{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,18]],"date-time":"2026-07-18T05:06:13Z","timestamp":1784351173984,"version":"3.55.0"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2022,4,21]],"date-time":"2022-04-21T00:00:00Z","timestamp":1650499200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,4,21]],"date-time":"2022-04-21T00:00:00Z","timestamp":1650499200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100003093","name":"Ministry of Education, Malaysia","doi-asserted-by":"crossref","award":["FRGS\/1\/2020\/STG06\/UCSI\/02\/1"],"award-info":[{"award-number":["FRGS\/1\/2020\/STG06\/UCSI\/02\/1"]}],"id":[{"id":"10.13039\/501100003093","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/100012046","name":"Vietnam Academy of Science and Technology","doi-asserted-by":"publisher","award":["DTCS-2022"],"award-info":[{"award-number":["DTCS-2022"]}],"id":[{"id":"10.13039\/100012046","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comp. Appl. Math."],"published-print":{"date-parts":[[2022,6]]},"DOI":"10.1007\/s40314-022-01823-4","type":"journal-article","created":{"date-parts":[[2022,4,21]],"date-time":"2022-04-21T18:05:57Z","timestamp":1650564357000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["New concepts of pentapartitioned neutrosophic graphs and applications for determining safest paths and towns in response to COVID-19"],"prefix":"10.1007","volume":"41","author":[{"given":"Shio Gai","family":"Quek","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ganeshsree","family":"Selvachandran","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"D.","family":"Ajay","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"P.","family":"Chellamani","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David","family":"Taniar","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hamido","family":"Fujita","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Phet","family":"Duong","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6356-0046","authenticated-orcid":false,"given":"Le Hoang","family":"Son","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nguyen Long","family":"Giang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2022,4,21]]},"reference":[{"key":"1823_CR1","first-page":"108","volume":"11","author":"D Ajay","year":"2020","unstructured":"Ajay D, Chellamani P (2020) Pythagorean neutrosophic fuzzy graphs. Int J Neutrosophic Sci 11:108\u2013114","journal-title":"Int J Neutrosophic Sci"},{"key":"1823_CR2","doi-asserted-by":"publisher","first-page":"177","DOI":"10.2298\/FIL1201177A","volume":"12","author":"M Akram","year":"2012","unstructured":"Akram M, Davvaz B (2012) Strong fuzzy graphs. Filomat 12:177\u2013196","journal-title":"Filomat"},{"key":"1823_CR3","doi-asserted-by":"publisher","first-page":"182","DOI":"10.1016\/j.ins.2012.06.024","volume":"218","author":"M Akram","year":"2013","unstructured":"Akram M, Dudek WA (2013) Intuitionistic fuzzy hypergraphs with applications. Inf Sci 218:182\u2013193","journal-title":"Inf Sci"},{"key":"1823_CR4","doi-asserted-by":"publisher","DOI":"10.1007\/s41066-020-00223-5","author":"M Akram","year":"2020","unstructured":"Akram M, Khan A (2020) Complex Pythagorean Dombi fuzzy graphs for decision making. Granul Comput. https:\/\/doi.org\/10.1007\/s41066-020-00223-5","journal-title":"Granul Comput"},{"issue":"8","key":"1823_CR5","doi-asserted-by":"publisher","first-page":"136","DOI":"10.3390\/math6080136","volume":"6","author":"M Akram","year":"2018","unstructured":"Akram M, Naz S (2018) Energy of Pythagorean fuzzy graphs with applications. Mathematics 6(8):136","journal-title":"Mathematics"},{"issue":"1","key":"1823_CR6","first-page":"8","volume":"25","author":"M Akram","year":"2020","unstructured":"Akram M, Saleem D, Al-Hawary T (2020a) Spherical fuzzy graphs with application to decision-making. Math Comput Appl 25(1):8","journal-title":"Math Comput Appl"},{"key":"1823_CR7","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1007\/s40747-019-0109-0","volume":"6","author":"M Akram","year":"2020","unstructured":"Akram M, Dar JM, Naz S (2020b) Pythagorean Dombi fuzzy graphs. Complex Intell Syst 6:29\u201354","journal-title":"Complex Intell Syst"},{"key":"1823_CR8","first-page":"86","volume":"10","author":"S Broumi","year":"2016","unstructured":"Broumi S, Talea M, Bakali A, Smarandache F (2016a) Single valued neutrosophic