{"id":"https://openalex.org/W7222423744","doi":"https://doi.org/10.48550/arxiv.2610.20648","title":"Connectivity and Diameter of Friends and Strangers Graphs Using Minimum Degree","display_name":"Connectivity and Diameter of Friends and Strangers Graphs Using Minimum Degree","publication_year":2026,"publication_date":"2026-10-12","ids":{"openalex":"https://openalex.org/W7222423744","doi":"https://doi.org/10.48550/arxiv.2610.20648"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.20648","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.20648","pdf_url":"https://arxiv.org/pdf/2610.20648","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/2610.20648","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5125691849","display_name":"Kai Mawhinney","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Mawhinney, Kai","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5125691849"],"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/T10374","display_name":"Advanced Graph Theory Research","score":0.8101000189781189,"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"}},"topics":[{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.8101000189781189,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.14749999344348907,"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":0.016499999910593033,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/graph-connectivity","display_name":"graph connectivity","score":0.8370000123977661},{"id":"https://openalex.org/keywords/graph-diameter","display_name":"graph diameter","score":0.9409999847412109},{"id":"https://openalex.org/keywords/minimum-degree","display_name":"minimum degree","score":0.8500000238418579},{"id":"https://openalex.org/keywords/transposition","display_name":"transposition","score":0.8100000023841858},{"id":"https://openalex.org/keywords/graph-products","display_name":"graph products","score":0.33500000834465027},{"id":"https://openalex.org/keywords/disconnected-graphs","display_name":"disconnected graphs","score":0.3409999907016754}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8273000121116638},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8259999752044678},{"id":"https://openalex.org/C48659774","wikidata":"https://www.wikidata.org/wiki/Q4907197","display_name":"Bijection, injection and surjection","level":3,"score":0.7494000196456909},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.6419000029563904},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.54339998960495},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.5196999907493591},{"id":"https://openalex.org/C76444178","wikidata":"https://www.wikidata.org/wiki/Q72897900","display_name":"Connectivity","level":3,"score":0.48179998993873596},{"id":"https://openalex.org/C99821215","wikidata":"https://www.wikidata.org/wiki/Q1136583","display_name":"Swap (finance)","level":2,"score":0.4406999945640564},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.43619999289512634},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.41130000352859497},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4036000072956085},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.39329999685287476},{"id":"https://openalex.org/C66882249","wikidata":"https://www.wikidata.org/wiki/Q169336","display_name":"Homogeneous","level":2,"score":0.31299999356269836},{"id":"https://openalex.org/C160446614","wikidata":"https://www.wikidata.org/wiki/Q1322892","display_name":"Chordal graph","level":3,"score":0.29760000109672546},{"id":"https://openalex.org/C193435613","wikidata":"https://www.wikidata.org/wiki/Q2997928","display_name":"Connected component","level":2,"score":0.27639999985694885},{"id":"https://openalex.org/C2983226441","wikidata":"https://www.wikidata.org/wiki/Q3115621","display_name":"Vertex