{"id":"https://openalex.org/W2149407242","doi":"https://doi.org/10.37236/1065","title":"Kernels of Directed Graph Laplacians","display_name":"Kernels of Directed Graph Laplacians","publication_year":2006,"publication_date":"2006-04-11","ids":{"openalex":"https://openalex.org/W2149407242","doi":"https://doi.org/10.37236/1065","mag":"2149407242"},"language":"en","primary_location":{"id":"doi:10.37236/1065","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1065","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v13i1r39/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v13i1r39/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5030281725","display_name":"John S. Caughman","orcid":"https://orcid.org/0000-0002-2692-9198"},"institutions":[{"id":"https://openalex.org/I126345244","display_name":"Portland State University","ror":"https://ror.org/00yn2fy02","country_code":"US","type":"education","lineage":["https://openalex.org/I126345244"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"J. S. Caughman","raw_affiliation_strings":["Department of Mathematics and Statistics Portland State University PO Box 751, Portland, OR 97207"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics Portland State University PO Box 751, Portland, OR 97207","institution_ids":["https://openalex.org/I126345244"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5059241473","display_name":"J. J. P. Veerman","orcid":"https://orcid.org/0000-0003-4551-5908"},"institutions":[{"id":"https://openalex.org/I126345244","display_name":"Portland State University","ror":"https://ror.org/00yn2fy02","country_code":"US","type":"education","lineage":["https://openalex.org/I126345244"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"J. J. P. Veerman","raw_affiliation_strings":["Department of Mathematics and Statistics Portland State University PO Box 751, Portland, OR 97207"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics Portland State University PO Box 751, Portland, OR 97207","institution_ids":["https://openalex.org/I126345244"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5030281725"],"corresponding_institution_ids":["https://openalex.org/I126345244"],"apc_list":null,"apc_paid":null,"fwci":4.1028,"has_fulltext":true,"cited_by_count":91,"citation_normalized_percentile":{"value":0.94332572,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"13","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.9998999834060669,"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":0.9998999834060669,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9995999932289124,"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"}},{"id":"https://openalex.org/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.9976999759674072,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8019247055053711},{"id":"https://openalex.org/keywords/adjacency-matrix","display_name":"Adjacency matrix","score":0.6786876916885376},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6580250859260559},{"id":"https://openalex.org/keywords/directed-graph","display_name":"Directed graph","score":0.6261844635009766},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.6087033152580261},{"id":"https://openalex.org/keywords/algebraic-connectivity","display_name":"Algebraic connectivity","score":0.5996302962303162},{"id":"https://openalex.org/keywords/laplacian-matrix","display_name":"Laplacian matrix","score":0.526342511177063},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.49968910217285156},{"id":"https://openalex.org/keywords/degree-matrix","display_name":"Degree matrix","score":0.49659615755081177},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.42420631647109985},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.40108492970466614},{"id":"https://openalex.org/keywords/line-graph","display_name":"Line graph","score":0.3866868019104004},{"id":"https://openalex.org/keywords/graph-power","display_name":"Graph power","score":0.14285850524902344}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8019247055053711},{"id":"https://openalex.org/C180356752","wikidata":"https://www.wikidata.org/wiki/Q727035","display_name":"Adjacency matrix","level":3,"score":0.6786876916885376},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6580250859260559},{"id":"https://openalex.org/C146380142","wikidata":"https://www.wikidata.org/wiki/Q1137726","display_name":"Directed graph","level":2,"score":0.6261844635009766},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.6087033152580261},{"id":"https://openalex.org/C7948225","wikidata":"https://www.wikidata.org/wiki/Q4723998","display_name":"Algebraic connectivity","level":4,"score":0.5996302962303162},{"id":"https://openalex.org/C115178988","wikidata":"https://www.wikidata.org/wiki/Q772067","display_name":"Laplacian