{"id":"https://openalex.org/W4405737831","doi":"https://doi.org/10.3390/sym17010005","title":"On Local Fractional Topological Indices and Entropies for Hyper-Chordal Ring Networks Using Local Fractional Metric Dimension","display_name":"On Local Fractional Topological Indices and Entropies for Hyper-Chordal Ring Networks Using Local Fractional Metric Dimension","publication_year":2024,"publication_date":"2024-12-24","ids":{"openalex":"https://openalex.org/W4405737831","doi":"https://doi.org/10.3390/sym17010005"},"language":"en","primary_location":{"id":"doi:10.3390/sym17010005","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17010005","pdf_url":"https://www.mdpi.com/2073-8994/17/1/5/pdf?version=1735052145","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/2073-8994/17/1/5/pdf?version=1735052145","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5082532885","display_name":"Shahzad Ali","orcid":"https://orcid.org/0000-0002-1302-1316"},"institutions":[{"id":"https://openalex.org/I174731842","display_name":"Islamia University of Bahawalpur","ror":"https://ror.org/002rc4w13","country_code":"PK","type":"education","lineage":["https://openalex.org/I174731842"]},{"id":"https://openalex.org/I4210102737","display_name":"Khwaja Fareed University of Engineering and Information Technology","ror":"https://ror.org/0161dyt30","country_code":"PK","type":"education","lineage":["https://openalex.org/I4210102737"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Shahzad Ali","raw_affiliation_strings":["Department of Mathematics, The Islamia University of Bahawalpur, Rahim Yar Kahn Campus, Rahim Yar Khan 64200, Pakistan","Institute of Mathematics, Khwaja Fareed University of Engineering and Information Technology, Rahim Yar Khan 64200, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, The Islamia University of Bahawalpur, Rahim Yar Kahn Campus, Rahim Yar Khan 64200, Pakistan","institution_ids":["https://openalex.org/I174731842"]},{"raw_affiliation_string":"Institute of Mathematics, Khwaja Fareed University of Engineering and Information Technology, Rahim Yar Khan 64200, Pakistan","institution_ids":["https://openalex.org/I4210102737"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5065100059","display_name":"Shahzaib Ashraf","orcid":"https://orcid.org/0000-0002-8616-8829"},"institutions":[{"id":"https://openalex.org/I4210102737","display_name":"Khwaja Fareed University of Engineering and Information Technology","ror":"https://ror.org/0161dyt30","country_code":"PK","type":"education","lineage":["https://openalex.org/I4210102737"]}],"countries":["PK"],"is_corresponding":true,"raw_author_name":"Shahzaib Ashraf","raw_affiliation_strings":["Institute of Mathematics, Khwaja Fareed University of Engineering and Information Technology, Rahim Yar Khan 64200, Pakistan"],"raw_orcid":"https://orcid.org/0000-0002-8616-8829","affiliations":[{"raw_affiliation_string":"Institute of Mathematics, Khwaja Fareed University of Engineering and Information Technology, Rahim Yar Khan 64200, Pakistan","institution_ids":["https://openalex.org/I4210102737"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5058791499","display_name":"Shahbaz Ali","orcid":"https://orcid.org/0000-0002-5998-0053"},"institutions":[{"id":"https://openalex.org/I174731842","display_name":"Islamia University of Bahawalpur","ror":"https://ror.org/002rc4w13","country_code":"PK","type":"education","lineage":["https://openalex.org/I174731842"]},{"id":"https://openalex.org/I4210102737","display_name":"Khwaja Fareed University of Engineering and Information Technology","ror":"https://ror.org/0161dyt30","country_code":"PK","type":"education","lineage":["https://openalex.org/I4210102737"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Shahbaz Ali","raw_affiliation_strings":["Department of Mathematics, The Islamia University of Bahawalpur, Rahim Yar Kahn Campus, Rahim Yar Khan 64200, Pakistan","Institute of Mathematics, Khwaja Fareed University of Engineering and Information Technology, Rahim Yar Khan 64200, Pakistan"],"raw_orcid":"https://orcid.org/0000-0002-5998-0053","affiliations":[{"raw_affiliation_string":"Department of Mathematics, The Islamia University of Bahawalpur, Rahim Yar Kahn Campus, Rahim Yar Khan 64200, Pakistan","institution_ids":["https://openalex.org/I174731842"]},{"raw_affiliation_string":"Institute of Mathematics, Khwaja Fareed University of