{"id":"https://openalex.org/W3081504174","doi":"https://doi.org/10.3390/e22090923","title":"An Efficient Algorithm to Count Tree-Like Graphs with a Given Number of Vertices and Self-Loops","display_name":"An Efficient Algorithm to Count Tree-Like Graphs with a Given Number of Vertices and Self-Loops","publication_year":2020,"publication_date":"2020-08-22","ids":{"openalex":"https://openalex.org/W3081504174","doi":"https://doi.org/10.3390/e22090923","mag":"3081504174","pmid":"https://pubmed.ncbi.nlm.nih.gov/33286692"},"language":"en","primary_location":{"id":"doi:10.3390/e22090923","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e22090923","pdf_url":"https://www.mdpi.com/1099-4300/22/9/923/pdf?version=1598260972","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"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":"Entropy","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/22/9/923/pdf?version=1598260972","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5069120004","display_name":"Naveed Ahmed Azam","orcid":"https://orcid.org/0000-0002-7941-3419"},"institutions":[{"id":"https://openalex.org/I22299242","display_name":"Kyoto University","ror":"https://ror.org/02kpeqv85","country_code":"JP","type":"education","lineage":["https://openalex.org/I22299242"]}],"countries":["JP"],"is_corresponding":true,"raw_author_name":"Naveed Ahmed Azam","raw_affiliation_strings":["Department of Applied Mathematics and Physics, Kyoto University, Kyoto 606-8502, Japan"],"raw_orcid":"https://orcid.org/0000-0002-7941-3419","affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics and Physics, Kyoto University, Kyoto 606-8502, Japan","institution_ids":["https://openalex.org/I22299242"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5084691948","display_name":"Aleksandar Shurbevski","orcid":"https://orcid.org/0000-0001-9224-6929"},"institutions":[{"id":"https://openalex.org/I22299242","display_name":"Kyoto University","ror":"https://ror.org/02kpeqv85","country_code":"JP","type":"education","lineage":["https://openalex.org/I22299242"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Aleksandar Shurbevski","raw_affiliation_strings":["Department of Applied Mathematics and Physics, Kyoto University, Kyoto 606-8502, Japan"],"raw_orcid":"https://orcid.org/0000-0001-9224-6929","affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics and Physics, Kyoto University, Kyoto 606-8502, Japan","institution_ids":["https://openalex.org/I22299242"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5073738140","display_name":"Hiroshi Nagamochi","orcid":"https://orcid.org/0000-0002-8332-1517"},"institutions":[{"id":"https://openalex.org/I22299242","display_name":"Kyoto University","ror":"https://ror.org/02kpeqv85","country_code":"JP","type":"education","lineage":["https://openalex.org/I22299242"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Hiroshi Nagamochi","raw_affiliation_strings":["Department of Applied Mathematics and Physics, Kyoto University, Kyoto 606-8502, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Applied Mathematics and Physics, Kyoto University, Kyoto 606-8502, Japan","institution_ids":["https://openalex.org/I22299242"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5069120004"],"corresponding_institution_ids":["https://openalex.org/I22299242"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2227},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2227},"fwci":0.4012,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.70916626,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"22","issue":"9","first_page":"923","last_page":"923"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11269","display_name":"Algorithms and Data Compression","score":0.9962999820709229,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T11269","display_name":"Algorithms and Data Compression","score":0.9962999820709229,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T12292","display_name":"Graph Theory and Algorithms","score":0.9871000051498413,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/T10317","display_name":"Advanced Database Systems and Queries","score":0.9861000180244446,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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":0.5880405902862549},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5177533030509949},{"id":"https://openalex.org/keywords/tree","display_name":"Tree (set theory)","score":0.41775646805763245},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.40984416007995605},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.39447587728500366},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.36053282022476196}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5880405902862549},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5177533030509949},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.41775646805763245},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.40984416007995605},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.39447587728500366},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.36053282022476196}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.3390/e22090923","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e22090923","pdf_url":"https://www.mdpi.com/1099-4300/22/9/923/pdf?version=1598260972","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"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":"Entropy","raw_type":"journal-article"},{"id":"pmid:33286692","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/33286692","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy (Basel, Switzerland)","raw_type":null},{"id":"pmh:oai:pubmedcentral.nih.gov:7597174","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/7597174","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"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":"Entropy (Basel)","raw_type":"Text"},{"id":"pmh:oai:doaj.org/article:d855e9e53231427e8bf8cbd171430580","is_oa":true,"landing_page_url":"https://doaj.org/article/d855e9e53231427e8bf8cbd171430580","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":"Entropy, Vol 22, Iss 9, p 923 (2020)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/1099-4300/22/9/923/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/e22090923","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":"Entropy; Volume 22; Issue 9; Pages: 923","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/e22090923","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e22090923","pdf_url":"https://www.mdpi.com/1099-4300/22/9/923/pdf?version=1598260972","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"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":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G4437838937","display_name":"Discrete Mathematics and its Applications","funder_award_id":"18J23484","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"},{"id":"https://openalex.org/G5626364646","display_name":"Theory design and implementation of practical optimization and enumeration algorithms over graph structure","funder_award_id":"20K11691","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"},{"id":"https://openalex.org/G7193312839","display_name":null,"funder_award_id":"JSPS KAKENHI Grant Number 18J23484","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"}],"funders":[{"id":"https://openalex.org/F4320334764","display_name":"Japan Society for the Promotion of Science","ror":"https://ror.org/00hhkn466"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3081504174.pdf","grobid_xml":"https://content.openalex.org/works/W3081504174.grobid-xml"},"referenced_works_count":19,"referenced_works":["https://openalex.org/W1966161686","https://openalex.org/W1972094496","https://openalex.org/W2006387233","https://openalex.org/W2035157229","https://openalex.org/W2049817705","https://openalex.org/W2064863523","https://openalex.org/W2065879466","https://openalex.org/W2132437698","https://openalex.org/W2153693853","https://openalex.org/W2160000945","https://openalex.org/W2745698998","https://openalex.org/W2947950688","https://openalex.org/W2992072991","https://openalex.org/W2998571806","https://openalex.org/W3021964567","https://openalex.org/W3024970380","https://openalex.org/W3192862300","https://openalex.org/W3194348808","https://openalex.org/W6643380523"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W2051487156","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2077600819","https://openalex.org/W2142036596","https://openalex.org/W2072657027","https://openalex.org/W2962838298"],"abstract_inverted_index":{"Graph":[0],"enumeration":[1],"with":[2,20,50,74,152],"given":[3],"constraints":[4],"is":[5],"an":[6,139],"interesting":[7],"problem":[8],"considered":[9],"to":[10,45,126],"be":[11,92,127],"one":[12],"of":[13,70,111,145,149],"the":[14,68,105,109,119,123,128,143],"fundamental":[15],"problems":[16],"in":[17,23,82,108],"graph":[18],"theory,":[19],"many":[21],"applications":[22],"natural":[24],"sciences":[25],"and":[26,31,39,55,79,85,121,138],"engineering":[27],"such":[28],"as":[29,95,104],"bio-informatics":[30],"computational":[32],"chemistry.":[33],"For":[34],"any":[35,153],"two":[36],"integers":[37],"n\u22651":[38],"\u0394\u22650,":[40],"we":[41,66,133],"propose":[42],"a":[43,96,115,135],"method":[44],"count":[46,67],"all":[47],"non-isomorphic":[48,71],"trees":[49,73],"n":[51,75],"vertices,":[52,76],"\u0394":[53,77],"self-loops,":[54],"no":[56,80],"multi-edges":[57],"based":[58],"on":[59,118,142],"dynamic":[60],"programming.":[61],"To":[62],"achieve":[63],"this":[64,131],"goal,":[65],"number":[69,144],"rooted":[72,97],"self-loops":[78],"multi-edges,":[81],"O(n2(n+\u0394(n+\u0394\u00b7min{n,\u0394})))":[83],"time":[84],"O(n2(\u03942+1))":[86],"space,":[87],"since":[88],"every":[89],"tree":[90,98],"can":[91],"uniquely":[93],"viewed":[94],"by":[99,113],"either":[100],"regarding":[101],"its":[102],"unicentroid":[103],"root,":[106],"or":[107],"case":[110],"bicentroid,":[112],"introducing":[114],"virtual":[116,124],"vertex":[117,125],"bicentroid":[120],"assuming":[122],"root.":[129],"By":[130],"result,":[132],"get":[134],"lower":[136],"bound":[137,141],"upper":[140],"tree-like":[146],"polymer":[147],"topologies":[148],"chemical":[150],"compounds":[151],"\u201ccycle":[154],"rank\u201d.":[155]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":2}],"updated_date":"2026-08-11T07:18:39.950985","created_date":"2025-10-10T00:00:00"}
