{"id":"https://openalex.org/W1487787302","doi":"https://doi.org/10.37236/2390","title":"Spectral Properties of Unitary Cayley Graphs of Finite Commutative Rings","display_name":"Spectral Properties of Unitary Cayley Graphs of Finite Commutative Rings","publication_year":2012,"publication_date":"2012-10-25","ids":{"openalex":"https://openalex.org/W1487787302","doi":"https://doi.org/10.37236/2390","mag":"1487787302"},"language":"en","primary_location":{"id":"doi:10.37236/2390","is_oa":true,"landing_page_url":"https://doi.org/10.37236/2390","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v19i4p13/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v19i4p13/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5088563078","display_name":"Xiaogang Liu","orcid":"https://orcid.org/0000-0002-0266-9774"},"institutions":[{"id":"https://openalex.org/I165779595","display_name":"University of Melbourne","ror":"https://ror.org/01ej9dk98","country_code":"AU","type":"education","lineage":["https://openalex.org/I165779595"]}],"countries":["AU"],"is_corresponding":true,"raw_author_name":"Xiaogang Liu","raw_affiliation_strings":["The University of Melbourne","The University of Melbourne,"],"affiliations":[{"raw_affiliation_string":"The University of Melbourne","institution_ids":["https://openalex.org/I165779595"]},{"raw_affiliation_string":"The University of Melbourne,","institution_ids":["https://openalex.org/I165779595"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5077967587","display_name":"Sanming Zhou","orcid":"https://orcid.org/0000-0001-9854-6076"},"institutions":[{"id":"https://openalex.org/I165779595","display_name":"University of Melbourne","ror":"https://ror.org/01ej9dk98","country_code":"AU","type":"education","lineage":["https://openalex.org/I165779595"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Sanming Zhou","raw_affiliation_strings":["The University of Melbourne","The University of Melbourne,"],"affiliations":[{"raw_affiliation_string":"The University of Melbourne","institution_ids":["https://openalex.org/I165779595"]},{"raw_affiliation_string":"The University of Melbourne,","institution_ids":["https://openalex.org/I165779595"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5088563078"],"corresponding_institution_ids":["https://openalex.org/I165779595"],"apc_list":null,"apc_paid":null,"fwci":2.0957,"has_fulltext":true,"cited_by_count":36,"citation_normalized_percentile":{"value":0.84667668,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":98},"biblio":{"volume":"19","issue":"4","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10849","display_name":"Finite Group Theory Research","score":0.9908999800682068,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11809","display_name":"Advanced NMR Techniques and Applications","score":0.9606999754905701,"subfield":{"id":"https://openalex.org/subfields/1607","display_name":"Spectroscopy"},"field":{"id":"https://openalex.org/fields/16","display_name":"Chemistry"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8925693035125732},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.809035062789917},{"id":"https://openalex.org/keywords/ramanujans-sum","display_name":"Ramanujan's sum","score":0.757312536239624},{"id":"https://openalex.org/keywords/cayley-graph","display_name":"Cayley graph","score":0.7209672331809998},{"id":"https://openalex.org/keywords/commutative-ring","display_name":"Commutative ring","score":0.5483631491661072},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.5273557305335999},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.48612356185913086},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4317959249019623},{"id":"https://openalex.org/keywords/voltage-graph","display_name":"Voltage graph","score":0.42715728282928467},{"id":"https://openalex.org/keywords/vertex-transitive-graph","display_name":"Vertex-transitive graph","score":0.41387200355529785},{"id":"https://openalex.org/keywords/commutative-property","display_name":"Commutative property","score":0.36197134852409363},{"id":"https://openalex.org/keywords/line-graph","display_name":"Line graph","score":0.353066086769104}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8925693035125732},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.809035062789917},{"id":"https://openalex.org/C131220774","wikidata":"https://www.wikidata.org/wiki/Q1441456","display_name":"Ramanujan's sum","level":2,"score":0.757312536239624},{"id":"https://openalex.org/C120204988","wikidata":"https://www.wikidata.org/wiki/Q859174","display_name":"Cayley graph","level":3,"score":0.7209672331809998},{"id":"https://openalex.org/C161491579","wikidata":"https://www.wikidata.org/wiki/Q858656","display_name":"Commutative ring","level":3,"score":0.5483631491661072},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.5273557305335999},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.48612356185913086},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4317959249019623},{"id":"https://openalex.org/C22149727","wikidata":"https://www.wikidata.org/wiki/Q7940747","display_name":"Voltage graph","level":4,"score":0.42715728282928467},{"id":"https://openalex.org/C113203676","wikidata":"https://www.wikidata.org/wiki/Q220997","display_name":"Vertex-transitive