Engineering and Information Technology, Rahim Yar Khan 64200, Pakistan","institution_ids":["https://openalex.org/I4210102737"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5115625799","display_name":"Abdullah Afzal","orcid":null},"institutions":[{"id":"https://openalex.org/I174731842","display_name":"Islamia University of Bahawalpur","ror":"https://ror.org/002rc4w13","country_code":"PK","type":"education","lineage":["https://openalex.org/I174731842"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Abdullah Afzal","raw_affiliation_strings":["Department of Mathematics, The Islamia University of Bahawalpur, Rahim Yar Kahn Campus, Rahim Yar Khan 64200, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, The Islamia University of Bahawalpur, Rahim Yar Kahn Campus, Rahim Yar Khan 64200, Pakistan","institution_ids":["https://openalex.org/I174731842"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5058870920","display_name":"Amal S. Alali","orcid":"https://orcid.org/0000-0001-7856-2861"},"institutions":[{"id":"https://openalex.org/I106778892","display_name":"Princess Nourah bint Abdulrahman University","ror":"https://ror.org/05b0cyh02","country_code":"SA","type":"education","lineage":["https://openalex.org/I106778892"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Amal S. Alali","raw_affiliation_strings":["Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"],"raw_orcid":"https://orcid.org/0000-0001-7856-2861","affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia","institution_ids":["https://openalex.org/I106778892"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5065100059"],"corresponding_institution_ids":["https://openalex.org/I4210102737"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.27690141,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"17","issue":"1","first_page":"5","last_page":"5"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.9987000226974487,"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"}},"topics":[{"id":"https://openalex.org/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.9987000226974487,"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/T11476","display_name":"Graph theory and applications","score":0.9955000281333923,"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/T13551","display_name":"Biochemical effects in animals","score":0.9485999941825867,"subfield":{"id":"https://openalex.org/subfields/2737","display_name":"Physiology"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/chordal-graph","display_name":"Chordal graph","score":0.7615641355514526},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7444776892662048},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.458219438791275},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4527079463005066},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.4490965008735657},{"id":"https://openalex.org/keywords/ring","display_name":"Ring (chemistry)","score":0.4351770579814911},{"id":"https://openalex.org/keywords/topology","display_name":"Topology (electrical circuits)","score":0.35703688859939575},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.34593430161476135}],"concepts":[{"id":"https://openalex.org/C160446614","wikidata":"https://www.wikidata.org/wiki/Q1322892","display_name":"Chordal graph","level":3,"score":0.7615641355514526},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7444776892662048},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.458219438791275},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4527079463005066},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.4490965008735657},{"id":"https://openalex.org/C2780378348","wikidata":"https://www.wikidata.org/wiki/Q25351438","display_name":"Ring (chemistry)","level":2,"score":0.4351770579814911},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.35703688859939575},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.34593430161476135},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C178790620","wikidata":"https://www.wikidata.org/wiki/Q11351","display_name":"Organic chemistry","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/sym17010005","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17010005","pdf_url":"https://www.mdpi.com/2073-8994/17/1/5/pdf?version=1735052145","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:b70c69ae302149dc8d2a30fe5e6bfd3d","is_oa":false,"landing_page_url":"https://doaj.org/article/b70c69ae302149dc8d2a30fe5e6bfd3d","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 