{"id":"https://openalex.org/W2899631890","doi":"https://doi.org/10.3390/sym10110608","title":"Maximum Detour\u2013Harary Index for Some Graph Classes","display_name":"Maximum Detour\u2013Harary Index for Some Graph Classes","publication_year":2018,"publication_date":"2018-11-07","ids":{"openalex":"https://openalex.org/W2899631890","doi":"https://doi.org/10.3390/sym10110608","mag":"2899631890"},"language":"en","primary_location":{"id":"doi:10.3390/sym10110608","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym10110608","pdf_url":"https://www.mdpi.com/2073-8994/10/11/608/pdf?version=1541584739","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/2073-8994/10/11/608/pdf?version=1541584739","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5072956430","display_name":"Wei Fang","orcid":"https://orcid.org/0000-0001-7674-1583"},"institutions":[{"id":"https://openalex.org/I184681353","display_name":"Anhui University of Science and Technology","ror":"https://ror.org/00q9atg80","country_code":"CN","type":"education","lineage":["https://openalex.org/I184681353"]},{"id":"https://openalex.org/I4210109416","display_name":"Anhui Science and Technology University","ror":"https://ror.org/01pn91c28","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210109416"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wei Fang","raw_affiliation_strings":["College of Information &amp; Network Engineering, Anhui Science and Technology University, Fengyang 233100, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Information &amp; Network Engineering, Anhui Science and Technology University, Fengyang 233100, China","institution_ids":["https://openalex.org/I184681353","https://openalex.org/I4210109416"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101892451","display_name":"Weihua Liu","orcid":"https://orcid.org/0000-0002-7002-8370"},"institutions":[{"id":"https://openalex.org/I4750791","display_name":"Henan Agricultural University","ror":"https://ror.org/04eq83d71","country_code":"CN","type":"education","lineage":["https://openalex.org/I4750791"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Wei-Hua Liu","raw_affiliation_strings":["College of Information and Management Science, Henan Agricultural University, Zhengzhou 450002, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Information and Management Science, Henan Agricultural University, Zhengzhou 450002, China","institution_ids":["https://openalex.org/I4750791"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5076981458","display_name":"Jia\u2010Bao Liu","orcid":"https://orcid.org/0000-0002-9620-7692"},"institutions":[{"id":"https://openalex.org/I118987531","display_name":"Anhui Jianzhu University","ror":"https://ror.org/0108wjw08","country_code":"CN","type":"education","lineage":["https://openalex.org/I118987531"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Jia-Bao Liu","raw_affiliation_strings":["School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China"],"raw_orcid":"https://orcid.org/0000-0002-9620-7692","affiliations":[{"raw_affiliation_string":"School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China","institution_ids":["https://openalex.org/I118987531"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101533350","display_name":"Fuyuan Chen","orcid":"https://orcid.org/0000-0002-0588-0479"},"institutions":[{"id":"https://openalex.org/I188935350","display_name":"Anhui University of Finance and Economics","ror":"https://ror.org/0152zzg30","country_code":"CN","type":"education","lineage":["https://openalex.org/I188935350"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Fu-Yuan Chen","raw_affiliation_strings":["Institute of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu 233030, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu 233030, China","institution_ids":["https://openalex.org/I188935350"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5016940753","display_name":"Zhen\u2010Mu Hong","orcid":"https://orcid.org/0000-0003-1248-0227"},"institutions":[{"id":"https://openalex.org/I188935350","display_name":"Anhui University of Finance and Economics","ror":"https://ror.org/0152zzg30","country_code":"CN","type":"education","lineage":["https://openalex.org/I188935350"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhen-Mu Hong","raw_affiliation_strings":["School of Finance, Anhui University of Finance and Economics, Bengbu 233030, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Finance, Anhui University of Finance and Economics, Bengbu 233030, China","institution_ids":["https://openalex.org/I188935350"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5069196389","display_name":"Zheng-Jiang Xia","orcid":"https://orcid.org/0000-0002-0173-3631"},"institutions":[{"id":"https://openalex.org/I188935350","display_name":"Anhui University of Finance and Economics","ror":"https://ror.org/0152zzg30","country_code":"CN","type":"education","lineage":["https://openalex.org/I188935350"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zheng-Jiang Xia","raw_affiliation_strings":["School of Finance, Anhui University of Finance and Economics, Bengbu 233030, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Finance, Anhui University of Finance and Economics, Bengbu 