{"id":"https://openalex.org/W3185407173","doi":"https://doi.org/10.3390/sym13081374","title":"On Normalized Laplacians, Degree-Kirchhoff Index and Spanning Tree of Generalized Phenylene","display_name":"On Normalized Laplacians, Degree-Kirchhoff Index and Spanning Tree of Generalized Phenylene","publication_year":2021,"publication_date":"2021-07-28","ids":{"openalex":"https://openalex.org/W3185407173","doi":"https://doi.org/10.3390/sym13081374","mag":"3185407173"},"language":"en","primary_location":{"id":"doi:10.3390/sym13081374","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13081374","pdf_url":"https://www.mdpi.com/2073-8994/13/8/1374/pdf?version=1627970857","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/2073-8994/13/8/1374/pdf?version=1627970857","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5077917349","display_name":"Umar Ali","orcid":null},"institutions":[{"id":"https://openalex.org/I143868143","display_name":"Anhui University","ror":"https://ror.org/05th6yx34","country_code":"CN","type":"education","lineage":["https://openalex.org/I143868143"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Umar Ali","raw_affiliation_strings":["School of Mathematical Sciences, Anhui University, Hefei 230601, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, Anhui University, Hefei 230601, China","institution_ids":["https://openalex.org/I143868143"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5103023353","display_name":"Hassan Raza","orcid":"https://orcid.org/0000-0003-2520-4379"},"institutions":[{"id":"https://openalex.org/I148128674","display_name":"University of Shanghai for Science and Technology","ror":"https://ror.org/00ay9v204","country_code":"CN","type":"education","lineage":["https://openalex.org/I148128674"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Hassan Raza","raw_affiliation_strings":["Business School, University of Shanghai for Science and Technology, Shanghai 230009, China"],"raw_orcid":"https://orcid.org/0000-0003-2520-4379","affiliations":[{"raw_affiliation_string":"Business School, University of Shanghai for Science and Technology, Shanghai 230009, China","institution_ids":["https://openalex.org/I148128674"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5102366457","display_name":"Yasir Ahmed","orcid":"https://orcid.org/0009-0003-8478-0541"},"institutions":[{"id":"https://openalex.org/I143868143","display_name":"Anhui University","ror":"https://ror.org/05th6yx34","country_code":"CN","type":"education","lineage":["https://openalex.org/I143868143"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yasir Ahmed","raw_affiliation_strings":["School of Mathematical Sciences, Anhui University, Hefei 230601, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, Anhui University, Hefei 230601, China","institution_ids":["https://openalex.org/I143868143"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5103023353"],"corresponding_institution_ids":["https://openalex.org/I148128674"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2227},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2227},"fwci":2.8188,"has_fulltext":true,"cited_by_count":7,"citation_normalized_percentile":{"value":0.91048415,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"13","issue":"8","first_page":"1374","last_page":"1374"},"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":0.996999979019165,"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/T10074","display_name":"Carbon Nanotubes in Composites","score":0.9894999861717224,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8009759783744812},{"id":"https://openalex.org/keywords/spanning-tree","display_name":"Spanning tree","score":0.7401303052902222},{"id":"https://openalex.org/keywords/laplacian-matrix","display_name":"Laplacian matrix","score":0.6874085068702698},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.6301536560058594},{"id":"https://openalex.org/keywords/diagonal","display_name":"Diagonal","score":0.6262413859367371},{"id":"https://openalex.org/keywords/multiplicative-function","display_name":"Multiplicative function","score":0.6137959957122803},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.6021901965141296},{"id":"https://openalex.org/keywords/laplace-operator","display_name":"Laplace operator","score":0.5538510084152222},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.532621443271637},{"id":"https://openalex.org/keywords/resistance-distance","display_name":"Resistance distance","score":0.5198281407356262},{"id":"https://openalex.org/keywords/characteristic-polynomial","display_name":"Characteristic polynomial","score":0.5140293836593628},{"id":"https://openalex.org/keywords/tree","display_name":"Tree (set theory)","score":0.4156368672847748},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.4106976389884949},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3511035442352295},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.32577502727508545},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.3249197006225586},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.27503979206085205},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.10239800810813904},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.0863206684589386}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8009759783744812},{"id":"https://openalex.org/C64331007","wikidata":"https://www.wikidata.org/wiki/Q831672","display_name":"Spanning tree","level":2,"score":0.7401303052902222},{"id":"https://openalex.org/C115178988","wikidata":"https://www.wikidata.org/wiki/Q772067","display_name":"Laplacian