17, Iss 1, p 5 (2024)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/sym17010005","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym17010005","pdf_url":"https://www.mdpi.com/2073-8994/17/1/5/pdf?version=1735052145","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":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4405737831.pdf","grobid_xml":"https://content.openalex.org/works/W4405737831.grobid-xml"},"referenced_works_count":51,"referenced_works":["https://openalex.org/W74903757","https://openalex.org/W1508174237","https://openalex.org/W1995875735","https://openalex.org/W2016485326","https://openalex.org/W2026014023","https://openalex.org/W2026118304","https://openalex.org/W2032558547","https://openalex.org/W2059749917","https://openalex.org/W2081422980","https://openalex.org/W2089400063","https://openalex.org/W2128261248","https://openalex.org/W2134851251","https://openalex.org/W2138978865","https://openalex.org/W2327198916","https://openalex.org/W2374819431","https://openalex.org/W2546767638","https://openalex.org/W2613425313","https://openalex.org/W2620748279","https://openalex.org/W2749146328","https://openalex.org/W2790852317","https://openalex.org/W2908632759","https://openalex.org/W2938479671","https://openalex.org/W2954675887","https://openalex.org/W2962773546","https://openalex.org/W2964252141","https://openalex.org/W3010709997","https://openalex.org/W3013142744","https://openalex.org/W3021044933","https://openalex.org/W3030025956","https://openalex.org/W3038927807","https://openalex.org/W3096880265","https://openalex.org/W3127964346","https://openalex.org/W3129229834","https://openalex.org/W3135501479","https://openalex.org/W3139498834","https://openalex.org/W3158689882","https://openalex.org/W3161343634","https://openalex.org/W3165843250","https://openalex.org/W3204899104","https://openalex.org/W4288097948","https://openalex.org/W4366780463","https://openalex.org/W4380267285","https://openalex.org/W4385172490","https://openalex.org/W4386495967","https://openalex.org/W4404761870","https://openalex.org/W6630300789","https://openalex.org/W6687151571","https://openalex.org/W6729031868","https://openalex.org/W6795261230","https://openalex.org/W6841184242","https://openalex.org/W6851966345"],"related_works":["https://openalex.org/W2626844239","https://openalex.org/W2065410741","https://openalex.org/W2133919704","https://openalex.org/W2020808467","https://openalex.org/W2252224195","https://openalex.org/W2020877234","https://openalex.org/W2940218203","https://openalex.org/W2272336114","https://openalex.org/W2073711843","https://openalex.org/W3158114875"],"abstract_inverted_index":{"An":[0],"algebraic":[1,15,40,47],"graph":[2,8,12,25,49],"is":[3,27,37],"defined":[4],"in":[5],"terms":[6],"of":[7,23,67,78,85,125],"theory":[9],"as":[10,57],"a":[11,24,28,30,33,58],"with":[13],"related":[14],"structures":[16],"or":[17,32],"characteristics.":[18],"If":[19],"the":[20,52,68,83,110,121,132,146],"vertex":[21],"set":[22],"G":[26,36],"group,":[29],"ring,":[31],"field,":[34],"then":[35],"called":[38],"an":[39,46],"structure":[41,48],"graph.":[42],"This":[43],"work":[44],"uses":[45],"based":[50,104],"on":[51,105],"modular":[53],"ring":[54,60,80,98,135],"Zn,":[55],"known":[56],"hyper-chordal":[59,79,97,134],"network.":[61],"The":[62,113],"lower":[63],"and":[64,139],"upper":[65],"bounds":[66],"local":[69,86,90,114],"fractional":[70,87,91],"metric":[71],"dimension":[72],"are":[73,94,107,117,129,142],"computed":[74,95,150],"for":[75,96,109,131],"certain":[76],"families":[77],"networks.":[81,99,112,136],"Utilizing":[82],"cardinalities":[84],"resolving":[88,92],"sets,":[89],"(LFR)M-polynomials":[93,106],"Further,":[100],"new":[101],"topological":[102,151],"indices":[103],"established":[108],"proposed":[111],"fraction":[115],"entropies":[116],"developed":[118],"by":[119],"modifying":[120],"first":[122],"three":[123],"kinds":[124],"Zagreb":[126],"entropies,":[127],"which":[128],"calculated":[130],"chosen":[133],"Furthermore,":[137],"numerical":[138],"graphical":[140],"comparisons":[141],"discussed":[143],"to":[144],"observe":[145],"order":[147],"between":[148],"newly":[149],"indices.":[152]},"counts_by_year":[],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2025-10-10T00:00:00"}