{"id":"https://openalex.org/W4417126216","doi":"https://doi.org/10.3390/axioms14120904","title":"Singularity Conditions and Probabilities of Tricyclic Graphs","display_name":"Singularity Conditions and Probabilities of Tricyclic Graphs","publication_year":2025,"publication_date":"2025-12-08","ids":{"openalex":"https://openalex.org/W4417126216","doi":"https://doi.org/10.3390/axioms14120904"},"language":"en","primary_location":{"id":"doi:10.3390/axioms14120904","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14120904","pdf_url":"https://www.mdpi.com/2075-1680/14/12/904/pdf?version=1765196684","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2075-1680/14/12/904/pdf?version=1765196684","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5050946162","display_name":"Haicheng Ma","orcid":"https://orcid.org/0000-0002-5503-4668"},"institutions":[{"id":"https://openalex.org/I116265982","display_name":"Qinghai University","ror":"https://ror.org/05h33bt13","country_code":"CN","type":"education","lineage":["https://openalex.org/I116265982"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Haicheng Ma","raw_affiliation_strings":["College of General Education, Guangdong University of Science and Technology, Dongguan 523083, China","School of Mathematics and Statistics, Qinghai Minzu University, Xining 810007, China"],"raw_orcid":"https://orcid.org/0000-0002-5503-4668","affiliations":[{"raw_affiliation_string":"College of General Education, Guangdong University of Science and Technology, Dongguan 523083, China","institution_ids":[]},{"raw_affiliation_string":"School of Mathematics and Statistics, Qinghai Minzu University, Xining 810007, China","institution_ids":["https://openalex.org/I116265982"]}]},{"author_position":"last","author":{"id":null,"display_name":"Xianghong He","orcid":"https://orcid.org/0009-0002-0131-2834"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Xianghong He","raw_affiliation_strings":["College of General Education, Guangdong University of Science and Technology, Dongguan 523083, China"],"raw_orcid":"https://orcid.org/0009-0002-0131-2834","affiliations":[{"raw_affiliation_string":"College of General Education, Guangdong University of Science and Technology, Dongguan 523083, China","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":{"value":1600,"currency":"CHF","value_usd":1782},"apc_paid":{"value":1600,"currency":"CHF","value_usd":1782},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.42973215,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"14","issue":"12","first_page":"904","last_page":"904"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.9805999994277954,"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":0.9805999994277954,"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/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.0015999999595806003,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12303","display_name":"Tensor decomposition and applications","score":0.0012000000569969416,"subfield":{"id":"https://openalex.org/subfields/2605","display_name":"Computational Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/adjacency-matrix","display_name":"Adjacency matrix","score":0.7732999920845032},{"id":"https://openalex.org/keywords/tricyclic","display_name":"Tricyclic","score":0.7555000185966492},{"id":"https://openalex.org/keywords/singularity","display_name":"Singularity","score":0.7498000264167786},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.5335999727249146},{"id":"https://openalex.org/keywords/adjacency-list","display_name":"Adjacency list","score":0.5196999907493591},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.40790000557899475},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.35269999504089355}],"concepts":[{"id":"https://openalex.org/C180356752","wikidata":"https://www.wikidata.org/wiki/Q727035","display_name":"Adjacency matrix","level":3,"score":0.7732999920845032},{"id":"https://openalex.org/C2781081902","wikidata":"https://www.wikidata.org/wiki/Q3539074","display_name":"Tricyclic","level":2,"score":0.7555000185966492},{"id":"https://openalex.org/C16171025","wikidata":"https://www.wikidata.org/wiki/Q863349","display_name":"Singularity","level":2,"score":0.7498000264167786},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6173999905586243},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5849999785423279},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.5335999727249146},{"id":"https://openalex.org/C110484373","wikidata":"https://www.wikidata.org/wiki/Q264398","display_name":"Adjacency list","level":2,"score":0.5196999907493591},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.43700000643730164},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.40790000557899475},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.35269999504089355},{"id":"https://openalex.org/C159867001","wikidata":"https://www.wikidata.org/wiki/Q904942","display_name":"Topological