matrix","level":3,"score":0.6874085068702698},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.6301536560058594},{"id":"https://openalex.org/C130367717","wikidata":"https://www.wikidata.org/wiki/Q189791","display_name":"Diagonal","level":2,"score":0.6262413859367371},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.6137959957122803},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.6021901965141296},{"id":"https://openalex.org/C165700671","wikidata":"https://www.wikidata.org/wiki/Q203484","display_name":"Laplace operator","level":2,"score":0.5538510084152222},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.532621443271637},{"id":"https://openalex.org/C612898","wikidata":"https://www.wikidata.org/wiki/Q7315618","display_name":"Resistance distance","level":5,"score":0.5198281407356262},{"id":"https://openalex.org/C42321380","wikidata":"https://www.wikidata.org/wiki/Q849705","display_name":"Characteristic polynomial","level":3,"score":0.5140293836593628},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.4156368672847748},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.4106976389884949},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3511035442352295},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.32577502727508545},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.3249197006225586},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.27503979206085205},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.10239800810813904},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0863206684589386},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.0},{"id":"https://openalex.org/C149530733","wikidata":"https://www.wikidata.org/wiki/Q5597091","display_name":"Graph power","level":4,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym13081374","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13081374","pdf_url":"https://www.mdpi.com/2073-8994/13/8/1374/pdf?version=1627970857","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:5e33f7e053414a9fa3121a2a85f17436","is_oa":true,"landing_page_url":"https://doaj.org/article/5e33f7e053414a9fa3121a2a85f17436","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":"Symmetry, Vol 13, Iss 8, p 1374 (2021)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/13/8/1374/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym13081374","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/sym13081374","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym13081374","pdf_url":"https://www.mdpi.com/2073-8994/13/8/1374/pdf?version=1627970857","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/W3185407173.pdf","grobid_xml":"https://content.openalex.org/works/W3185407173.grobid-xml"},"referenced_works_count":38,"referenced_works":["https://openalex.org/W634966607","https://openalex.org/W1484040084","https://openalex.org/W1578099820","https://openalex.org/W1980698230","https://openalex.org/W1983773166","https://openalex.org/W1995058487","https://openalex.org/W2006975882","https://openalex.org/W2012513251","https://openalex.org/W2055963997","https://openalex.org/W2081084405","https://openalex.org/W2135650427","https://openalex.org/W2137108941","https://openalex.org/W2318794083","https://openalex.org/W2337153326","https://openalex.org/W2338127217","https://openalex.org/W2418022162","https://openalex.org/W2469362354","https://openalex.org/W2555597088","https://openalex.org/W2559645233","https://openalex.org/W2806654778","https://openalex.org/W2886205120","https://openalex.org/W2903684107","https://openalex.org/W2921238808","https://openalex.org/W2951092953","https://openalex.org/W2962868843","https://openalex.org/W2976860982","https://openalex.org/W3005941619","https://openalex.org/W3039052301","https://openalex.org/W3143364932","https://openalex.org/W3163927547","https://openalex.org/W4238431945","https://openalex.org/W4247782380","https://openalex.org/W4285719527","https://openalex.org/W6703554538","https://openalex.org/W6716756615","https://openalex.org/W6720336310","https://openalex.org/W6729870989","https://openalex.org/W6792258961"],"related_works":["https://openalex.org/W4289083261","https://openalex.org/W2883186538","https://openalex.org/W3004345458","https://openalex.org/W2113991431","https://openalex.org/W2570009360","https://openalex.org/W2402951419","https://openalex.org/W2051180862","https://openalex.org/W1993536369","https://openalex.org/W2039169612","https://openalex.org/W2287993572"],"abstract_inverted_index":{"The":[0,14],"normalized":[1,36,45],"Laplacian":[2,37,46],"is":[3,70],"extremely":[4],"important":[5],"for":[6],"analyzing":[7],"the":[8,35,44,52,67,74,79,90,93],"structural":[9],"properties":[10],"of":[11,17,21,48,51,54,61,81,84],"non-regular":[12],"graphs.":[13],"molecular":[15],"graph":[16],"generalized":[18,85],"phenylene":[19,86],"consists":[20],"n":[22],"hexagons":[23],"and":[24,59,78,95],"2n":[25],"squares,":[26],"denoted":[27],"by":[28,33],"Ln6,4,4.":[29],"In":[30],"this":[31],"paper,":[32],"using":[34],"polynomial":[38],"decomposition":[39],"theorem,":[40],"we":[41],"have":[42],"investigated":[43],"spectrum":[47],"Ln6,4,4":[49],"consisting":[50],"eigenvalues":[53],"symmetric":[55],"tri-diagonal":[56],"matrices":[57],"LA":[58],"LS":[60],"order":[62],"4n+1.":[63],"As":[64],"an":[65],"application,":[66],"significant":[68],"formula":[69],"obtained":[71],"to":[72],"calculate":[73],"multiplicative":[75],"degree-Kirchhoff":[76],"index":[77],"number":[80],"spanning":[82],"trees":[83],"network":[87],"based":[88],"on":[89],"relationships":[91],"between":[92],"coefficients":[94],"roots.":[96]},"counts_by_year":[{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2}],"updated_date":"2026-07-22T07:51:19.307946","created_date":"2025-10-10T00:00:00"}