graph","level":5,"score":0.41387200355529785},{"id":"https://openalex.org/C183778304","wikidata":"https://www.wikidata.org/wiki/Q165474","display_name":"Commutative property","level":2,"score":0.36197134852409363},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.353066086769104}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.37236/2390","is_oa":true,"landing_page_url":"https://doi.org/10.37236/2390","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v19i4p13/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.358.518","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.358.518","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.combinatorics.org/ojs/index.php/eljc/article/viewFile/v19i4p13/pdf/","raw_type":"text"}],"best_oa_location":{"id":"doi:10.37236/2390","is_oa":true,"landing_page_url":"https://doi.org/10.37236/2390","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v19i4p13/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2190904872","display_name":null,"funder_award_id":"MIFRS","funder_id":"https://openalex.org/F4320320974","funder_display_name":"University of Melbourne"},{"id":"https://openalex.org/G5714184577","display_name":null,"funder_award_id":"Fellowship","funder_id":"https://openalex.org/F4320334704","funder_display_name":"Australian Research Council"},{"id":"https://openalex.org/G7402294210","display_name":null,"funder_award_id":"FT110100629","funder_id":"https://openalex.org/F4320334704","funder_display_name":"Australian Research Council"}],"funders":[{"id":"https://openalex.org/F4320320974","display_name":"University of Melbourne","ror":"https://ror.org/01ej9dk98"},{"id":"https://openalex.org/F4320334704","display_name":"Australian Research Council","ror":"https://ror.org/05mmh0f86"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1487787302.pdf","grobid_xml":"https://content.openalex.org/works/W1487787302.grobid-xml"},"referenced_works_count":43,"referenced_works":["https://openalex.org/W247697463","https://openalex.org/W593236292","https://openalex.org/W1479863711","https://openalex.org/W1530740238","https://openalex.org/W1534540780","https://openalex.org/W1546802965","https://openalex.org/W1927695923","https://openalex.org/W1964294838","https://openalex.org/W1964824028","https://openalex.org/W1972151784","https://openalex.org/W1998961442","https://openalex.org/W2009876987","https://openalex.org/W2018047324","https://openalex.org/W2025453683","https://openalex.org/W2025864448","https://openalex.org/W2049707341","https://openalex.org/W2067964978","https://openalex.org/W2075838781","https://openalex.org/W2090425189","https://openalex.org/W2099526591","https://openalex.org/W2118545174","https://openalex.org/W2131467982","https://openalex.org/W2149397526","https://openalex.org/W2161108572","https://openalex.org/W2525209814","https://openalex.org/W2742091132","https://openalex.org/W2986001730","https://openalex.org/W3000216555","https://openalex.org/W3148190347","https://openalex.org/W4237658155","https://openalex.org/W4238591275","https://openalex.org/W4285719527","https://openalex.org/W4296292757","https://openalex.org/W6609457598","https://openalex.org/W6640230470","https://openalex.org/W6641285925","https://openalex.org/W6655018717","https://openalex.org/W6656719795","https://openalex.org/W6662980539","https://openalex.org/W6673154547","https://openalex.org/W6675409862","https://openalex.org/W6683773253","https://openalex.org/W6779915889"],"related_works":["https://openalex.org/W3024284632","https://openalex.org/W2377531823","https://openalex.org/W2084790024","https://openalex.org/W4375831034","https://openalex.org/W156501607","https://openalex.org/W2373941240","https://openalex.org/W2385269323","https://openalex.org/W2380035736","https://openalex.org/W2362129634","https://openalex.org/W2072785941"],"abstract_inverted_index":{"Let":[0],"$R$":[1,21],"be":[2,73,87],"a":[3,65,76],"finite":[4],"commutative":[5],"ring.":[6],"The":[7],"unitary":[8],"Cayley":[9],"graph":[10,17,40,97],"of":[11,34,36,47,50,84,94,98,105],"$R$,":[12],"denoted":[13],"$G_R$,":[14,99],"is":[15,31,41,56],"the":[16,32,44,82,92,95,102],"with":[18],"vertex":[19],"set":[20,24,33],"and":[22,67,75,78,100,107],"edge":[23],"$\\left\\{\\{a,b\\}:a,b\\in":[25],"R,":[26],"a-b\\in":[27],"R^\\times\\right\\}$,":[28],"where":[29],"$R^\\times$":[30],"units":[35],"$R$.":[37],"An":[38],"$r$-regular":[39],"Ramanujan":[42],"if":[43],"absolute":[45],"value":[46],"every":[48],"eigenvalue":[49],"it":[51],"other":[52],"than":[53],"$\\pm":[54],"r$":[55],"at":[57],"most":[58],"$2\\sqrt{r-1}$.":[59],"In":[60],"this":[61],"paper":[62],"we":[63],"give":[64],"necessary":[66,77],"sufficient":[68,79],"condition":[69,80],"for":[70,81],"$G_R$":[71,85,106],"to":[72,86],"Ramanujan,":[74],"complement":[83],"Ramanujan.":[88],"We":[89],"also":[90],"determine":[91],"energy":[93],"line":[96,109],"compute":[101],"spectral":[103],"moments":[104],"its":[108],"graph.":[110]},"counts_by_year":[{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":5},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":5},{"year":2015,"cited_by_count":2},{"year":2013,"cited_by_count":3}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}