{"id":"https://openalex.org/W4296760951","doi":"https://doi.org/10.3390/sym14091900","title":"The g-Extra Connectivity of the Strong Product of Paths and Cycles","display_name":"The g-Extra Connectivity of the Strong Product of Paths and Cycles","publication_year":2022,"publication_date":"2022-09-11","ids":{"openalex":"https://openalex.org/W4296760951","doi":"https://doi.org/10.3390/sym14091900"},"language":"en","primary_location":{"id":"doi:10.3390/sym14091900","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14091900","pdf_url":"https://www.mdpi.com/2073-8994/14/9/1900/pdf?version=1663757617","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/14/9/1900/pdf?version=1663757617","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5043590120","display_name":"Qinze Zhu","orcid":null},"institutions":[{"id":"https://openalex.org/I96908189","display_name":"Xinjiang University","ror":"https://ror.org/059gw8r13","country_code":"CN","type":"education","lineage":["https://openalex.org/I96908189"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Qinze Zhu","raw_affiliation_strings":["College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China","institution_ids":["https://openalex.org/I96908189"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5085199355","display_name":"Yingzhi Tian","orcid":"https://orcid.org/0000-0002-1595-9834"},"institutions":[{"id":"https://openalex.org/I96908189","display_name":"Xinjiang University","ror":"https://ror.org/059gw8r13","country_code":"CN","type":"education","lineage":["https://openalex.org/I96908189"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Yingzhi Tian","raw_affiliation_strings":["College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China"],"raw_orcid":"https://orcid.org/0000-0002-1595-9834","affiliations":[{"raw_affiliation_string":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China","institution_ids":["https://openalex.org/I96908189"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5085199355"],"corresponding_institution_ids":["https://openalex.org/I96908189"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.5704,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.68581032,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"14","issue":"9","first_page":"1900","last_page":"1900"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10829","display_name":"Interconnection Networks and Systems","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/T10829","display_name":"Interconnection Networks and Systems","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/1705","display_name":"Computer Networks and Communications"},"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/T10018","display_name":"Advancements in Battery Materials","score":0.9871000051498413,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10083","display_name":"Graphene research and applications","score":0.9861999750137329,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8149943351745605},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7731114625930786},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.6415541768074036},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.5318275094032288},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.5106740593910217},{"id":"https://openalex.org/keywords/cardinality","display_name":"Cardinality (data modeling)","score":0.4476635456085205},{"id":"https://openalex.org/keywords/connectivity","display_name":"Connectivity","score":0.4286521375179291},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3838481307029724},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.16193121671676636},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.07960423827171326}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8149943351745605},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7731114625930786},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.6415541768074036},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.5318275094032288},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.5106740593910217},{"id":"https://openalex.org/C87117476","wikidata":"https://www.wikidata.org/wiki/Q362383","display_name":"Cardinality (data modeling)","level":2,"score":0.4476635456085205},{"id":"https://openalex.org/C76444178","wikidata":"https://www.wikidata.org/wiki/Q72897900","display_name":"Connectivity","level":3,"score":0.4286521375179291},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3838481307029724},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.16193121671676636},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.07960423827171326},{"id":"https://openalex.org/C124101348","wikidata":"https://www.wikidata.org/wiki/Q172491","display_name":"Data mining","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym14091900","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14091900","pdf_url":"https://www.mdpi.com/2073-8994/14/9/1900/pdf?version=1663757617","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:75909856f06b4925bf3bb4c47e33c39d","is_oa":true,"landing_page_url":"https://doaj.org/article/75909856f06b4925bf3bb4c47e33c39d","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 14, Iss 9, p 1900 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/14/9/1900/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym14091900","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"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":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym14091900","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym14091900","pdf_url":"https://www.mdpi.com/2073-8994/14/9/1900/pdf?version=1663757617","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G8752940626","display_name":"k-\u70b9\u8fde\u901a\u56fe\u4e2d\u4fdd\u6301\u8fde\u901a\u5ea6\u7684\u5b50\u6811\u7684\u7814\u7a76","funder_award_id":"11861066","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":true,"grobid_xml":false},"content_urls":{"pdf":"https://content.openalex.org/works/W4296760951.pdf"},"referenced_works_count":29,"referenced_works":["https://openalex.org/W159485782","https://openalex.org/W560799365","https://openalex.org/W1860999107","https://openalex.org/W1963704255","https://openalex.org/W1971967319","https://openalex.org/W1980648360","https://openalex.org/W1984370079","https://openalex.org/W1989412486","https://openalex.org/W2025587779","https://openalex.org/W2033449646","https://openalex.org/W2058759738","https://openalex.org/W2081640546","https://openalex.org/W2088535325","https://openalex.org/W2106155325","https://openalex.org/W2124839136","https://openalex.org/W2145331678","https://openalex.org/W2169989080","https://openalex.org/W2209020398","https://openalex.org/W2523017701","https://openalex.org/W2554120345","https://openalex.org/W2765252281","https://openalex.org/W2963565037","https://openalex.org/W3120865572","https://openalex.org/W4224885795","https://openalex.org/W4232896595","https://openalex.org/W4233701160","https://openalex.org/W4237486254","https://openalex.org/W6730392116","https://openalex.org/W6745041502"],"related_works":["https://openalex.org/W4313216298","https://openalex.org/W2071797893","https://openalex.org/W2097063382","https://openalex.org/W2384431448","https://openalex.org/W3002741190","https://openalex.org/W3037640829","https://openalex.org/W2893201198","https://openalex.org/W2884117239","https://openalex.org/W3189256254","https://openalex.org/W2055455751"],"abstract_inverted_index":{"Let":[0],"G":[1,18,42,52,54],"be":[2,8],"a":[3,9,21,48,117,120],"connected":[4],"graph":[5,17],"and":[6,28,65,80,86,90,94,119,122],"g":[7],"non-negative":[10],"integer.":[11],"A":[12],"vertex":[13,76],"set":[14,77,83],"S":[15],"of":[16,31,41,47,51,106,110,116,126],"is":[19,26,43,71,78,84],"called":[20],"g-extra":[22,39,49,59,104],"cut":[23,50],"if":[24,53],"G\u2212S":[25,32],"disconnected":[27],"each":[29],"component":[30],"has":[33,55],"at":[34,56],"least":[35,57],"g+1":[36],"vertices.":[37],"The":[38],"connectivity":[40,105],"the":[44,67,103,107,113,123],"minimum":[45],"cardinality":[46],"one":[58],"cut.":[60],"For":[61],"two":[62,111,127],"graphs":[63],"G1=(V1,E1)":[64],"G2=(V2,E2),":[66],"strong":[68,108,114,124],"product":[69,109,115,125],"G1\u22a0G2":[70],"defined":[72],"as":[73],"follows:":[74],"its":[75,81],"V1\u00d7V2":[79],"edge":[82],"{(x1,x2)(y1,y2)|x1=x2":[85],"y1y2\u2208E2;":[87],"or":[88,92],"y1=y2":[89],"x1x2\u2208E1;":[91],"x1x2\u2208E1":[93],"y1y2\u2208E2},":[95],"where":[96],"(x1,x2),(y1,y2)\u2208V1\u00d7V2.":[97],"In":[98],"this":[99],"paper,":[100],"we":[101],"obtain":[102],"paths,":[112],"path":[118],"cycle,":[121],"cycles.":[128]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":3}],"updated_date":"2026-06-06T09:05:17.133730","created_date":"2025-10-10T00:00:00"}