{"id":"https://openalex.org/W4415671216","doi":"https://doi.org/10.1007/s00037-025-00270-x","title":"On a Hierarchy of Spectral Isomorphism Invariants","display_name":"On a Hierarchy of Spectral Isomorphism Invariants","publication_year":2025,"publication_date":"2025-10-29","ids":{"openalex":"https://openalex.org/W4415671216","doi":"https://doi.org/10.1007/s00037-025-00270-x"},"language":"en","primary_location":{"id":"doi:10.1007/s00037-025-00270-x","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00037-025-00270-x","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00037-025-00270-x.pdf","source":{"id":"https://openalex.org/S193995496","display_name":"Computational Complexity","issn_l":"1016-3328","issn":["1016-3328","1420-8954"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320186","host_organization_name":"Birkh\u00e4user","host_organization_lineage":["https://openalex.org/P4310320186","https://openalex.org/P4310319900"],"host_organization_lineage_names":["Birkh\u00e4user","Springer Science+Business Media"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"computational complexity","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/s00037-025-00270-x.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5003212467","display_name":"V. Arvind","orcid":"https://orcid.org/0000-0002-1988-7866"},"institutions":[{"id":"https://openalex.org/I19149307","display_name":"Chennai Mathematical Institute","ror":"https://ror.org/04zp24820","country_code":"IN","type":"education","lineage":["https://openalex.org/I19149307"]},{"id":"https://openalex.org/I59611059","display_name":"Institute of Mathematical Sciences","ror":"https://ror.org/05078rg59","country_code":"IN","type":"facility","lineage":["https://openalex.org/I59611059"]}],"countries":["IN"],"is_corresponding":true,"raw_author_name":"V. Arvind","raw_affiliation_strings":["The Institute of Mathematical Sciences (HBNI) and Chennai, Mathematical Institute, Chennai, 600 113, India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"The Institute of Mathematical Sciences (HBNI) and Chennai, Mathematical Institute, Chennai, 600 113, India","institution_ids":["https://openalex.org/I59611059","https://openalex.org/I19149307"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5056964913","display_name":"Frank Fuhlbr\u00fcck","orcid":"https://orcid.org/0000-0001-6007-0922"},"institutions":[{"id":"https://openalex.org/I39343248","display_name":"Humboldt-Universit\u00e4t zu Berlin","ror":"https://ror.org/01hcx6992","country_code":"DE","type":"education","lineage":["https://openalex.org/I39343248"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Frank Fuhlbr\u00fcck","raw_affiliation_strings":["Institut f\u00fcr Informatik, Humboldt Universit\u00e4t zu Berlin, Unter den Linden 6, Berlin, 10099, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Informatik, Humboldt Universit\u00e4t zu Berlin, Unter den Linden 6, Berlin, 10099, Germany","institution_ids":["https://openalex.org/I39343248"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5103016415","display_name":"Johannes K\u00f6bler","orcid":"https://orcid.org/0000-0002-1215-7270"},"institutions":[{"id":"https://openalex.org/I39343248","display_name":"Humboldt-Universit\u00e4t zu Berlin","ror":"https://ror.org/01hcx6992","country_code":"DE","type":"education","lineage":["https://openalex.org/I39343248"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Johannes K\u00f6bler","raw_affiliation_strings":["Institut f\u00fcr Informatik, Humboldt Universit\u00e4t zu Berlin, Unter den Linden 6, Berlin, 10099, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Informatik, Humboldt Universit\u00e4t zu Berlin, Unter den Linden 6, Berlin, 10099, Germany","institution_ids":["https://openalex.org/I39343248"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5064842516","display_name":"Oleg Verbitsky","orcid":"https://orcid.org/0000-0002-9524-1901"},"institutions":[{"id":"https://openalex.org/I39343248","display_name":"Humboldt-Universit\u00e4t zu Berlin","ror":"https://ror.org/01hcx6992","country_code":"DE","type":"education","lineage":["https://openalex.org/I39343248"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Oleg Verbitsky","raw_affiliation_strings":["Institut f\u00fcr Informatik, Humboldt Universit\u00e4t zu Berlin, Unter den Linden 6, Berlin, 10099, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Informatik, Humboldt Universit\u00e4t zu Berlin, Unter den Linden 6, Berlin, 