{"id":"https://openalex.org/W4324149075","doi":"https://doi.org/10.3390/sym15030718","title":"Kirchhoff Index and Additive Kirchhoff Index Based on Multiplicative Degree for a Random Polyomino Chain","display_name":"Kirchhoff Index and Additive Kirchhoff Index Based on Multiplicative Degree for a Random Polyomino Chain","publication_year":2023,"publication_date":"2023-03-14","ids":{"openalex":"https://openalex.org/W4324149075","doi":"https://doi.org/10.3390/sym15030718"},"language":"en","primary_location":{"id":"doi:10.3390/sym15030718","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15030718","pdf_url":"https://www.mdpi.com/2073-8994/15/3/718/pdf?version=1678760324","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/2073-8994/15/3/718/pdf?version=1678760324","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5084233260","display_name":"Meilian Li","orcid":null},"institutions":[{"id":"https://openalex.org/I181361081","display_name":"Longyan University","ror":"https://ror.org/0483s5p06","country_code":"CN","type":"education","lineage":["https://openalex.org/I181361081"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Meilian Li","raw_affiliation_strings":["School of Mathematics and Information Engineering, Longyan University, Longyan 364012, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Information Engineering, Longyan University, Longyan 364012, China","institution_ids":["https://openalex.org/I181361081"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5087152064","display_name":"Muhammad Asif","orcid":"https://orcid.org/0000-0001-5684-4737"},"institutions":[{"id":"https://openalex.org/I191208505","display_name":"Xiamen University","ror":"https://ror.org/00mcjh785","country_code":"CN","type":"education","lineage":["https://openalex.org/I191208505"]},{"id":"https://openalex.org/I4210100043","display_name":"Superior University","ror":"https://ror.org/00yh88643","country_code":"PK","type":"education","lineage":["https://openalex.org/I4210100043"]}],"countries":["CN","PK"],"is_corresponding":true,"raw_author_name":"Muhammad Asif","raw_affiliation_strings":["Faculty of Sciences, The Superior University, Lahore 54000, Pakistan","School of Mathematical Sciences, Xiamen University, Xiamen 361005, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Faculty of Sciences, The Superior University, Lahore 54000, Pakistan","institution_ids":["https://openalex.org/I4210100043"]},{"raw_affiliation_string":"School of Mathematical Sciences, Xiamen University, Xiamen 361005, China","institution_ids":["https://openalex.org/I191208505"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5014716837","display_name":"Haidar Ali","orcid":"https://orcid.org/0000-0002-9815-7158"},"institutions":[{"id":"https://openalex.org/I195024194","display_name":"Riphah International University","ror":"https://ror.org/02kdm5630","country_code":"PK","type":"education","lineage":["https://openalex.org/I195024194"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Haidar Ali","raw_affiliation_strings":["Department of Mathematics, Riphah International University, Faisalabad 38000, Pakistan"],"raw_orcid":"https://orcid.org/0000-0002-9815-7158","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Riphah International University, Faisalabad 38000, Pakistan","institution_ids":["https://openalex.org/I195024194"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5087527528","display_name":"Fizza Mahmood","orcid":null},"institutions":[{"id":"https://openalex.org/I4210158239","display_name":"Government College Women University Faisalabad","ror":"https://ror.org/05rq0zy06","country_code":"PK","type":"education","lineage":["https://openalex.org/I4210158239"]}],"countries":["PK"],"is_corresponding":false,"raw_author_name":"Fizza Mahmood","raw_affiliation_strings":["Department of Mathematics, Government College Women University, Faisalabad 38000, Pakistan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Government College Women University, Faisalabad 38000, Pakistan","institution_ids":["https://openalex.org/I4210158239"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5020257260","display_name":"Parvez Ali","orcid":"https://orcid.org/0000-0001-7363-4563"},"institutions":[{"id":"https://openalex.org/I156216236","display_name":"Qassim University","ror":"https://ror.org/01wsfe280","country_code":"SA","type":"education","lineage":["https://openalex.org/I156216236"]}],"countries":["SA"],"is_corresponding":false,"raw_author_name":"Parvez Ali","raw_affiliation_strings":["Department of Mechanical Engineering, College of Engineering, Qassim University, Unaizah 56215, Saudi Arabia"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mechanical Engineering, College of Engineering, Qassim University, Unaizah 56215, Saudi Arabia","institution_ids":["https://openalex.org/I156216236"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5087152064"],"corresponding_institution_ids":["https://openalex.org/I191208505","https://openalex.org/I4210100043"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":1.0631,"has_fulltext":false,"cited_by_count":2,"citation_normalized_percentile":{"value":0.7123437,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":95},"biblio":{"volume":"15","issue":"3","first_page":"718","last_page":"718"},"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/T12536","display_name":"Topological and Geometric Data Analysis","score":0.9865999817848206,"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/T10096","display_name":"Metal-Organic