{"id":"https://openalex.org/W2578578423","doi":"https://doi.org/10.1137/16m1072607","title":"Algebraic Properties of Generalized Graph Laplacians: Resistor Networks, Critical Groups, and Homological Algebra","display_name":"Algebraic Properties of Generalized Graph Laplacians: Resistor Networks, Critical Groups, and Homological Algebra","publication_year":2018,"publication_date":"2018-01-01","ids":{"openalex":"https://openalex.org/W2578578423","doi":"https://doi.org/10.1137/16m1072607","mag":"2578578423"},"language":"en","primary_location":{"id":"doi:10.1137/16m1072607","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m1072607","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Discrete Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1604.07075","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"David Jekel","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"David Jekel","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":null,"display_name":"Avi Levy","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Avi Levy","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":null,"display_name":"Will Dana","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Will Dana","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":null,"display_name":"Austin Stromme","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Austin Stromme","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":null,"display_name":"Collin Litterell","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Collin Litterell","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":5,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.2137,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.51201347,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"32","issue":"2","first_page":"1040","last_page":"1110"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12536","display_name":"Topological and Geometric Data Analysis","score":0.6462000012397766,"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/T12536","display_name":"Topological and Geometric Data Analysis","score":0.6462000012397766,"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/T11476","display_name":"Graph theory and applications","score":0.22849999368190765,"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/T10064","display_name":"Complex Network Analysis Techniques","score":0.01889999955892563,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/spectral-graph-theory","display_name":"Spectral graph theory","score":0.6169999837875366},{"id":"https://openalex.org/keywords/algebraic-connectivity","display_name":"Algebraic connectivity","score":0.574400007724762},{"id":"https://openalex.org/keywords/laplacian-matrix","display_name":"Laplacian matrix","score":0.5231000185012817},{"id":"https://openalex.org/keywords/homological-algebra","display_name":"Homological algebra","score":0.45989999175071716},{"id":"https://openalex.org/keywords/algebraic-graph-theory","display_name":"Algebraic graph theory","score":0.4453999996185303},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.4174000024795532},{"id":"https://openalex.org/keywords/laplace-operator","display_name":"Laplace operator","score":0.4047999978065491},{"id":"https://openalex.org/keywords/adjacency-list","display_name":"Adjacency list","score":0.3919999897480011},{"id":"https://openalex.org/keywords/graph-property","display_name":"Graph property","score":0.3871999979019165},{"id":"https://openalex.org/keywords/graph-algebra","display_name":"Graph algebra","score":0.3840999901294708}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8410000205039978},{"id":"https://openalex.org/C74003402","wikidata":"https://www.wikidata.org/wiki/Q3180727","display_name":"Spectral graph theory","level":5,"score":0.6169999837875366},{"id":"https://openalex.org/C7948225","wikidata":"https://www.wikidata.org/wiki/Q4723998","display_name":"Algebraic connectivity","level":4,"score":0.574400007724762},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5414000153541565},{"id":"https://openalex.org/C115178988","wikidata":"https://www.wikidata.org/wiki/Q772067","display_name":"Laplacian matrix","level":3,"score":0.5231000185012817},{"id":"https://openalex.org/C18364862","wikidata":"https://www.wikidata.org/wiki/Q579978","display_name":"Homological algebra","level":3,"score":0.45989999175071716},{"id":"https://openalex.org/C25799778","wikidata":"https://www.wikidata.org/wiki/Q4724003","display_name":"Algebraic graph theory","level":3,"score":0.4453999996185303},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.4174000024795532},{"id":"https://openalex.org/C165700671","wikidata":"https://www.wikidata.org/wiki/Q203484","display_name":"Laplace operator","level":2,"score":0.4047999978065491},{"id":"https://openalex.org/C110484373","wikidata":"https://www.wikidata.org/wiki/Q264398","display_name":"Adjacency list","level":2,"score":0.3919999897480011},{"id":"https://openalex.org/C64339825","wikidata":"https://www.wikidata.org/wiki/Q722659","display_name":"Graph property","level":5,"score":0.3871999979019165},{"id":"https://openalex.org/C45961005","wikidata":"https://www.wikidata.org/wiki/Q5597072","display_name":"Graph algebra","level":5,"score":0.3840999901294708},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.38269999623298645},{"id":"https://openalex.org/C180356752","wikidata":"https://www.wikidata.org/wiki/Q727035","display_name":"Adjacency matrix","level":3,"score":0.3824000060558319},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.3580000102519989},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.3346000015735626},{"id":"https://openalex.org/C156772000","wikidata":"https://www.wikidata.org/wiki/Q864475","display_name":"Functor","level":2,"score":0.32989999651908875},{"id":"https://openalex.org/C627467","wikidata":"https://www.wikidata.org/wiki/Q599027","display_name":"Harmonic