10099, Germany","institution_ids":["https://openalex.org/I39343248"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5003212467"],"corresponding_institution_ids":["https://openalex.org/I19149307","https://openalex.org/I59611059"],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":{"value":2290,"currency":"EUR","value_usd":2890},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.35019661,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"34","issue":"2","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":0.8510000109672546,"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":0.8510000109672546,"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":0.023399999365210533,"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":0.02019999921321869,"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/graph-isomorphism","display_name":"Graph isomorphism","score":0.6722000241279602},{"id":"https://openalex.org/keywords/hierarchy","display_name":"Hierarchy","score":0.586899995803833},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.508400022983551},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.4878999888896942},{"id":"https://openalex.org/keywords/adjacency-list","display_name":"Adjacency list","score":0.4584999978542328},{"id":"https://openalex.org/keywords/isomorphism","display_name":"Isomorphism (crystallography)","score":0.4426000118255615},{"id":"https://openalex.org/keywords/spectral-graph-theory","display_name":"Spectral graph theory","score":0.41290000081062317},{"id":"https://openalex.org/keywords/characterization","display_name":"Characterization (materials science)","score":0.40790000557899475}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7864000201225281},{"id":"https://openalex.org/C61665672","wikidata":"https://www.wikidata.org/wiki/Q303100","display_name":"Graph isomorphism","level":4,"score":0.6722000241279602},{"id":"https://openalex.org/C31170391","wikidata":"https://www.wikidata.org/wiki/Q188619","display_name":"Hierarchy","level":2,"score":0.586899995803833},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.508400022983551},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.4878999888896942},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.46549999713897705},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4636000096797943},{"id":"https://openalex.org/C110484373","wikidata":"https://www.wikidata.org/wiki/Q264398","display_name":"Adjacency list","level":2,"score":0.4584999978542328},{"id":"https://openalex.org/C203436722","wikidata":"https://www.wikidata.org/wiki/Q902950","display_name":"Isomorphism (crystallography)","level":3,"score":0.4426000118255615},{"id":"https://openalex.org/C74003402","wikidata":"https://www.wikidata.org/wiki/Q3180727","display_name":"Spectral graph theory","level":5,"score":0.41290000081062317},{"id":"https://openalex.org/C2780841128","wikidata":"https://www.wikidata.org/wiki/Q5073781","display_name":"Characterization (materials science)","level":2,"score":0.40790000557899475},{"id":"https://openalex.org/C180356752","wikidata":"https://www.wikidata.org/wiki/Q727035","display_name":"Adjacency matrix","level":3,"score":0.38830000162124634},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.3734000027179718},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.35409998893737793},{"id":"https://openalex.org/C64339825","wikidata":"https://www.wikidata.org/wiki/Q722659","display_name":"Graph property","level":5,"score":0.35339999198913574},{"id":"https://openalex.org/C12426560","wikidata":"https://www.wikidata.org/wiki/Q189569","display_name":"Basis (linear algebra)","level":2,"score":0.3174000084400177},{"id":"https://openalex.org/C56654003","wikidata":"https://www.wikidata.org/wiki/Q7858715","display_name":"Two-graph","level":5,"score":0.31619998812675476},{"id":"https://openalex.org/C2985906921","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Spectral properties","level":2,"score":0.30809998512268066},{"id":"https://openalex.org/C7036158","wikidata":"https://www.wikidata.org/wiki/Q3385162","display_name":"Graph homomorphism","level":5,"score":0.27230000495910645},{"id":"https://openalex.org/C18116961","wikidata":"https://www.wikidata.org/wiki/Q734209","display_name":"Generating set of a group","level":2,"score":0.26910001039505005},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.2524000108242035}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s00037-025-00270-x","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00037-025-00270-x","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00037-025-00270-x.pdf","source":{"id":"https://openalex.org/S193995496","display_name":"Computational