Spectral graph theory is a vast and expanding area of combinatorics. We start these notes by introducing and motivating classical matrices associated with a graph, and then show how to derive combinatorial properties of a graph from the eigenvalues of these matrices. We then examine more modern results such as polynomial interlacing and high dimensional expanders
{"title":"Spectral graph theory","authors":"Amol Sahebrao Hinge","doi":"10.1090/cbms/129/05","DOIUrl":"https://doi.org/10.1090/cbms/129/05","url":null,"abstract":"Spectral graph theory is a vast and expanding area of combinatorics. We start these notes by introducing and motivating classical matrices associated with a graph, and then show how to derive combinatorial properties of a graph from the eigenvalues of these matrices. We then examine more modern results such as polynomial interlacing and high dimensional expanders","PeriodicalId":440287,"journal":{"name":"Zeta and 𝐿-functions in Number Theory and\n Combinatorics","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130096951","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"𝐿-functions in geometry","authors":"","doi":"10.1090/cbms/129/03","DOIUrl":"https://doi.org/10.1090/cbms/129/03","url":null,"abstract":"","PeriodicalId":440287,"journal":{"name":"Zeta and 𝐿-functions in Number Theory and\n Combinatorics","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115908672","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Number theoretic zeta and 𝐿-functions","authors":"","doi":"10.1090/cbms/129/01","DOIUrl":"https://doi.org/10.1090/cbms/129/01","url":null,"abstract":"","PeriodicalId":440287,"journal":{"name":"Zeta and 𝐿-functions in Number Theory and\n Combinatorics","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115262906","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Explicit constructions of Ramanujan\u0000 graphs","authors":"","doi":"10.1090/cbms/129/06","DOIUrl":"https://doi.org/10.1090/cbms/129/06","url":null,"abstract":"","PeriodicalId":440287,"journal":{"name":"Zeta and 𝐿-functions in Number Theory and\n Combinatorics","volume":"87 1-3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123637778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Selberg zeta function","authors":"D. Hejhal","doi":"10.1007/BFB0079610","DOIUrl":"https://doi.org/10.1007/BFB0079610","url":null,"abstract":"","PeriodicalId":440287,"journal":{"name":"Zeta and 𝐿-functions in Number Theory and\n Combinatorics","volume":"200 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132943375","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we introduce the definition of the Ihara zeta function and its determinant formula. We provide a detailed proof of the determinant formula by using the edge zeta function. This paper is based on an article [1] written by Audrey A. Terras with the addition of details and a focus on the proof.
本文介绍了Ihara zeta函数的定义及其行列式。利用边zeta函数给出了行列式的详细证明。本文基于Audrey a . Terras撰写的一篇文章b[1]，并增加了细节和对证明的关注。
{"title":"The Ihara zeta function","authors":"Tongyu Yang, A. Terras","doi":"10.1090/cbms/129/04","DOIUrl":"https://doi.org/10.1090/cbms/129/04","url":null,"abstract":"In this paper, we introduce the definition of the Ihara zeta function and its determinant formula. We provide a detailed proof of the determinant formula by using the edge zeta function. This paper is based on an article [1] written by Audrey A. Terras with the addition of details and a focus on the proof.","PeriodicalId":440287,"journal":{"name":"Zeta and 𝐿-functions in Number Theory and\n Combinatorics","volume":"89 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134339368","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Zeta and 𝐿-functions of complexes","authors":"","doi":"10.1090/cbms/129/08","DOIUrl":"https://doi.org/10.1090/cbms/129/08","url":null,"abstract":"","PeriodicalId":440287,"journal":{"name":"Zeta and 𝐿-functions in Number Theory and\n Combinatorics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129119303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Artin 𝐿-functions and prime distributions for\u0000 graphs","authors":"","doi":"10.1090/cbms/129/07","DOIUrl":"https://doi.org/10.1090/cbms/129/07","url":null,"abstract":"","PeriodicalId":440287,"journal":{"name":"Zeta and 𝐿-functions in Number Theory and\n Combinatorics","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128394900","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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