233030, China","institution_ids":["https://openalex.org/I188935350"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5101892451"],"corresponding_institution_ids":["https://openalex.org/I4750791"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2227},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2227},"fwci":0.0,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.21747951,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"10","issue":"11","first_page":"608","last_page":"608"},"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":0.9951000213623047,"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":0.9754999876022339,"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":0.8126649856567383},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.6002606749534607},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5548416972160339},{"id":"https://openalex.org/keywords/index","display_name":"Index (typography)","score":0.49119827151298523},{"id":"https://openalex.org/keywords/path","display_name":"Path (computing)","score":0.45045924186706543},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.21363532543182373},{"id":"https://openalex.org/keywords/world-wide-web","display_name":"World Wide Web","score":0.0414750874042511}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8126649856567383},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.6002606749534607},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5548416972160339},{"id":"https://openalex.org/C2777382242","wikidata":"https://www.wikidata.org/wiki/Q6017816","display_name":"Index (typography)","level":2,"score":0.49119827151298523},{"id":"https://openalex.org/C2777735758","wikidata":"https://www.wikidata.org/wiki/Q817765","display_name":"Path (computing)","level":2,"score":0.45045924186706543},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.21363532543182373},{"id":"https://openalex.org/C136764020","wikidata":"https://www.wikidata.org/wiki/Q466","display_name":"World Wide Web","level":1,"score":0.0414750874042511},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/sym10110608","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym10110608","pdf_url":"https://www.mdpi.com/2073-8994/10/11/608/pdf?version=1541584739","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:4f84a9cf0e074cca93903858d84480cb","is_oa":true,"landing_page_url":"https://doaj.org/article/4f84a9cf0e074cca93903858d84480cb","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 10, Iss 11, p 608 (2018)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/sym10110608","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym10110608","pdf_url":"https://www.mdpi.com/2073-8994/10/11/608/pdf?version=1541584739","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/G1293752319","display_name":null,"funder_award_id":"11601001","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3065972664","display_name":null,"funder_award_id":"11601006","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G925363208","display_name":null,"funder_award_id":"11601002","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2899631890.pdf","grobid_xml":"https://content.openalex.org/works/W2899631890.grobid-xml"},"referenced_works_count":41,"referenced_works":["https://openalex.org/W74903757","https://openalex.org/W1022727114","https://openalex.org/W1966170506","https://openalex.org/W1980510534","https://openalex.org/W1999644567","https://openalex.org/W1999928129","https://openalex.org/W2016496752","https://openalex.org/W2028818487","https://openalex.org/W2038716921","https://openalex.org/W2051025159","https://openalex.org/W2061067020","https://openalex.org/W2068042639","https://openalex.org/W2075970225","https://openalex.org/W2076274021","https://openalex.org/W2079015178","https://openalex.org/W2083651186","https://openalex.org/W2186293682","https://openalex.org/W2187893287","https://openalex.org/W2189319813","https://openalex.org/W2229925045","https://openalex.org/W2318794083","https://openalex.org/W2340120343","https://openalex.org/W2514804622","https://openalex.org/W2748674079","https://openalex.org/W2806577560","https://openalex.org/W2808847576","https://openalex.org/W2810124054","https://openalex.org/W2844802808","https://openalex.org/W2964155563","https://openalex.org/W2980287569","https://openalex.org/W4285719527","https://openalex.org/W6686564536","https://openalex.org/W6686951396","https://openalex.org/W6687480473","https://openalex.org/W6689615075","https://openalex.org/W6704126733","https://openalex.org/W6716183041","https://openalex.org/W6722314686","https://openalex.org/W6726369951","https://openalex.org/W6730594845","https://openalex.org/W7036401774"],"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":{"The":[0],"definition":[1],"of":[2,56],"a":[3,37],"Detour\u2013Harary":[4,72],"index":[5,73],"is":[6,36,51],"\u03c9":[7],"H":[8],"(":[9,21,26,44],"G":[10,22,31,35,49],")":[11,23,32,50],"=":[12],"1":[13,24],"2":[14],"\u2211":[15],"u":[16,27,45,62],",":[17,28,33,46],"v":[18,29,47],"\u2208":[19],"V":[20],"l":[25,43],"|":[30,48],"where":[34],"simple":[38],"and":[39,42,63,79],"connected":[40],"graph,":[41],"equal":[52],"to":[53],"the":[54,57,70],"length":[55],"longest":[58],"path":[59],"between":[60],"vertices":[61],"v.":[64],"In":[65],"this":[66],"paper,":[67],"we":[68],"obtained":[69],"maximum":[71],"about":[74],"unicyclic":[75],"graphs,":[76,78],"bicyclic":[77],"cacti,":[80],"respectively.":[81]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1}],"updated_date":"2026-07-22T07:51:19.307946","created_date":"2025-10-10T00:00:00"}
