu, Shiping Xia, Qing Combinatorics Differential Geometry An amply regular graph is a regular graph such that any two adjacent vertices have $α$ common neighbors and any two vertices with distance $2$ have $β$ common neighbors. We prove a sharp lower bound estimate for the Lin--Lu--Yau curvature of any amply regular graph with girth $3$ and $β>α$. The proof involves new ideas relating discrete Ricci curvature with local matching properties: This includes a novel construction of a regular bipartite graph from the local structure and related distance estimates. As a consequence, we obtain sharp diameter and eigenvalue bounds for amply regular graphs. |
| title | Bounding the diameter and eigenvalues of amply regular graphs via Lin-Lu-Yau curvature |
| topic | Combinatorics Differential Geometry |
| url | https://arxiv.org/abs/2210.02988 |