{"id":"https://openalex.org/W4362669394","doi":"https://doi.org/10.3390/sym15040871","title":"Ordering Unicyclic Connected Graphs with Girth g \u2265 3 Having Greatest SK Indices","display_name":"Ordering Unicyclic Connected Graphs with Girth g \u2265 3 Having Greatest SK Indices","publication_year":2023,"publication_date":"2023-04-06","ids":{"openalex":"https://openalex.org/W4362669394","doi":"https://doi.org/10.3390/sym15040871"},"language":"en","primary_location":{"id":"doi:10.3390/sym15040871","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15040871","pdf_url":"https://www.mdpi.com/2073-8994/15/4/871/pdf?version=1681117825","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/15/4/871/pdf?version=1681117825","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100460917","display_name":"Hui Wang","orcid":"https://orcid.org/0000-0003-2633-6015"},"institutions":[{"id":"https://openalex.org/I4210156464","display_name":"Anhui Xinhua University","ror":"https://ror.org/053d7x641","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210156464"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wang Hui","raw_affiliation_strings":["General Education Department, Anhui Xinhua University, Hefei 230088, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"General Education Department, Anhui Xinhua University, Hefei 230088, China","institution_ids":["https://openalex.org/I4210156464"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5024186062","display_name":"Adnan Aslam","orcid":"https://orcid.org/0000-0003-4523-8023"},"institutions":[{"id":"https://openalex.org/I142732210","display_name":"University of Engineering and Technology Lahore","ror":"https://ror.org/0051w2v06","country_code":"PK","type":"education","lineage":["https://openalex.org/I142732210"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Adnan Aslam","raw_affiliation_strings":["Department of Natural Sciences and Humanities, University of Engineering and Technology, Lahore 54000, Pakistan"],"raw_orcid":"https://orcid.org/0000-0003-4523-8023","affiliations":[{"raw_affiliation_string":"Department of Natural Sciences and Humanities, University of Engineering and Technology, Lahore 54000, Pakistan","institution_ids":["https://openalex.org/I142732210"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5077340518","display_name":"Salma Kanwal","orcid":"https://orcid.org/0000-0002-7560-5317"},"institutions":[{"id":"https://openalex.org/I161237661","display_name":"Lahore College for Women University","ror":"https://ror.org/02bf6br77","country_code":"PK","type":"education","lineage":["https://openalex.org/I161237661"]}],"countries":["PK"],"is_corresponding":true,"raw_author_name":"Salma Kanwal","raw_affiliation_strings":["Department of Mathematics, Lahore College for Women University, Lahore 54000, Pakistan"],"raw_orcid":"https://orcid.org/0000-0002-7560-5317","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Lahore College for Women University, Lahore 54000, Pakistan","institution_ids":["https://openalex.org/I161237661"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5089170808","display_name":"Mahnoor Akram","orcid":null},"institutions":[{"id":"https://openalex.org/I161237661","display_name":"Lahore College for Women University","ror":"https://ror.org/02bf6br77","country_code":"PK","type":"education","lineage":["https://openalex.org/I161237661"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Mahnoor Akram","raw_affiliation_strings":["Department of Mathematics, Lahore College for Women University, Lahore 54000, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Lahore College for Women University, Lahore 54000, Pakistan","institution_ids":["https://openalex.org/I161237661"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5111126611","display_name":"Tahira Sumbal Shaikh","orcid":null},"institutions":[{"id":"https://openalex.org/I161237661","display_name":"Lahore College for Women University","ror":"https://ror.org/02bf6br77","country_code":"PK","type":"education","lineage":["https://openalex.org/I161237661"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Tahira Sumbal Shaikh","raw_affiliation_strings":["Department of Mathematics, Lahore College for Women University, Lahore 54000, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Lahore College for Women University, Lahore 54000, Pakistan","institution_ids":["https://openalex.org/I161237661"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5080609106","display_name":"Xuewu Zuo","orcid":"https://orcid.org/0000-0002-5804-5370"},"institutions":[{"id":"https://openalex.org/I4210156464","display_name":"Anhui Xinhua University","ror":"https://ror.org/053d7x641","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210156464"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xuewu Zuo","raw_affiliation_strings":["General Education Department, Anhui Xinhua University, Hefei 230088, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"General Education Department, Anhui Xinhua University, Hefei 