graphs. J New Theory 10:86\u2013101","journal-title":"J New Theory"},{"key":"1823_CR9","first-page":"74","volume":"11","author":"S Broumi","year":"2016","unstructured":"Broumi S, Bakali A, Talea M, Smarandache F (2016b) Isolated single valued neutrosophic graphs. Neutrosophic Sets Syst 11:74\u201378","journal-title":"Neutrosophic Sets Syst"},{"key":"1823_CR10","doi-asserted-by":"publisher","first-page":"184","DOI":"10.4028\/www.scientific.net\/AMM.841.184","volume":"841","author":"S Broumi","year":"2016","unstructured":"Broumi S, Smarandache F, Talea M, Bakali A (2016c) An introduction to bipolar single valued neutrosophic graph theory. Appl Mech Mater 841:184\u2013191","journal-title":"Appl Mech Mater"},{"key":"1823_CR11","first-page":"84","volume":"11","author":"S Broumi","year":"2016","unstructured":"Broumi S, Talea M, Bakali A, Smarandache F (2016d) On bipolar single valued neutrosophic graphs. J New Theory 11:84\u2013102","journal-title":"J New Theory"},{"key":"1823_CR12","doi-asserted-by":"publisher","first-page":"2475","DOI":"10.3233\/IFS-152017","volume":"30","author":"R Chatterjee","year":"2016","unstructured":"Chatterjee R, Majumdar P, Samanta SK (2016a) On some similarity measures and entropy on quadripartitioned single valued neutrosophic sets. J Intell Fuzzy Syst 30:2475\u20132485","journal-title":"J Intell Fuzzy Syst"},{"key":"1823_CR13","first-page":"35","volume":"14","author":"R Chatterjee","year":"2016","unstructured":"Chatterjee R, Majumdar P, Samanta SK (2016b) Interval-valued possibility quadripartitioned single valued neutrosophic soft sets and some uncertainty based measures on them. Neutrosophic Sets Syst 14:35\u201343","journal-title":"Neutrosophic Sets Syst"},{"key":"1823_CR14","doi-asserted-by":"publisher","first-page":"8857","DOI":"10.1007\/s00500-019-04417-1","volume":"24","author":"R Chatterjee","year":"2020","unstructured":"Chatterjee R, Majumdar P, Samanta SK (2020) A multi-criteria group decision making algorithm with quadripartitioned neutrosophic weighted aggregation operators using quadripartitioned neutrosophic numbers in IPQSVNSS environment. Soft Comput 24:8857\u20138880","journal-title":"Soft Comput"},{"issue":"9","key":"1823_CR15","doi-asserted-by":"publisher","first-page":"373","DOI":"10.3390\/sym10090373","volume":"10","author":"A Dey","year":"2018","unstructured":"Dey A, Son LH, Kumar PKK, Selvachandran G, Quek SG (2018) New concepts on vertex and edge coloring of simple vague graphs. Symmetry 10(9):373","journal-title":"Symmetry"},{"issue":"6","key":"1823_CR16","doi-asserted-by":"publisher","first-page":"1338","DOI":"10.1007\/s12559-020-09765-x","volume":"12","author":"H Ero\u011flu","year":"2020","unstructured":"Ero\u011flu H, \u015eahin R (2020) A neutrosophic VIKOR method-based decision-making with an improved distance measure and score function: case study of selection for renewable energy alternatives. Cogn Comput 12(6):1338\u20131355","journal-title":"Cogn Comput"},{"key":"1823_CR17","first-page":"33","volume":"12","author":"AN Gani","year":"2008","unstructured":"Gani AN, Radha K (2008) On regular fuzzy graphs. J Phys Sci 12:33\u201340","journal-title":"J Phys Sci"},{"issue":"9","key":"1823_CR18","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1177\/1550147719877613","volume":"15","author":"M Gulistan","year":"2019","unstructured":"Gulistan M, Mohammad M, Karaaslan F, Kadry S, Khan S, Wahab HA (2019) Neutrosophic cubic Heronian mean operators with applications in multiple attribute group decision-making using cosine similarity functions. Int J Distrib Sens Netw 15(9):1\u201321","journal-title":"Int J Distrib Sens Netw"},{"issue":"1","key":"1823_CR19","doi-asserted-by":"publisher","first-page":"55","DOI":"10.30948\/afmi.2017.14.1.55","volume":"14","author":"A Hassan","year":"2017","unstructured":"Hassan A, Malik MA, Broumi S, Bakali A, Talea M, Smarandache F (2017) Special types of bipolar single valued neutrosophic graphs. Ann Fuzzy Math Inform 14(1):55\u201373","journal-title":"Ann Fuzzy Math Inform"},{"key":"1823_CR20","doi-asserted-by":"publisher","first-page":"1263","DOI":"10.3390\/sym12081263","volume":"12","author":"D Karaba\u0161evi\u0107","year":"2020","unstructured":"Karaba\u0161evi\u0107 D, Stanujki\u0107 D, Zavadskas EK, Stanimirovi\u0107 P, Popovi\u0107 G, Predi\u0107 B, Uluta\u015f A (2020) A novel extension of the TOPSIS method adapted for