matrix","level":3,"score":0.526342511177063},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.49968910217285156},{"id":"https://openalex.org/C162199024","wikidata":"https://www.wikidata.org/wiki/Q3085391","display_name":"Degree matrix","level":5,"score":0.49659615755081177},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.42420631647109985},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.40108492970466614},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.3866868019104004},{"id":"https://openalex.org/C149530733","wikidata":"https://www.wikidata.org/wiki/Q5597091","display_name":"Graph power","level":4,"score":0.14285850524902344},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"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/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.37236/1065","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1065","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v13i1r39/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},{"id":"pmh:oai:pdxscholar.library.pdx.edu:mth_fac-1029","is_oa":false,"landing_page_url":"https://pdxscholar.library.pdx.edu/mth_fac/30","pdf_url":null,"source":{"id":"https://openalex.org/S4377196300","display_name":"PDXScholar  (Portland State University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I126345244","host_organization_name":"Portland State University","host_organization_lineage":["https://openalex.org/I126345244"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Mathematics and Statistics Faculty Publications and Presentations","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.175.8622","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.175.8622","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.emis.de/journals/EJC/Volume_13/PDF/v13i1r39.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.37236/1065","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1065","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v13i1r39/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2149407242.pdf","grobid_xml":"https://content.openalex.org/works/W2149407242.grobid-xml"},"referenced_works_count":7,"referenced_works":["https://openalex.org/W193717806","https://openalex.org/W2001864568","https://openalex.org/W2073108694","https://openalex.org/W2093520943","https://openalex.org/W2610857016","https://openalex.org/W3099358910","https://openalex.org/W6668541753"],"related_works":["https://openalex.org/W2362717856","https://openalex.org/W2891080236","https://openalex.org/W1557246723","https://openalex.org/W1977857744","https://openalex.org/W2531476721","https://openalex.org/W2028183796","https://openalex.org/W2105661768","https://openalex.org/W2128448454","https://openalex.org/W2124916376","https://openalex.org/W2030274157"],"abstract_inverted_index":{"Let":[0],"$G$":[1,35],"denote":[2],"a":[3,55,107,133],"directed":[4,25,59,124,156],"graph":[5],"with":[6,158],"adjacency":[7],"matrix":[8,12,18],"$Q$":[9],"and":[10,67,85,98,105,151],"in-degree":[11],"$D$.":[13],"We":[14,126],"consider":[15],"the":[16,24,38,41,45,79,96,111,118,123,137,145],"Kirchhoff":[17,31],"$L=D-Q$,":[19],"sometimes":[20],"referred":[21],"to":[22,58,155],"as":[23,61],"Laplacian.":[26],"A":[27],"classical":[28],"result":[29,73],"of":[30,40,47,50,88,101,113,120,122,148],"asserts":[32],"that":[33,95,106],"when":[34],"is":[36],"undirected,":[37],"multiplicity":[39],"eigenvalue":[42],"0":[43,102],"equals":[44],"number":[46,119],"connected":[48],"components":[49],"$G$.":[51],"This":[52],"fact":[53],"has":[54,74],"meaningful":[56],"generalization":[57],"graphs,":[60],"was":[62],"recently":[63],"observed":[64],"by":[65,131],"Chebotarev":[66],"Agaev":[68],"in":[69,78,92,144],"2005.":[70],"Since":[71],"this":[72,93,114],"many":[75],"important":[76],"applications":[77],"sciences,":[80],"we":[81],"offer":[82],"an":[83],"independent":[84],"self-contained":[86],"proof":[87],"their":[89,129],"theorem,":[90],"showing":[91],"paper":[94],"algebraic":[97],"geometric":[99],"multiplicities":[100],"are":[103,142],"equal,":[104],"graph-theoretic":[108],"property":[109],"determines":[110],"dimension":[112],"eigenspace":[115],"\u2013":[116],"namely,":[117],"reaches":[121],"graph.":[125],"also":[127],"extend":[128],"results":[130,141],"deriving":[132],"natural":[134],"basis":[135],"for":[136],"corresponding":[138],"eigenspace.":[139],"The":[140],"proved":[143],"general":[146],"context":[147],"stochastic":[149],"matrices,":[150],"apply":[152],"equally":[153],"well":[154],"graphs":[157],"non-negative":[159],"edge":[160],"weights.":[161]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":12},{"year":2023,"cited_by_count":8},{"year":2022,"cited_by_count":6},{"year":2021,"cited_by_count":9},{"year":2020,"cited_by_count":7},{"year":2019,"cited_by_count":5},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":5},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":5},{"year":2012,"cited_by_count":2}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}
