{"id":"https://openalex.org/W4415991293","doi":"https://doi.org/10.3390/axioms14110825","title":"On Distance Laplacian Energy of Unicyclic and Bicyclic Graphs","display_name":"On Distance Laplacian Energy of Unicyclic and Bicyclic Graphs","publication_year":2025,"publication_date":"2025-11-07","ids":{"openalex":"https://openalex.org/W4415991293","doi":"https://doi.org/10.3390/axioms14110825"},"language":"en","primary_location":{"id":"doi:10.3390/axioms14110825","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14110825","pdf_url":null,"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://doi.org/10.3390/axioms14110825","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100380680","display_name":"Dan Li","orcid":"https://orcid.org/0000-0001-8598-873X"},"institutions":[{"id":"https://openalex.org/I96908189","display_name":"Xinjiang University","ror":"https://ror.org/059gw8r13","country_code":"CN","type":"education","lineage":["https://openalex.org/I96908189"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Dan Li","raw_affiliation_strings":["College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China","institution_ids":["https://openalex.org/I96908189"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5111264705","display_name":"Shiqi Zhou","orcid":"https://orcid.org/0009-0005-6063-1247"},"institutions":[{"id":"https://openalex.org/I96908189","display_name":"Xinjiang University","ror":"https://ror.org/059gw8r13","country_code":"CN","type":"education","lineage":["https://openalex.org/I96908189"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Shiqi Zhou","raw_affiliation_strings":["College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China","institution_ids":["https://openalex.org/I96908189"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I96908189"],"apc_list":{"value":1600,"currency":"CHF","value_usd":1782},"apc_paid":{"value":1600,"currency":"CHF","value_usd":1782},"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.38352957,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"14","issue":"11","first_page":"825","last_page":"825"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.989799976348877,"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":0.989799976348877,"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/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.0015999999595806003,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.0007999999797903001,"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/graph","display_name":"Graph","score":0.503600001335144},{"id":"https://openalex.org/keywords/laplacian-matrix","display_name":"Laplacian matrix","score":0.49309998750686646},{"id":"https://openalex.org/keywords/energy","display_name":"Energy (signal processing)","score":0.4410000145435333},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.44040000438690186},{"id":"https://openalex.org/keywords/wiener-index","display_name":"Wiener index","score":0.40610000491142273},{"id":"https://openalex.org/keywords/connectivity","display_name":"Connectivity","score":0.4043000042438507},{"id":"https://openalex.org/keywords/laplace-operator","display_name":"Laplace operator","score":0.36160001158714294}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7001000046730042},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.656499981880188},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.503600001335144},{"id":"https://openalex.org/C115178988","wikidata":"https://www.wikidata.org/wiki/Q772067","display_name":"Laplacian matrix","level":3,"score":0.49309998750686646},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.48240000009536743},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.4410000145435333},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.44040000438690186},{"id":"https://openalex.org/C16640446","wikidata":"https://www.wikidata.org/wiki/Q909065","display_name":"Wiener index","level":3,"score":0.40610000491142273},{"id":"https://openalex.org/C76444178","wikidata":"https://www.wikidata.org/wiki/Q72897900","display_name":"Connectivity","level":3,"score":0.4043000042438507},{"id":"https://openalex.org/C165700671","wikidata":"https://www.wikidata.org/wiki/Q203484","display_name":"Laplace operator","level":2,"score":0.36160001158714294},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.3273000121116638},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.30640000104904175},{"id":"https://openalex.org/C159867001","wikidata":"https://www.wikidata.org/wiki/Q904942","display_name":"Topological index","level":2,"score":0.2930000126361847},{"id":"https://openalex.org/C78913703","wikidata":"https://www.wikidata.org/wiki/Q5597087","display_name":"Graph energy","level":5,"score":0.2815000116825104},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.2597000002861023},{"id":"https://openalex.org/C111208986","wikidata":"https://www.wikidata.org/wiki/Q901698","display_name":"Distance matrix","level":2,"score":0.257099986076355},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.25540000200271606}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/axioms14110825","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14110825","pdf_url":null,"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:52067571caf640f488a5e57434df122a","is_oa":true,"landing_page_url":"https://doaj.org/article/52067571caf640f488a5e57434df122a","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 11, p 825 (2025)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/axioms14110825","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms14110825","pdf_url":null,"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/G1103760527","display_name":null,"funder_award_id":"2024D01C41","funder_id":"https://openalex.org/F4320311028","funder_display_name":"Natural Science Foundation of Xinjiang Province"},{"id":"https://openalex.org/G1143433260","display_name":null,"funder_award_id":"11901498","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G6294865353","display_name":null,"funder_award_id":"12361071","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320311028","display_name":"Natural Science Foundation of Xinjiang Province","ror":null},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":9,"referenced_works":["https://openalex.org/W1987987167","https://openalex.org/W2048396151","https://openalex.org/W2055041342","https://openalex.org/W2299459697","https://openalex.org/W2790264134","https://openalex.org/W2791326320","https://openalex.org/W2962683955","https://openalex.org/W3032975195","https://openalex.org/W4391256074"],"related_works":[],"abstract_inverted_index":{"For":[0],"a":[1,66,72],"connected":[2],"graph":[3,39,52],"G,":[4],"let":[5],"DL(G)":[6],"be":[7,14],"its":[8,15],"distance":[9],"Laplacian":[10],"matrix":[11],"and":[12,54,78,93,113,125],"\u03bb1(G)\u2265\u03bb2(G)\u2265\u2026\u2265\u03bbn\u22121(G)&gt;\u03bbn(G)=0":[13],"DL":[16,19,62,127],"eigenvalues.":[17],"The":[18],"energy":[20,40,49,63,128],"of":[21,33,47,65,68,74,90,104,111,116,122],"G":[22],"is":[23,29,42],"defined":[24],"as":[25],"DLEG=\u2211i=1n\u03bbi(G)\u22122WGn,":[26],"where":[27],"W(G)":[28],"the":[30,48,60,84,88,95,102,105,109,114,120,126],"Wiener":[31],"index":[32],"G.":[34],"An":[35],"important":[36],"problem":[37],"in":[38],"studies":[41],"to":[43],"determine":[44],"exact":[45],"formulations":[46,64],"for":[50,98],"specific":[51],"classes":[53],"their":[55,79,99,117],"complements.":[56,80,100],"This":[57],"paper":[58],"gives":[59],"precise":[61],"class":[67,73],"bicyclic":[69],"graphs":[70,76,85,106],"C2(p,q),":[71],"unicyclic":[75],"C1(p,q),":[77],"Moreover,":[81],"we":[82],"order":[83],"C2(p,q)":[86],"on":[87,108,119],"basic":[89,110],"\u03bb1,":[91,123],"\u03bbn\u22121,":[92],"consider":[94],"same":[96],"problems":[97],"And":[101],"ordering":[103,115],"C1(p,q)":[107],"\u03bbn\u22121":[112,124],"complements":[118],"basics":[121],"are":[129],"obtained.":[130]},"counts_by_year":[],"updated_date":"2026-07-23T08:03:31.855105","created_date":"2025-11-07T00:00:00"}