the use of single-valued neutrosophic sets and Hamming distance for e-commerce development strategies selection. Symmetry 12:1263","journal-title":"Symmetry"},{"key":"1823_CR21","unstructured":"Kauffman A (1973) Introduction a la Theorie des Sous-emsembles Flous, 1, Masson et Cie"},{"key":"1823_CR22","doi-asserted-by":"publisher","DOI":"10.1007\/s12559-020-09802-9","author":"NEM Khalifa","year":"2021","unstructured":"Khalifa NEM, Smarandache F, Manogaran G, Loey M (2021) A study of the neutrosophic set significance on deep transfer learning models: an experimental case on a limited covid-19 chest X-ray dataset. Cognit Comput. https:\/\/doi.org\/10.1007\/s12559-020-09802-9","journal-title":"Cognit Comput"},{"issue":"3","key":"1823_CR23","doi-asserted-by":"publisher","first-page":"1245","DOI":"10.3233\/IFS-130810","volume":"26","author":"P Majumdar","year":"2014","unstructured":"Majumdar P, Samanta SK (2014) On similarity and entropy of neutrosophic sets. J Intell Fuzzy Syst 26(3):1245\u20131252","journal-title":"J Intell Fuzzy Syst"},{"key":"1823_CR24","first-page":"184","volume":"36","author":"R Mallick","year":"2020","unstructured":"Mallick R, Pramanik S (2020) Pentapartitioned neutrosophic set and its properties. Neutrosophic Sets Syst 36:184\u2013192","journal-title":"Neutrosophic Sets Syst"},{"issue":"1","key":"1823_CR25","doi-asserted-by":"publisher","first-page":"321","DOI":"10.37418\/amsj.9.1.26","volume":"9","author":"SY Mohamed","year":"2020","unstructured":"Mohamed SY, Ali AM (2020) Energy of spherical fuzzy graphs. Adv Math Sci J 9(1):321\u2013332","journal-title":"Adv Math Sci J"},{"key":"1823_CR26","first-page":"159","volume":"79","author":"JN Mordeson","year":"1994","unstructured":"Mordeson JN, Peng CS (1994) Operations on fuzzy graphs information. Sciences 79:159\u2013170","journal-title":"Sciences"},{"key":"1823_CR27","doi-asserted-by":"publisher","first-page":"2137","DOI":"10.3233\/JIFS-161944","volume":"32","author":"S Naz","year":"2017","unstructured":"Naz S, Rashmanlou H, Malik MA (2017) Operations on single valued neutrosophic graphs with application. J Intell Fuzzy Syst 32:2137\u20132151","journal-title":"J Intell Fuzzy Syst"},{"issue":"3","key":"1823_CR28","doi-asserted-by":"publisher","first-page":"50","DOI":"10.3390\/axioms7030050","volume":"7","author":"S Naz","year":"2018","unstructured":"Naz S, Akram M, Smarandache F (2018) Certain notions of energy in single-valued neutrosophic graphs. Axioms 7(3):50","journal-title":"Axioms"},{"key":"1823_CR29","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1007\/3-540-34783-6_15","volume-title":"Intuitionistic fuzzy graphs. Computational intelligence theory and applications","author":"R Parvathi","year":"2006","unstructured":"Parvathi R, Karunambigai MG (2006) Intuitionistic fuzzy graphs. Computational intelligence theory and applications. Springer, Berlin, pp 139\u2013150"},{"key":"1823_CR30","doi-asserted-by":"publisher","first-page":"336","DOI":"10.1016\/j.asoc.2014.08.070","volume":"25","author":"JJ Peng","year":"2014","unstructured":"Peng JJ, Wang JQ, Zhang HY, Chen XH (2014) An outranking approach for multi-criteria decision-making problems with simplified neutrosophic sets. Appl Soft Comput 25:336\u2013346","journal-title":"Appl Soft Comput"},{"issue":"6","key":"1823_CR31","doi-asserted-by":"publisher","first-page":"190","DOI":"10.3390\/sym10060190","volume":"10","author":"SG Quek","year":"2018","unstructured":"Quek SG, Broumi S, Selvachandran G, Bakali A, Talea M, Smarandache F (2018) Some results on the graph theory for complex neutrosophic sets. Symmetry 10(6):190","journal-title":"Symmetry"},{"key":"1823_CR32","doi-asserted-by":"crossref","unstructured":"Rosenfeld A (1975) Fuzzy graphs, fuzzy sets and their applications to cognitive and decision processes. In: Proceeding of the U.S. Japan Seminar, University of California, Berkeley, California 1974. Academic Press, New York, p 77\u201395","DOI":"10.1016\/B978-0-12-775260-0.50008-6"},{"key":"1823_CR33","volume-title":"A unifying field in logics, neutrosophy: neutrosophic probability, set and logic","author":"F Smarandache","year":"1999","unstructured":"Smarandache F (1999) A unifying field in logics, neutrosophy: neutrosophic probability, set and logic. American Research Press, Rehobooth"},{"key":"1823_CR34","first-page":"143","volume":"4","author":"F Smarandache","year":"2013","unstructured":"Smarandache