230088, China","institution_ids":["https://openalex.org/I4210156464"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5077340518"],"corresponding_institution_ids":["https://openalex.org/I161237661"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.5381,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.57735931,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"15","issue":"4","first_page":"871","last_page":"871"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":1.0,"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":1.0,"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/T11298","display_name":"Synthesis and Properties of Aromatic Compounds","score":0.9945999979972839,"subfield":{"id":"https://openalex.org/subfields/1605","display_name":"Organic Chemistry"},"field":{"id":"https://openalex.org/fields/16","display_name":"Chemistry"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.98089998960495,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8240270018577576},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7556730508804321},{"id":"https://openalex.org/keywords/girth","display_name":"Girth (graph theory)","score":0.6519305109977722},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.5463578701019287},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.41449517011642456}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8240270018577576},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7556730508804321},{"id":"https://openalex.org/C2777434295","wikidata":"https://www.wikidata.org/wiki/Q959831","display_name":"Girth (graph theory)","level":2,"score":0.6519305109977722},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.5463578701019287},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.41449517011642456}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym15040871","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15040871","pdf_url":"https://www.mdpi.com/2073-8994/15/4/871/pdf?version=1681117825","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:9a329e759b114c9381e54a4280045542","is_oa":false,"landing_page_url":"https://doaj.org/article/9a329e759b114c9381e54a4280045542","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 15, Iss 4, p 871 (2023)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/15/4/871/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym15040871","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym15040871","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15040871","pdf_url":"https://www.mdpi.com/2073-8994/15/4/871/pdf?version=1681117825","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"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":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G4821398558","display_name":null,"funder_award_id":"KJ2021A1154","funder_id":"https://openalex.org/F4320334897","funder_display_name":"Natural Science Foundation of Anhui Province"}],"funders":[{"id":"https://openalex.org/F4320334897","display_name":"Natural Science Foundation of Anhui Province","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4362669394.pdf","grobid_xml":"https://content.openalex.org/works/W4362669394.grobid-xml"},"referenced_works_count":21,"referenced_works":["https://openalex.org/W1983169983","https://openalex.org/W1992105048","https://openalex.org/W2008281970","https://openalex.org/W2044591523","https://openalex.org/W2045993700","https://openalex.org/W2065534534","https://openalex.org/W2077669295","https://openalex.org/W2325277747","https://openalex.org/W2527007236","https://openalex.org/W2601334426","https://openalex.org/W2737121322","https://openalex.org/W2886772203","https://openalex.org/W3037200217","https://openalex.org/W3134775138","https://openalex.org/W3160388010","https://openalex.org/W3161475662","https://openalex.org/W3207039476","https://openalex.org/W6741012158","https://openalex.org/W6779904230","https://openalex.org/W6795271121","https://openalex.org/W6802616226"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2077600819","https://openalex.org/W2142036596","https://openalex.org/W2072657027","https://openalex.org/W2962838298","https://openalex.org/W2600246793"],"abstract_inverted_index":{"For":[0],"a":[1,61,110],"graph,":[2],"the":[3,9,12,15,18,20,26,29,32,35,38,44,47,50,53,56,67,83,103,106],"SK":[4,69,88],"index":[5,22,40],"is":[6,23,41,102],"equal":[7,24,42],"to":[8,25,43,66],"half":[10,27,45],"of":[11,14,17,28,31,34,46,49,52,55,105],"sum":[13,51],"degrees":[16,33,54],"vertices,":[19,36],"SK1":[21],"product":[30],"and":[37,63,78,86,95],"SK2":[39],"square":[48],"vertices.":[57],"This":[58],"paper":[59],"shows":[60],"simple":[62],"unified":[64],"approach":[65],"greatest":[68],"indices":[70,89],"for":[71],"unicyclic":[72],"graphs":[73,81],"by":[74],"using":[75],"some":[76],"transformations":[77],"characterizes":[79],"these":[80],"with":[82],"first,":[84],"second,":[85],"third":[87],"having":[90],"order":[91],"r":[92],"\u2265":[93,98],"5":[94],"girth":[96,101],"g":[97],"3,":[99],"where":[100],"length":[104],"shortest":[107],"cycle":[108],"in":[109],"graph.":[111]},"counts_by_year":[{"year":2024,"cited_by_count":1}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2025-10-10T00:00:00"}