{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:33:03Z","timestamp":1760236383492,"version":"build-2065373602"},"reference-count":5,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,11,18]],"date-time":"2021-11-18T00:00:00Z","timestamp":1637193600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003093","name":"Ministry of Higher Education","doi-asserted-by":"publisher","award":["FRGS\/1\/2020\/STG06\/UTM\/01\/1"],"award-info":[{"award-number":["FRGS\/1\/2020\/STG06\/UTM\/01\/1"]}],"id":[{"id":"10.13039\/501100003093","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Fuzzy topological topographic mapping (FTTM) is a mathematical model which consists of a set of homeomorphic topological spaces designed to solve the neuro magnetic inverse problem. A sequence of FTTM, FTTMn, is an extension of FTTM that is arranged in a symmetrical form. The special characteristic of FTTM, namely the homeomorphisms between its components, allows the generation of new FTTM. The generated FTTMs can be represented as pseudo graphs. A graph of pseudo degree zero is a special type of graph where each of the FTTM components differs from the one adjacent to it. Previous researchers have investigated and conjectured the number of generated FTTM pseudo degree zero with respect to n number of components and k number of versions. In this paper, the conjecture is proven analytically for the first time using a newly developed grid-based method. Some definitions and properties of the novel grid-based method are introduced and developed along the way. The developed definitions and properties of the method are then assembled to prove the conjecture. The grid-based technique is simple yet offers some visualization features of the conjecture.<\/jats:p>","DOI":"10.3390\/sym13112203","type":"journal-article","created":{"date-parts":[[2021,11,19]],"date-time":"2021-11-19T02:43:31Z","timestamp":1637289811000},"page":"2203","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Extended Graph of the Fuzzy Topographic Topological Mapping Model"],"prefix":"10.3390","volume":"13","author":[{"given":"Muhammad Zillullah","family":"Mukaram","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, Faculty of Science, University Teknologi Malaysia, Johor Bahru 81310, Johor, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2675-7952","authenticated-orcid":false,"given":"Tahir","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Faculty of Science, University Teknologi Malaysia, Johor Bahru 81310, Johor, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Norma","family":"Alias","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Faculty of Science, University Teknologi Malaysia, Johor Bahru 81310, Johor, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3228-5413","authenticated-orcid":false,"given":"Noorsufia Abd","family":"Shukor","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Faculty of Science, University Teknologi Malaysia, Johor Bahru 81310, Johor, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Faridah","family":"Mustapha","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Faculty of Science, University Teknologi Malaysia, Johor Bahru 81310, Johor, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"7519643","DOI":"10.1155\/2021\/7519643","article-title":"Graph of Fuzzy Topographic Topological Mapping in Relation to k-Fibonacci Sequence","volume":"2021","author":"Shukor","year":"2021","journal-title":"J. Math."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Mukaram, M.Z., Ahmad, T., and Alias, N. (2018, January 10\u201312). Graph of Pseudo Degree Zero Generated by FTTMnk. Proceedings of the International Conference on Mathematical Sciences and Technology 2018 (Mathtech2018): Innovative Technologies for Mathematics & Mathematics for Technological Innovation, Penang, Malaysia.","DOI":"10.1063\/1.5136361"},{"key":"ref_3","first-page":"363","article-title":"Domination in interval-valued fuzzy graphs","volume":"6","author":"Debnath","year":"2013","journal-title":"Ann. Fuzzy Math. Inform."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2571","DOI":"10.3233\/JIFS-181522","article-title":"Results on generalized intuitionistic fuzzy hypergroupoids","volume":"36","author":"Konwar","year":"2019","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_5","unstructured":"Elsafi, M.S.A.E. (2014). Combinatorial Analysis of N-tuple Polygonal Sequence of Fuzzy Topographic Topological Mapping. [Ph.D. Thesis, University Teknologi Malaysia]."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/11\/2203\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:32:11Z","timestamp":1760167931000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/11\/2203"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,18]]},"references-count":5,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2021,11]]}},"alternative-id":["sym13112203"],"URL":"https:\/\/doi.org\/10.3390\/sym13112203","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,11,18]]}}}