{"id":"https://openalex.org/W4387184669","doi":"https://doi.org/10.1007/s00493-023-00068-x","title":"A Solution to Babai\u2019s Problems on Digraphs with Non-diagonalizable Adjacency Matrix","display_name":"A Solution to Babai\u2019s Problems on Digraphs with Non-diagonalizable Adjacency Matrix","publication_year":2023,"publication_date":"2023-09-29","ids":{"openalex":"https://openalex.org/W4387184669","doi":"https://doi.org/10.1007/s00493-023-00068-x"},"language":"en","primary_location":{"id":"doi:10.1007/s00493-023-00068-x","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00493-023-00068-x","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00493-023-00068-x.pdf","source":{"id":"https://openalex.org/S72366220","display_name":"COMBINATORICA","issn_l":"0209-9683","issn":["0209-9683","1439-6912"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorica","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00493-023-00068-x.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100330839","display_name":"Yuxuan Li","orcid":"https://orcid.org/0000-0002-7941-5887"},"institutions":[{"id":"https://openalex.org/I165779595","display_name":"University of Melbourne","ror":"https://ror.org/01ej9dk98","country_code":"AU","type":"education","lineage":["https://openalex.org/I165779595"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Yuxuan Li","raw_affiliation_strings":["School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia"],"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia","institution_ids":["https://openalex.org/I165779595"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5012479169","display_name":"Binzhou Xia","orcid":"https://orcid.org/0000-0002-8031-2981"},"institutions":[{"id":"https://openalex.org/I165779595","display_name":"University of Melbourne","ror":"https://ror.org/01ej9dk98","country_code":"AU","type":"education","lineage":["https://openalex.org/I165779595"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Binzhou Xia","raw_affiliation_strings":["School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia"],"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia","institution_ids":["https://openalex.org/I165779595"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5077967587","display_name":"Sanming Zhou","orcid":"https://orcid.org/0000-0001-9854-6076"},"institutions":[{"id":"https://openalex.org/I165779595","display_name":"University of Melbourne","ror":"https://ror.org/01ej9dk98","country_code":"AU","type":"education","lineage":["https://openalex.org/I165779595"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Sanming Zhou","raw_affiliation_strings":["School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia"],"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia","institution_ids":["https://openalex.org/I165779595"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5101164031","display_name":"Wenying Zhu","orcid":null},"institutions":[{"id":"https://openalex.org/I25254941","display_name":"Beijing Normal University","ror":"https://ror.org/022k4wk35","country_code":"CN","type":"education","lineage":["https://openalex.org/I25254941"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Wenying Zhu","raw_affiliation_strings":["Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China"],"affiliations":[{"raw_affiliation_string":"Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China","institution_ids":["https://openalex.org/I25254941"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5101164031"],"corresponding_institution_ids":["https://openalex.org/I25254941"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.25,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"44","issue":"1","first_page":"179","last_page":"203"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10849","display_name":"Finite Group Theory Research","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T10849","display_name":"Finite Group Theory Research","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T11476","display_name":"Graph theory and applications","score":0.9995999932289124,"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/T10374","display_name":"Advanced Graph Theory Research","score":0.9962000250816345,"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/diagonalizable-matrix","display_name":"Diagonalizable matrix","score":0.8214904069900513},{"id":"https://openalex.org/keywords/digraph","display_name":"Digraph","score":0.766538143157959},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7618417739868164},{"id":"https://openalex.org/keywords/adjacency-matrix","display_name":"Adjacency matrix","score":0.6963908076286316},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6410050392150879},{"id":"https://openalex.org/keywords/transitive-relation","display_name":"Transitive relation","score":0.5860815048217773},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.5747132301330566},{"id":"https://openalex.org/keywords/adjacency-list","display_name":"Adjacency list","score":0.5340828895568848},{"id":"https://openalex.org/keywords/directed-graph","display_name":"Directed graph","score":0.4192863702774048},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.41838985681533813},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.3462305963039398},{"id":"https://openalex.org/keywords/symmetric-matrix","display_name":"Symmetric matrix","score":0.16432228684425354},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.1213890016078949}],"concepts":[{"id":"https://openalex.org/C30072841","wikidata":"https://www.wikidata.org/wiki/Q1767080","display_name":"Diagonalizable