F (2013) n-valued refined neutrosophic logic and its applications to physics. Prog Phys 4:143\u2013146","journal-title":"Prog Phys"},{"key":"1823_CR35","doi-asserted-by":"publisher","first-page":"283","DOI":"10.1007\/s10489-020-01706-3","volume":"51","author":"RP Tan","year":"2021","unstructured":"Tan RP (2021) Decision-making method based on new entropy and refined single-valued neutrosophic sets and its application in typhoon disaster assessment. Appl Intell 51:283\u2013307","journal-title":"Appl Intell"},{"issue":"3","key":"1823_CR36","doi-asserted-by":"publisher","first-page":"523","DOI":"10.1007\/s13042-016-0552-9","volume":"9","author":"ZP Tian","year":"2018","unstructured":"Tian ZP, Wang J, Zhang HY, Wang JQ (2018) Multi-criteria decision-making based on generalized prioritized aggregation operators under simplified neutrosophic uncertain linguistic environment. Int J Mach Learn Cybern 9(3):523\u2013539","journal-title":"Int J Mach Learn Cybern"},{"key":"1823_CR37","first-page":"410","volume":"4","author":"H Wang","year":"2010","unstructured":"Wang H, Smarandache F, Zhang YQ, Sunderraman R (2010) Single valued neutrosophic sets. Multispace Multistructure 4:410\u2013413","journal-title":"Multispace Multistructure"},{"issue":"6","key":"1823_CR38","doi-asserted-by":"publisher","first-page":"1104","DOI":"10.1007\/s40815-016-0180-2","volume":"18","author":"XH Wu","year":"2016","unstructured":"Wu XH, Wang J, Peng JJ, Chen XH (2016) Cross-entropy and prioritized aggregation operator with simplified neutrosophic sets and their application in multi-criteria decision-making problems. Int J Fuzzy Syst 18(6):1104\u20131116","journal-title":"Int J Fuzzy Syst"},{"key":"1823_CR39","doi-asserted-by":"publisher","DOI":"10.1007\/s40815-020-00983-x","author":"J Ye","year":"2020","unstructured":"Ye J, Song J, Du S (2020) Correlation coefficients of consistency neutrosophic sets regarding neutrosophic multi-valued sets and their multi-attribute decision-making method. Int J Fuzzy Syst. https:\/\/doi.org\/10.1007\/s40815-020-00983-x","journal-title":"Int J Fuzzy Syst"},{"key":"1823_CR40","doi-asserted-by":"publisher","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","volume":"8","author":"LA Zadeh","year":"1965","unstructured":"Zadeh LA (1965) Fuzzy sets. Inf Control 8:338\u2013353","journal-title":"Inf Control"},{"key":"1823_CR41","doi-asserted-by":"publisher","first-page":"615","DOI":"10.1007\/s00521-015-1882-3","volume":"27","author":"H Zhang","year":"2016","unstructured":"Zhang H, Wang J, Chen X (2016) An outranking approach for multi-criteria decision-making problems with interval-valued neutrosophic sets. Neural Comput Appl 27:615\u2013627","journal-title":"Neural Comput Appl"},{"issue":"5","key":"1823_CR42","doi-asserted-by":"publisher","first-page":"470","DOI":"10.3390\/math7050470","volume":"7","author":"C Zuo","year":"2019","unstructured":"Zuo C, Pal A, Dey A (2019) New concepts of picture fuzzy graphs with application. Mathematics 7(5):470","journal-title":"Mathematics"}],"container-title":["Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-022-01823-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40314-022-01823-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40314-022-01823-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,6,9]],"date-time":"2022-06-09T21:24:09Z","timestamp":1654809849000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40314-022-01823-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,21]]},"references-count":42,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2022,6]]}},"alternative-id":["1823"],"URL":"https:\/\/doi.org\/10.1007\/s40314-022-01823-4","relation":{},"ISSN":["2238-3603","1807-0302"],"issn-type":[{"value":"2238-3603","type":"print"},{"value":"1807-0302","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,4,21]]},"assertion":[{"value":"3 October 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 October 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 March 2022","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 April 2022","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that there is no conflict of interests. This research does not involve any human or animal activities. All authors have checked and agreed the submission.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}],"article-number":"151"}}