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arXiv · 2210.02988

Bounding the diameter and eigenvalues of amply regular graphs via Lin-Lu-Yau curvature

Xueping Huang·Shiping Liu·Qing Xia
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Abstract

An amply regular graph is a regular graph such that any two adjacent vertices have $\alpha$ common neighbors and any two vertices with distance $2$ have $\beta$ common neighbors. We prove a sharp lower bound estimate for the Lin--Lu--Yau curvature of any amply regular graph with girth $3$ and $\beta>\alpha$. 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.

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Xueping Huang, Shiping Liu, Qing Xia. 2022-10-06. Bounding the diameter and eigenvalues of amply regular graphs via Lin-Lu-Yau curvature. https://arxiv.org/abs/2210.02988

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