{"id":"https://openalex.org/W7204257635","doi":"https://doi.org/10.48550/arxiv.2608.24861","title":"A Vertex-Localized Positive Square-Energy Strengthening of Tur\u00e1n's Theorem","display_name":"A Vertex-Localized Positive Square-Energy Strengthening of Tur\u00e1n's Theorem","publication_year":2026,"publication_date":"2026-08-25","ids":{"openalex":"https://openalex.org/W7204257635","doi":"https://doi.org/10.48550/arxiv.2608.24861"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.24861","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.24861","pdf_url":"https://arxiv.org/pdf/2608.24861","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2608.24861","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5139135143","display_name":"Abhay Jayarajan","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Jayarajan, Abhay","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5102731329","display_name":"M. Rajesh Kannan","orcid":"https://orcid.org/0000-0001-8038-1795","observed_orcids":["https://orcid.org/0000-0001-8038-1795"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Kannan, M. Rajesh","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5052555886","display_name":"Shivaramakrishna Pragada","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Pragada, Shivaramakrishna","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5122826038","display_name":"Rahul Roy","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Roy, Rahul","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"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/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":0.9990000128746033,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.00019999999494757503,"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":9.999999747378752e-05,"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/turan-s-theorem","display_name":"Tur\u00e1n's theorem","score":0.9800000190734863},{"id":"https://openalex.org/keywords/adjacency-eigenvalues","display_name":"adjacency matrix eigenvalues","score":0.7250000238418579},{"id":"https://openalex.org/keywords/clique-number","display_name":"clique number","score":0.7509999871253967},{"id":"https://openalex.org/keywords/complete-multipartite-graphs","display_name":"complete multipartite graphs","score":0.859000027179718},{"id":"https://openalex.org/keywords/isolated-vertices","display_name":"isolated vertices","score":0.7289999723434448},{"id":"https://openalex.org/keywords/graph-energy","display_name":"graph energy","score":0.3659999966621399}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.791100025177002},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5008000135421753},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4970000088214874},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.49399998784065247},{"id":"https://openalex.org/C45555294","wikidata":"https://www.wikidata.org/wiki/Q28113351","display_name":"Inequality","level":2,"score":0.4666000008583069},{"id":"https://openalex.org/C2777045944","wikidata":"https://www.wikidata.org/wiki/Q12170198","display_name":"Zh\u00e0ng","level":3,"score":0.44760000705718994},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.43650001287460327},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.40869998931884766},{"id":"https://openalex.org/C2777035058","wikidata":"https://www.wikidata.org/wiki/Q1662634","display_name":"Clique","level":2,"score":0.39100000262260437},{"id":"https://openalex.org/C32972407","wikidata":"https://www.wikidata.org/wiki/Q1187620","display_name":"Perfect graph","level":5,"score":0.32600000500679016},{"id":"https://openalex.org/C2776548862","wikidata":"https://www.wikidata.org/wiki/Q2694495","display_name":"Existence theorem","level":2,"score":0.30820000171661377},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.2648000121116638}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.24861","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.24861","pdf_url":"https://arxiv.org/pdf/2608.24861","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2608.24861","doi":"https://doi.org/10.48550/arxiv.2608.24861","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.24861","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2608.24861","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.24861","pdf_url":"https://arxiv.org/pdf/2608.24861","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320306227","display_name":"Arthritis National Research Foundation","ror":"https://ror.org/03abbex33"},{"id":"https://openalex.org/F4320320719","display_name":"Department of Science and Technology, Ministry of Science and Technology, India","ror":"https://ror.org/0101xrq71"},{"id":"https://openalex.org/F4320320767","display_name":"University Grants Commission","ror":"https://ror.org/04p800546"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204257635.pdf","grobid_xml":"https://content.openalex.org/works/W7204257635.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7169859200","https://openalex.org/W4416960217","https://openalex.org/W4403795305","https://openalex.org/W7203755424","https://openalex.org/W7169887649","https://openalex.org/W7211986434","https://openalex.org/W4402380364","https://openalex.org/W4415255454","https://openalex.org/W4417128720","https://openalex.org/W7163549230"],"abstract_inverted_index":{"Let":[0,18],"$G$":[1],"be":[2],"a":[3,25,62,77],"graph":[4,66],"of":[5,24,72,79],"order":[6,23],"$n$":[7],"with":[8],"the":[9,21,32],"adjacency":[10],"eigenvalues":[11],"$\u03bb_1(G)":[12],"\\geq":[13,15],"\\dots":[14],"\u03bb_n(G)":[16],"$.":[17],"$c(v)$":[19],"denote":[20],"maximum":[22],"clique":[26],"containing":[27],"vertex":[28],"$v$.":[29],"We":[30,47],"prove":[31],"vertex-localized":[33],"positive":[34],"square-energy":[35],"inequality":[36],"\\[":[37,44],"\\sqrt{s_+(G)}":[38],"\\leq":[39],"\\sum_{v\\in":[40],"V}\\left(1-\\frac1{c(v)}\\right),":[41],"\\]":[42,46],"where":[43],"s_+(G)=\\sum_{\u03bb_i(G)>0}\u03bb_i(G)^2.":[45],"also":[48],"characterize":[49],"equality.":[50],"Apart":[51],"from":[52,61],"edgeless":[53],"graphs,":[54],"equality":[55],"holds":[56],"precisely":[57],"for":[58],"graphs":[59],"obtained":[60],"complete":[63],"regular":[64],"multipartite":[65],"by":[67],"adding":[68],"an":[69],"arbitrary":[70],"number":[71],"isolated":[73],"vertices.":[74],"This":[75],"settles":[76],"conjecture":[78],"Kannan,":[80],"Kumar":[81],"and":[82],"Pragada.":[83]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-08-27T00:00:00"}