Frameworks: Synthesis and Applications","score":0.9853000044822693,"subfield":{"id":"https://openalex.org/subfields/1604","display_name":"Inorganic Chemistry"},"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/polyomino","display_name":"Polyomino","score":0.8523310422897339},{"id":"https://openalex.org/keywords/multiplicative-function","display_name":"Multiplicative function","score":0.8463676571846008},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.8256630301475525},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6770206093788147},{"id":"https://openalex.org/keywords/index","display_name":"Index (typography)","score":0.6670765280723572},{"id":"https://openalex.org/keywords/chain","display_name":"Chain (unit)","score":0.4258460998535156},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.34500807523727417},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.20141011476516724},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.15390560030937195},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.14816439151763916},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.11983489990234375}],"concepts":[{"id":"https://openalex.org/C69919132","wikidata":"https://www.wikidata.org/wiki/Q1144819","display_name":"Polyomino","level":3,"score":0.8523310422897339},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.8463676571846008},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.8256630301475525},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6770206093788147},{"id":"https://openalex.org/C2777382242","wikidata":"https://www.wikidata.org/wiki/Q6017816","display_name":"Index (typography)","level":2,"score":0.6670765280723572},{"id":"https://openalex.org/C199185054","wikidata":"https://www.wikidata.org/wiki/Q552299","display_name":"Chain (unit)","level":2,"score":0.4258460998535156},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.34500807523727417},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.20141011476516724},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.15390560030937195},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.14816439151763916},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.11983489990234375},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.0},{"id":"https://openalex.org/C136764020","wikidata":"https://www.wikidata.org/wiki/Q466","display_name":"World Wide Web","level":1,"score":0.0},{"id":"https://openalex.org/C1276947","wikidata":"https://www.wikidata.org/wiki/Q333","display_name":"Astronomy","level":1,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym15030718","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15030718","pdf_url":"https://www.mdpi.com/2073-8994/15/3/718/pdf?version=1678760324","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:7305990750f44409b8f79bdde4150d85","is_oa":false,"landing_page_url":"https://doaj.org/article/7305990750f44409b8f79bdde4150d85","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 3, p 718 (2023)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/15/3/718/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym15030718","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; Volume 15; Issue 3; Pages: 718","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym15030718","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym15030718","pdf_url":"https://www.mdpi.com/2073-8994/15/3/718/pdf?version=1678760324","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":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4324149075.pdf"},"referenced_works_count":19,"referenced_works":["https://openalex.org/W1875795917","https://openalex.org/W1969888675","https://openalex.org/W2016358293","https://openalex.org/W2019635364","https://openalex.org/W2067696638","https://openalex.org/W2076124375","https://openalex.org/W2081084405","https://openalex.org/W2090247235","https://openalex.org/W2095398244","https://openalex.org/W2559645233","https://openalex.org/W2772618122","https://openalex.org/W2890400649","https://openalex.org/W2990230913","https://openalex.org/W3013052188","https://openalex.org/W3165695461","https://openalex.org/W4247782380","https://openalex.org/W4295078734","https://openalex.org/W6639440540","https://openalex.org/W7061961335"],"related_works":["https://openalex.org/W4288734188","https://openalex.org/W209156724","https://openalex.org/W2135008840","https://openalex.org/W4213044019","https://openalex.org/W148666007","https://openalex.org/W2103874366","https://openalex.org/W2572231869","https://openalex.org/W2962715651","https://openalex.org/W2119324012","https://openalex.org/W2950159329"],"abstract_inverted_index":{"Several":[0],"topological":[1],"indices":[2],"are":[3],"known":[4],"to":[5,24,49],"have":[6],"widespread":[7],"implications":[8],"in":[9],"a":[10,72],"variety":[11],"of":[12,44,67],"research":[13],"areas.":[14],"Over":[15],"the":[16,18,37,41,45,51,56,64,68,77],"years,":[17],"Kirchhoff":[19],"index":[20,54,80],"has":[21],"turned":[22],"out":[23],"be":[25],"an":[26],"extremely":[27],"significant":[28],"and":[29,55,81],"efficient":[30],"index.":[31,59,84],"In":[32],"this":[33],"paper,":[34],"we":[35],"propose":[36],"exact":[38],"formulas":[39],"for":[40,71],"expected":[42,69],"values":[43,70],"random":[46,73],"polyomino":[47,74],"chain":[48,75],"construct":[50],"multiplicative":[52,78],"degree-Kirchhoff":[53,58,79,83],"additive":[57,82],"We":[60],"also":[61],"carefully":[62],"examine":[63],"highest":[65],"degree":[66],"through":[76]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
