{"id":"https://openalex.org/W7169814005","doi":"https://doi.org/10.48550/arxiv.2607.15941","title":"Generalized Nordhaus--Gaddum Inequalities for Eigenvalues","display_name":"Generalized Nordhaus--Gaddum Inequalities for Eigenvalues","publication_year":2026,"publication_date":"2026-07-17","ids":{"openalex":"https://openalex.org/W7169814005","doi":"https://doi.org/10.48550/arxiv.2607.15941"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2607.15941","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2607.15941","pdf_url":"https://arxiv.org/pdf/2607.15941","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2607.15941","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5141170438","display_name":"Sahil Agarwal","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I155781252","display_name":"University of South Carolina","ror":"https://ror.org/02b6qw903","country_code":"US","type":"education","lineage":["https://openalex.org/I155781252"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Agarwal, Sahil","raw_affiliation_strings":["University of South Carolina , Columbia , SC 29208 "],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of South Carolina , Columbia , SC 29208 ","institution_ids":["https://openalex.org/I155781252"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5141169912","display_name":"Carter Antley","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I155781252","display_name":"University of South Carolina","ror":"https://ror.org/02b6qw903","country_code":"US","type":"education","lineage":["https://openalex.org/I155781252"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Antley, Carter","raw_affiliation_strings":["University of South Carolina , Columbia , SC 29208 "],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of South Carolina , Columbia , SC 29208 ","institution_ids":["https://openalex.org/I155781252"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5141201267","display_name":"Joseph Aulenbacher","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I155781252","display_name":"University of South Carolina","ror":"https://ror.org/02b6qw903","country_code":"US","type":"education","lineage":["https://openalex.org/I155781252"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Aulenbacher, Joseph","raw_affiliation_strings":["University of South Carolina , Columbia , SC 29208 "],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of South Carolina , Columbia , SC 29208 ","institution_ids":["https://openalex.org/I155781252"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5111143279","display_name":"George Brooks","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I155781252","display_name":"University of South Carolina","ror":"https://ror.org/02b6qw903","country_code":"US","type":"education","lineage":["https://openalex.org/I155781252"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Brooks, George","raw_affiliation_strings":["University of South Carolina , Columbia , SC 29208 "],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of South Carolina , Columbia , SC 29208 ","institution_ids":["https://openalex.org/I155781252"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5071191859","display_name":"Ian Gonz\u00e1lez","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I155781252","display_name":"University of South Carolina","ror":"https://ror.org/02b6qw903","country_code":"US","type":"education","lineage":["https://openalex.org/I155781252"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Gonzalez, Ian","raw_affiliation_strings":["University of South Carolina , Columbia , SC 29208 "],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of South Carolina , Columbia , SC 29208 ","institution_ids":["https://openalex.org/I155781252"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5120500127","display_name":"Luke Hawranick","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I155781252","display_name":"University of South Carolina","ror":"https://ror.org/02b6qw903","country_code":"US","type":"education","lineage":["https://openalex.org/I155781252"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Hawranick, Luke","raw_affiliation_strings":["University of South Carolina , Columbia , SC 29208 "],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of South Carolina , Columbia , SC 29208 ","institution_ids":["https://openalex.org/I155781252"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5067520532","display_name":"William Linz","orcid":"https://orcid.org/0009-0006-0579-0576","observed_orcids":["https://orcid.org/0009-0006-0579-0576"]},"institutions":[{"id":"https://openalex.org/I155781252","display_name":"University of South Carolina","ror":"https://ror.org/02b6qw903","country_code":"US","type":"education","lineage":["https://openalex.org/I155781252"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Linz, William","raw_affiliation_strings":["University of South Carolina , Columbia , SC 29208 "],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of South Carolina , Columbia , SC 29208 ","institution_ids":["https://openalex.org/I155781252"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5146383599","display_name":"Linyuan Lu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I155781252","display_name":"University of South Carolina","ror":"https://ror.org/02b6qw903","country_code":"US","type":"education","lineage":["https://openalex.org/I155781252"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Lu, Linyuan","raw_affiliation_strings":["University of South Carolina , Columbia , SC 29208 "],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of South Carolina , Columbia , SC 29208 ","institution_ids":["https://openalex.org/I155781252"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5141185883","display_name":"Aiden Williams","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I155781252","display_name":"University of South Carolina","ror":"https://ror.org/02b6qw903","country_code":"US","type":"education","lineage":["https://openalex.org/I155781252"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Williams, Aiden","raw_affiliation_strings":["University of South Carolina , Columbia , SC 29208 "],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of South Carolina , Columbia , SC 29208 ","institution_ids":["https://openalex.org/I155781252"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I155781252"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"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":0.9990000128746033,"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/T11476","display_name":"Graph theory and applications","score":0.9990000128746033,"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/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.0003000000142492354,"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"}},{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.00019999999494757503,"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/adjacency-eigenvalues","display_name":"adjacency matrix eigenvalues","score":0.9639999866485596},{"id":"https://openalex.org/keywords/spectral-radius","display_name":"spectral radius","score":0.7480000257492065},{"id":"https://openalex.org/keywords/graph-complement","display_name":"graph complement","score":0.8410000205039978},{"id":"https://openalex.org/keywords/spectral-gap","display_name":"spectral gap","score":0.5009999871253967},{"id":"https://openalex.org/keywords/extremal-graph-theory","display_name":"extremal graph theory","score":0.722000002861023},{"id":"https://openalex.org/keywords/eigenvalue-bounds","display_name":"eigenvalue bounds","score":0.6520000100135803}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8600999712944031},{"id":"https://openalex.org/C140532419","wikidata":"https://www.wikidata.org/wiki/Q249748","display_name":"Spectral radius","level":3,"score":0.7027999758720398},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.6800000071525574},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5091000199317932},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.47369998693466187},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4388999938964844},{"id":"https://openalex.org/C110484373","wikidata":"https://www.wikidata.org/wiki/Q264398","display_name":"Adjacency list","level":2,"score":0.3946000039577484},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.3686000108718872},{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.34360000491142273},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.3370000123977661},{"id":"https://openalex.org/C180356752","wikidata":"https://www.wikidata.org/wiki/Q727035","display_name":"Adjacency matrix","level":3,"score":0.335099995136261},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.2833000123500824},{"id":"https://openalex.org/C2780841128","wikidata":"https://www.wikidata.org/wiki/Q5073781","display_name":"Characterization (materials science)","level":2,"score":0.28110000491142273},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.27810001373291016},{"id":"https://openalex.org/C25343380","wikidata":"https://www.wikidata.org/wiki/Q277521","display_name":"Relation (database)","level":2,"score":0.27160000801086426},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.26249998807907104},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.2531000077724457}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2607.15941","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2607.15941","pdf_url":"https://arxiv.org/pdf/2607.15941","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2607.15941","doi":"https://doi.org/10.48550/arxiv.2607.15941","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2607.15941","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2607.15941","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2607.15941","pdf_url":"https://arxiv.org/pdf/2607.15941","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G8921292286","display_name":"RTG: Mathematical Foundation of Data Science at University of South Carolina","funder_award_id":"2038080","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7169814005.pdf","grobid_xml":"https://content.openalex.org/works/W7169814005.grobid-xml"},"referenced_works_count":5,"referenced_works":["https://openalex.org/W6891895912","https://openalex.org/W7140174796","https://openalex.org/W7147099471","https://openalex.org/W7147345047","https://openalex.org/W7148251236"],"related_works":["https://openalex.org/W1982906615","https://openalex.org/W2950191773","https://openalex.org/W7099372244","https://openalex.org/W2736657833","https://openalex.org/W7095231570","https://openalex.org/W2024841505","https://openalex.org/W2940644921","https://openalex.org/W2489868914","https://openalex.org/W2105661768","https://openalex.org/W2108014672"],"abstract_inverted_index":{"For":[0],"a":[1,39],"graph":[2],"$G$,":[3],"let":[4],"$":[5],"\u03bb_1(G)\\ge":[6],"\u03bb_2(G)\\ge":[7],"\\cdots":[8],"\\ge":[9],"\u03bb_n(G)$":[10],"denote":[11],"the":[12,44,48,64,96,112,124],"adjacency":[13],"eigenvalues":[14],"of":[15,43,67,99,115,128],"$G$.":[16],"We":[17,62,77,109],"prove":[18,78],"that":[19,54],"for":[20,47,72,85,104],"all":[21,86],"looped":[22],"graphs":[23,129],"$G$":[24],"on":[25,81,95,123],"$n$":[26],"vertices,":[27],"\\[\u03bb_1(G)":[28],"+":[29,56,83,102],"\u03bb_2(\\overline{G})":[30],"\\le":[31,58],"\\frac87":[32],"n.":[33],"\\]":[34,71],"Our":[35],"method":[36],"also":[37,91,110],"gives":[38],"short":[40],"new":[41],"proof":[42],"Nordhaus-Gaddum":[45,117],"result":[46],"spectral":[49,126],"radius":[50],"proved":[51],"by":[52,130],"Terpai":[53],"$\u03bb_1(G)":[55],"\u03bb_1(\\overline{G})":[57],"\\frac43n":[59],"-":[60],"1$.":[61],"investigate":[63],"asymptotic":[65],"maximum":[66,125],"\\[":[68],"\u03bb_i(G)+\u03bb_j(\\overline":[69],"G)":[70],"fixed":[73,105],"$i$":[74,106],"and":[75,90,107,133],"$j$.":[76,108],"general":[79,93],"bounds":[80,94],"$\u03bb_i(G)":[82],"\u03bb_{j}(\\overline{G})$":[84],"pairs":[87],"$(i,":[88],"j)$":[89],"give":[92],"related":[97],"problem":[98],"minimizing":[100],"$\u03bb_{n-i+1}(G)":[101],"\u03bb_{n-j+1}(\\overline{G})$":[103],"show":[111],"close":[113],"relation":[114],"these":[116],"type":[118],"problems":[119],"to":[120],"recent":[121],"work":[122],"gaps":[127],"Brooks,":[131],"Linz":[132],"Lu.":[134]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-07-21T00:00:00"}