Complexity","issn_l":"1016-3328","issn":["1016-3328","1420-8954"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320186","host_organization_name":"Birkh\u00e4user","host_organization_lineage":["https://openalex.org/P4310320186","https://openalex.org/P4310319900"],"host_organization_lineage_names":["Birkh\u00e4user","Springer Science+Business Media"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"computational complexity","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s00037-025-00270-x","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00037-025-00270-x","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00037-025-00270-x.pdf","source":{"id":"https://openalex.org/S193995496","display_name":"Computational Complexity","issn_l":"1016-3328","issn":["1016-3328","1420-8954"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320186","host_organization_name":"Birkh\u00e4user","host_organization_lineage":["https://openalex.org/P4310320186","https://openalex.org/P4310319900"],"host_organization_lineage_names":["Birkh\u00e4user","Springer Science+Business Media"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"computational complexity","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G5465230307","display_name":null,"funder_award_id":"KO 1053/8-2","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"}],"funders":[{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4415671216.pdf","grobid_xml":"https://content.openalex.org/works/W4415671216.grobid-xml"},"referenced_works_count":21,"referenced_works":["https://openalex.org/W1529660315","https://openalex.org/W1530202807","https://openalex.org/W1970491586","https://openalex.org/W1971347998","https://openalex.org/W2000173816","https://openalex.org/W2018047324","https://openalex.org/W2026820792","https://openalex.org/W2051748083","https://openalex.org/W2069473647","https://openalex.org/W2104550457","https://openalex.org/W2171567099","https://openalex.org/W2181072414","https://openalex.org/W2409645877","https://openalex.org/W2524594508","https://openalex.org/W2963968673","https://openalex.org/W3091502386","https://openalex.org/W3124501589","https://openalex.org/W3211555008","https://openalex.org/W4213018468","https://openalex.org/W4237658155","https://openalex.org/W4316652379"],"related_works":[],"abstract_inverted_index":{"Abstract":[0],"We":[1,42],"consider":[2],"a":[3,44,62,165],"hierarchy":[4,50,149],"of":[5,29,38,48,53,71,89,98,114,122,126,164],"graph":[6,166],"invariants":[7,13,73],"that":[8,103],"naturally":[9],"extends":[10],"the":[11,23,27,39,54,69,81,86,99,115,118,127,144,148,151,169,172],"spectral":[12,162],"defined":[14],"by":[15,26,137],"F\u00fcrer":[16],"(Lin.":[17],"Alg.":[18],"Appl.":[19],"2010)":[20],"based":[21],"on":[22],"angles":[24],"formed":[25],"set":[28],"standard":[30],"basis":[31],"vectors":[32],"and":[33,91,117,150,171],"their":[34,138],"projections":[35],"onto":[36],"eigenspaces":[37],"adjacency":[40],"matrix.":[41],"provide":[43],"purely":[45],"combinatorial":[46],"characterization":[47],"this":[49],"in":[51,74,78,112,124],"terms":[52,113],"walk":[55],"counts.":[56],"This":[57],"allows":[58],"us":[59],"to":[60,65,110],"give":[61],"complete":[63],"answer":[64],"F\u00fcrer's":[66],"question":[67],"about":[68],"strength":[70],"his":[72],"distinguishing":[75],"non-isomorphic":[76],"graphs":[77,106,134],"comparison":[79],"with":[80],"2-dimensional":[82],"Weisfeiler-Leman":[83,152],"algorithm,":[84],"extending":[85],"recent":[87],"work":[88],"Rattan":[90],"Seppelt":[92],"(SODA":[93],"2023).":[94],"As":[95],"another":[96],"application":[97],"characterization,":[100],"we":[101,142],"prove":[102],"almost":[104,132],"all":[105,133],"are":[107,135],"determined":[108,136],"up":[109],"isomorphism":[111],"spectrum":[116],"angles,":[119],"which":[120],"is":[121],"interest":[123],"view":[125],"long-standing":[128],"open":[129],"problem":[130],"whether":[131],"eigenvalues":[139],"alone.":[140],"Finally,":[141],"describe":[143],"exact":[145],"relationship":[146],"between":[147],"algorithms":[153],"for":[154],"small":[155],"dimensions,":[156],"as":[157,168],"also":[158],"some":[159],"other":[160],"important":[161],"characteristics":[163],"such":[167],"generalized":[170],"main":[173],"spectra.":[174]},"counts_by_year":[],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-29T00:00:00"}