function","level":2,"score":0.3264000117778778},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3075000047683716},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.30649998784065247},{"id":"https://openalex.org/C78913703","wikidata":"https://www.wikidata.org/wiki/Q5597087","display_name":"Graph energy","level":5,"score":0.3037000000476837},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3034999966621399},{"id":"https://openalex.org/C612898","wikidata":"https://www.wikidata.org/wiki/Q7315618","display_name":"Resistance distance","level":5,"score":0.30219998955726624},{"id":"https://openalex.org/C182419690","wikidata":"https://www.wikidata.org/wiki/Q205464","display_name":"Algebraic structure","level":2,"score":0.299699991941452},{"id":"https://openalex.org/C2731732","wikidata":"https://www.wikidata.org/wiki/Q7301119","display_name":"Real algebraic geometry","level":3,"score":0.29809999465942383},{"id":"https://openalex.org/C78313660","wikidata":"https://www.wikidata.org/wiki/Q727659","display_name":"Commutative algebra","level":2,"score":0.29269999265670776},{"id":"https://openalex.org/C22149727","wikidata":"https://www.wikidata.org/wiki/Q7940747","display_name":"Voltage graph","level":4,"score":0.2879999876022339},{"id":"https://openalex.org/C148607811","wikidata":"https://www.wikidata.org/wiki/Q746863","display_name":"Algebraic function","level":3,"score":0.2815000116825104},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.27630001306533813},{"id":"https://openalex.org/C77462122","wikidata":"https://www.wikidata.org/wiki/Q15176852","display_name":"Intersection theory","level":5,"score":0.27000001072883606},{"id":"https://openalex.org/C204710682","wikidata":"https://www.wikidata.org/wiki/Q3036983","display_name":"Integral graph","level":5,"score":0.2662000060081482},{"id":"https://openalex.org/C68363185","wikidata":"https://www.wikidata.org/wiki/Q180969","display_name":"Algebraic geometry","level":2,"score":0.2648000121116638},{"id":"https://openalex.org/C184363387","wikidata":"https://www.wikidata.org/wiki/Q3001219","display_name":"Stable curve","level":4,"score":0.2558000087738037},{"id":"https://openalex.org/C199793520","wikidata":"https://www.wikidata.org/wiki/Q4723996","display_name":"Algebraic cycle","level":3,"score":0.25369998812675476},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.25189998745918274},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.25099998712539673}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/16m1072607","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m1072607","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Discrete Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1604.07075","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1604.07075","pdf_url":"https://arxiv.org/pdf/1604.07075","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1604.07075","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1604.07075","pdf_url":"https://arxiv.org/pdf/1604.07075","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":[],"awards":[{"id":"https://openalex.org/G7547726215","display_name":null,"funder_award_id":"DMS-1460937","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":false,"pdf":false},"content_urls":null,"referenced_works_count":26,"referenced_works":["https://openalex.org/W60687185","https://openalex.org/W1775635241","https://openalex.org/W1910083634","https://openalex.org/W1964658353","https://openalex.org/W1967877035","https://openalex.org/W1969007985","https://openalex.org/W1969823477","https://openalex.org/W1974447478","https://openalex.org/W1982253010","https://openalex.org/W1989823380","https://openalex.org/W1995854863","https://openalex.org/W1998018314","https://openalex.org/W2000105188","https://openalex.org/W2013190996","https://openalex.org/W2019212211","https://openalex.org/W2037190943","https://openalex.org/W2048667371","https://openalex.org/W2059711768","https://openalex.org/W2064237326","https://openalex.org/W2082513731","https://openalex.org/W2083387674","https://openalex.org/W2111293718","https://openalex.org/W2142867942","https://openalex.org/W2158469522","https://openalex.org/W2324792688","https://openalex.org/W4235905547"],"related_works":[],"abstract_inverted_index":{"We":[0,54,75],"propose":[1],"an":[2,116,129],"algebraic":[3,79,117],"framework":[4],"for":[5,131,139],"generalized":[6,37,50],"graph":[7,30],"Laplacians":[8],"which":[9],"unifies":[10],"the":[11,16,20,23,63,66,86,103,120,140,146,150],"study":[12,76],"of":[13,22,59,72,105,119,134,142,152,157],"resistor":[14,106],"networks,":[15],"critical":[17,51,147],"group,":[18],"and":[19,25,69,94,136,149],"eigenvalues":[21,154],"Laplacian":[24,38,153],"adjacency":[26],"matrices.":[27],"Given":[28],"a":[29,36,43,49],"with":[31,35,40,122],"boundary":[32,123],"$G$":[33],"together":[34],"$L$":[39],"entries":[41],"in":[42,145,155],"commutative":[44],"ring":[45],"$R$,":[46],"we":[47,98],"define":[48],"group":[52,148],"$\\Upsilon_R(G,L)$.":[53],"relate":[55],"$\\Upsilon_R(G,L)$":[56],"to":[57,108,115],"spaces":[58],"harmonic":[60,90,111],"functions":[61],"on":[62,85],"network":[64,87],"using":[65],"Hom,":[67],"Tor,":[68],"Ext":[70],"functors":[71],"homological":[73],"algebra.":[74],"how":[77],"these":[78],"objects":[80],"transform":[81],"under":[82],"combinatorial":[83],"operations":[84,101],"$(G,L)$,":[88],"including":[89],"morphisms,":[91],"layer-stripping,":[92],"duality,":[93],"symmetry.":[95],"In":[96],"particular,":[97],"use":[99],"layer-stripping":[100],"from":[102],"theory":[104],"networks":[107],"systematize":[109],"discrete":[110],"continuation.":[112],"This":[113],"leads":[114],"characterization":[118],"graphs":[121],"that":[124],"can":[125],"be":[126],"completely":[127],"layer-stripped,":[128],"algorithm":[130],"simplifying":[132],"computation":[133],"$\\Upsilon_R(G,L)$,":[135],"upper":[137],"bounds":[138],"number":[141],"invariant":[143],"factors":[144],"multiplicity":[151],"terms":[156],"geometric":[158],"quantities.":[159]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2021,"cited_by_count":1}],"updated_date":"2026-04-09T08:11:56.329763","created_date":"2017-01-26T00:00:00"}