connectivity","level":4,"score":0.27559998631477356}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.20648","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.20648","pdf_url":"https://arxiv.org/pdf/2610.20648","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.2610.20648","doi":"https://doi.org/10.48550/arxiv.2610.20648","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.20648","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:2610.20648","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.20648","pdf_url":"https://arxiv.org/pdf/2610.20648","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":[],"funders":[{"id":"https://openalex.org/F4320309370","display_name":"University of Pennsylvania","ror":"https://ror.org/00b30xv10"},{"id":"https://openalex.org/F4320309636","display_name":"University of Minnesota","ror":"https://ror.org/03grvy078"},{"id":"https://openalex.org/F4320311500","display_name":"University of Minnesota Duluth","ror":"https://ror.org/01hy4qx27"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7222423744.pdf","grobid_xml":"https://content.openalex.org/works/W7222423744.grobid-xml"},"referenced_works_count":15,"referenced_works":["https://openalex.org/W1986164452","https://openalex.org/W1996584990","https://openalex.org/W1997089200","https://openalex.org/W2008048468","https://openalex.org/W2044418619","https://openalex.org/W2048036606","https://openalex.org/W3084114785","https://openalex.org/W3087158933","https://openalex.org/W3182958786","https://openalex.org/W4376127016","https://openalex.org/W4402900143","https://openalex.org/W4405920823","https://openalex.org/W4415332478","https://openalex.org/W4417226595","https://openalex.org/W7204534184"],"related_works":["https://openalex.org/W4415332478","https://openalex.org/W4402900143","https://openalex.org/W4417226595","https://openalex.org/W3182958786","https://openalex.org/W7204534184","https://openalex.org/W4221157124","https://openalex.org/W3087158933","https://openalex.org/W4411929964","https://openalex.org/W4309734294","https://openalex.org/W4405920823"],"abstract_inverted_index":{"Given":[0],"graphs":[1,72,127,181],"$X,Y$":[2,69,124],"on":[3,149],"$n$":[4],"vertices,":[5],"Defant":[6],"and":[7,12,25,29,45,51,53,73,103,109,141,166,168,183],"Kravitz":[8],"defined":[9],"the":[10,84,129,135,150,161,185,192],"friends":[11],"strangers":[13],"graph,":[14],"denoted":[15],"$\\mathrm{FS}(X,Y)$.":[16],"The":[17,80],"vertices":[18,100],"of":[19,65,137,153,164,171,194,198],"$\\mathrm{FS}(X,Y)$":[20,34,77,138,154,199],"are":[21,31,47,55,70,125],"bijections":[22,27],"$\u03c3:V(X)\\to":[23],"V(Y)$,":[24],"two":[26],"$\u03c3$":[28],"$\u03c3'$":[30],"adjacent":[32,48,56,99],"in":[33,49,57,83,114,155],"if":[35,68,123],"they":[36],"differ":[37],"by":[38],"a":[39,63,147],"transposition":[40],"$(ab)$":[41],"such":[42,95],"that":[43,67,88,96,122,142,177,191],"$a$":[44,108],"$b$":[46,110],"$X$":[50,182],"$\u03c3(a)$":[52],"$\u03c3(b)$":[54],"$Y$.":[58],"Our":[59],"main":[60,81],"result":[61],"resolves":[62,169],"conjecture":[64],"Bangachev":[66],"connected":[71,126,196],"$2\\min\\{\u03b4(X),\u03b4(Y)\\}+3\\max\\{\u03b4(X),\u03b4(Y)\\}\\ge":[74],"3n$,":[75],"then":[76,134],"is":[78,86,139,144,200],"connected.":[79],"step":[82],"proof":[85],"showing":[87],"there":[89],"exists":[90],"an":[91],"absolute":[92],"constant":[93],"$C$":[94,117],"for":[97,179],"any":[98,104,195],"$a,b\\in":[101],"V(X)$":[102],"vertex":[105],"$\u03c3\\in":[106],"V(\\mathrm{FS}(X,Y))$,":[107],"can":[111],"swap":[112],"positions":[113],"at":[115],"most":[116],"moves.":[118],"Furthermore,":[119],"we":[120,175],"prove":[121],"satisfying":[128],"above":[130],"minimum":[131,187],"degree":[132,188],"condition,":[133],"diameter":[136,152,162,193],"$O(n^2/\\min\\{\u03b4(X),\u03b4(Y)\\})$,":[140],"this":[143,156],"sharp":[145],"as":[146],"bound":[148],"maximum":[151],"regime.":[157],"This":[158],"work":[159],"improves":[160],"results":[163],"Akella":[165],"Li,":[167],"many":[170],"their":[172],"questions.":[173],"Moreover,":[174],"show":[176],"even":[178],"disconnected":[180],"$Y$,":[184],"same":[186],"condition":[189],"implies":[190],"component":[197],"$O(n^2)$.":[201]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-17T00:00:00"}