matrix","level":4,"score":0.8214904069900513},{"id":"https://openalex.org/C2779145032","wikidata":"https://www.wikidata.org/wiki/Q191494","display_name":"Digraph","level":2,"score":0.766538143157959},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7618417739868164},{"id":"https://openalex.org/C180356752","wikidata":"https://www.wikidata.org/wiki/Q727035","display_name":"Adjacency matrix","level":3,"score":0.6963908076286316},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6410050392150879},{"id":"https://openalex.org/C191399111","wikidata":"https://www.wikidata.org/wiki/Q64861","display_name":"Transitive relation","level":2,"score":0.5860815048217773},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.5747132301330566},{"id":"https://openalex.org/C110484373","wikidata":"https://www.wikidata.org/wiki/Q264398","display_name":"Adjacency list","level":2,"score":0.5340828895568848},{"id":"https://openalex.org/C146380142","wikidata":"https://www.wikidata.org/wiki/Q1137726","display_name":"Directed graph","level":2,"score":0.4192863702774048},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.41838985681533813},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.3462305963039398},{"id":"https://openalex.org/C54848796","wikidata":"https://www.wikidata.org/wiki/Q339011","display_name":"Symmetric matrix","level":3,"score":0.16432228684425354},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.1213890016078949},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s00493-023-00068-x","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00493-023-00068-x","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00493-023-00068-x.pdf","source":{"id":"https://openalex.org/S72366220","display_name":"COMBINATORICA","issn_l":"0209-9683","issn":["0209-9683","1439-6912"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorica","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s00493-023-00068-x","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00493-023-00068-x","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00493-023-00068-x.pdf","source":{"id":"https://openalex.org/S72366220","display_name":"COMBINATORICA","issn_l":"0209-9683","issn":["0209-9683","1439-6912"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorica","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G8589651859","display_name":null,"funder_award_id":"China","funder_id":"https://openalex.org/F4320322725","funder_display_name":"China Scholarship Council"}],"funders":[{"id":"https://openalex.org/F4320320974","display_name":"University of Melbourne","ror":"https://ror.org/01ej9dk98"},{"id":"https://openalex.org/F4320321432","display_name":"Beijing Normal University","ror":"https://ror.org/022k4wk35"},{"id":"https://openalex.org/F4320322725","display_name":"China Scholarship Council","ror":"https://ror.org/04atp4p48"}],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4387184669.pdf"},"referenced_works_count":15,"referenced_works":["https://openalex.org/W388525125","https://openalex.org/W606453859","https://openalex.org/W1488435683","https://openalex.org/W1495591881","https://openalex.org/W1976677460","https://openalex.org/W1980268457","https://openalex.org/W1999850363","https://openalex.org/W2042601287","https://openalex.org/W2045743529","https://openalex.org/W2120980679","https://openalex.org/W2125731539","https://openalex.org/W2594299571","https://openalex.org/W3039948875","https://openalex.org/W4234295943","https://openalex.org/W4238591275"],"related_works":["https://openalex.org/W1976986376","https://openalex.org/W2350235717","https://openalex.org/W2950318982","https://openalex.org/W3083664609","https://openalex.org/W4316126606","https://openalex.org/W1986517672","https://openalex.org/W2354198990","https://openalex.org/W4294156822","https://openalex.org/W3103368965","https://openalex.org/W4387184669"],"abstract_inverted_index":{"Abstract":[0],"The":[1],"fact":[2,23],"that":[3],"the":[4,53,63,76,88],"adjacency":[5,70,101,146],"matrix":[6],"of":[7,44,65,91,118,140,144],"every":[8],"finite":[9],"graph":[10,19],"is":[11,32],"diagonalizable":[12],"plays":[13],"a":[14,93,97],"fundamental":[15],"role":[16],"in":[17,27,47,75],"spectral":[18],"theory.":[20],"Since":[21],"this":[22,48,107],"does":[24],"not":[25,104],"hold":[26],"general":[28],"for":[29,39,62,122],"digraphs,":[30,142],"it":[31,37],"natural":[33],"to":[34,52],"ask":[35],"whether":[36],"holds":[38],"digraphs":[40,67,121],"with":[41,68],"certain":[42],"level":[43],"symmetry.":[45],"Interest":[46],"question":[49],"dates":[50],"back":[51],"early":[54],"1980":[55],"s,":[56],"when":[57],"P.":[58],"J.":[59],"Cameron":[60],"asked":[61],"existence":[64],"arc-transitive":[66],"non-diagonalizable":[69],"matrix.":[71],"This":[72],"was":[73],"answered":[74],"affirmative":[77],"by":[78,113],"Babai":[79,86],"(J":[80],"Graph":[81],"Theory":[82],"9:363\u2013370,":[83],"1985).":[84],"Then":[85],"posed":[87],"open":[89],"problems":[90,112],"constructing":[92,114],"2-arc-transitive":[94],"digraph":[95,99],"and":[96,136],"vertex-primitive":[98,141],"whose":[100,145],"matrices":[102,147],"are":[103,148],"diagonalizable.":[105],"In":[106],"paper,":[108],"we":[109],"solve":[110],"Babai\u2019s":[111],"an":[115,137],"infinite":[116,138],"family":[117,139],"s":[119],"-arc-transitive":[120],"each":[123],"integer":[124],"$$s\\ge":[125],"2$$":[126],"<mml:math":[127],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[128],"<mml:mrow>":[129],"<mml:mi>s</mml:mi>":[130],"<mml:mo>\u2265</mml:mo>":[131],"<mml:mn>2</mml:mn>":[132],"</mml:mrow>":[133],"</mml:math>":[134],",":[135],"both":[143],"non-diagonalizable.":[149]},"counts_by_year":[],"updated_date":"2026-03-12T08:34:05.389933","created_date":"2025-10-10T00:00:00"}