{"title":"Bounding the Diameter and Eigenvalues of Amply Regular Graphs via Lin–Lu–Yau Curvature","authors":"Xueping Huang, Shiping Liu, Qing Xia","doi":"10.1007/s00493-024-00113-3","DOIUrl":null,"url":null,"abstract":"<p>An amply regular graph is a regular graph such that any two adjacent vertices have <span>\\(\\alpha \\)</span> common neighbors and any two vertices with distance 2 have <span>\\(\\beta \\)</span> common neighbors. We prove a sharp lower bound estimate for the Lin–Lu–Yau curvature of any amply regular graph with girth 3 and <span>\\(\\beta >\\alpha \\)</span>. 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.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Combinatorica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00493-024-00113-3","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
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.
期刊介绍:
COMBINATORICA publishes research papers in English in a variety of areas of combinatorics and the theory of computing, with particular emphasis on general techniques and unifying principles. Typical but not exclusive topics covered by COMBINATORICA are
- Combinatorial structures (graphs, hypergraphs, matroids, designs, permutation groups).
- Combinatorial optimization.
- Combinatorial aspects of geometry and number theory.
- Algorithms in combinatorics and related fields.
- Computational complexity theory.
- Randomization and explicit construction in combinatorics and algorithms.