index","level":2,"score":0.31520000100135803},{"id":"https://openalex.org/C74133993","wikidata":"https://www.wikidata.org/wiki/Q3115472","display_name":"Indifference graph","level":3,"score":0.3131999969482422},{"id":"https://openalex.org/C146380142","wikidata":"https://www.wikidata.org/wiki/Q1137726","display_name":"Directed graph","level":2,"score":0.3127000033855438},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.3061000108718872},{"id":"https://openalex.org/C78913703","wikidata":"https://www.wikidata.org/wiki/Q5597087","display_name":"Graph energy","level":5,"score":0.2768000066280365},{"id":"https://openalex.org/C68103157","wikidata":"https://www.wikidata.org/wiki/Q569347","display_name":"Graph product","level":5,"score":0.26910001039505005},{"id":"https://openalex.org/C12843","wikidata":"https://www.wikidata.org/wiki/Q201721","display_name":"Gravitational singularity","level":2,"score":0.2596000134944916},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.25690001249313354}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/axioms14120904","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14120904","pdf_url":"https://www.mdpi.com/2075-1680/14/12/904/pdf?version=1765196684","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:46a7deb0032a456881e3901cab9d2e88","is_oa":true,"landing_page_url":"https://doaj.org/article/46a7deb0032a456881e3901cab9d2e88","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms, Vol 14, Iss 12, p 904 (2025)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/axioms14120904","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14120904","pdf_url":"https://www.mdpi.com/2075-1680/14/12/904/pdf?version=1765196684","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G4779907239","display_name":null,"funder_award_id":"2022-ZJ-924","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G6338983319","display_name":null,"funder_award_id":"GKY-2022BSQD-36","funder_id":"https://openalex.org/F4320318654","funder_display_name":"Guangdong University of Science and Technology"},{"id":"https://openalex.org/G8059353764","display_name":null,"funder_award_id":"GKY-2022CQTD-5","funder_id":"https://openalex.org/F4320318654","funder_display_name":"Guangdong University of Science and Technology"},{"id":"https://openalex.org/G8737091757","display_name":"\u56fe\u7684\u7ebf\u6027\u53c2\u6570\u548c\u5339\u914d\u591a\u9879\u5f0f\u7814\u7a76","funder_award_id":"11561056","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320314997","display_name":"Strong","ror":"https://ror.org/041vyzr56"},{"id":"https://openalex.org/F4320318654","display_name":"Guangdong University of Science and Technology","ror":null},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4417126216.pdf","grobid_xml":"https://content.openalex.org/works/W4417126216.grobid-xml"},"referenced_works_count":28,"referenced_works":["https://openalex.org/W154962059","https://openalex.org/W1590837392","https://openalex.org/W1975006923","https://openalex.org/W1976956961","https://openalex.org/W1978707241","https://openalex.org/W1995366886","https://openalex.org/W1997431816","https://openalex.org/W2010453253","https://openalex.org/W2043661920","https://openalex.org/W2053883806","https://openalex.org/W2084962260","https://openalex.org/W2109556613","https://openalex.org/W2493319874","https://openalex.org/W2918013612","https://openalex.org/W2963028288","https://openalex.org/W2972451988","https://openalex.org/W3024522820","https://openalex.org/W3188648783","https://openalex.org/W4200376908","https://openalex.org/W4205676416","https://openalex.org/W4212897146","https://openalex.org/W4221030474","https://openalex.org/W4310213905","https://openalex.org/W4319738587","https://openalex.org/W4390233571","https://openalex.org/W4403736364","https://openalex.org/W4405921562","https://openalex.org/W4411499345"],"related_works":[],"abstract_inverted_index":{"A":[0],"graph":[1],"G":[2],"is":[3,12],"said":[4],"to":[5],"be":[6],"singular":[7,14,68],"if":[8],"its":[9],"adjacency":[10,42],"matrix":[11],"a":[13],"matrix.":[15],"In":[16],"this":[17],"paper,":[18],"by":[19],"analyzing":[20],"the":[21,38,46,52,65],"structure":[22],"of":[23,29,40,54,58,67],"Sachs":[24],"subgraphs":[25],"on":[26],"three":[27,56],"types":[28,57],"tricyclic":[30,59,71],"graphs":[31,69,72],"\u03d6(a1,a2,b1,b2,c1,c2),":[32,73],"\u03c3(a1,a2,b1,b2,c),":[33,74],"and":[34,36,48,75,79],"\u03c4(a1,a2,a3,a4)":[35,76],"calculating":[37],"determinants":[39],"their":[41],"matrices,":[43],"we":[44,62],"investigate":[45],"necessary":[47],"sufficient":[49],"conditions":[50],"for":[51],"singularity":[53],"these":[55],"graphs.":[60],"Furthermore,":[61],"derive":[63],"that":[64],"probabilities":[66],"among":[70],"are":[77],"217512,113256,":[78],"67128,":[80],"respectively.":[81]},"counts_by_year":[],"updated_date":"2026-07-23T08:03:31.855105","created_date":"2025-12-08T00:00:00"}
