{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:36:43Z","timestamp":1760233003742,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,12,15]],"date-time":"2022-12-15T00:00:00Z","timestamp":1671062400000},"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, Malaysia","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 that consists of a set of homeomorphic topological spaces designed to solve the neuro magnetic inverse problem. The key to the model is its topological structure that can accommodate electrical or magnetic recorded brain signal. A sequence of FTTM, FTTMn, is an extension of FTTM whereby its form can be arranged in a symmetrical form, i.e., polygon. 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 pseudo-graph consists of vertices that signify the generated FTTM and edges that connect their incidence components. A graph of pseudo degree zero, G0(FTTMnk\u00a0), however, is a special type of graph where each of the FTTM components differs from its adjacent. A researcher posted a conjecture on G03(FTTMn3) in 2014, and it was finally proven in 2021 by researchers who used their novel grid-based method. In this paper, the extended G03(FTTMn3), namely, the conjecture on G04(FTTMn4) that was posed in 2018, is narrated and proven using simple mathematical induction.<\/jats:p>","DOI":"10.3390\/sym14122645","type":"journal-article","created":{"date-parts":[[2022,12,15]],"date-time":"2022-12-15T02:20:57Z","timestamp":1671070857000},"page":"2645","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Extended Graph of Fuzzy Topographic Topological Mapping Model: G04(FTTMn4)"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3228-5413","authenticated-orcid":false,"given":"Noorsufia Abd","family":"Shukor","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"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2533-8740","authenticated-orcid":false,"given":"Amidora","family":"Idris","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":"Siti Rahmah","family":"Awang","sequence":"additional","affiliation":[{"name":"Faculty of Management, University Teknologi Malaysia, Johor Bahru 81310, Johor, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Muhammad Zillullah","family":"Mukaram","sequence":"additional","affiliation":[{"name":"Pt Visi Global Teknologi, Yayasan Darul Marfu\u2019, Lantai 3. No 43 Jalan Haji Zaenudin, Gandaria Utara, Kebayoran Baru 12140, Jakarta Selatan, Indonesia"}],"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"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,15]]},"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","first-page":"35","article-title":"Homeomorphisms of Fuzzy Topographic Topological Mapping (FTTM)","volume":"21","author":"Ahmad","year":"2005","journal-title":"Matematika"},{"key":"ref_3","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]."},{"key":"ref_4","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_5","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_6","doi-asserted-by":"crossref","unstructured":"Zhu, J., Li, B., Zhang, Z., Zhao, L., and Li, H. (2022). High-Order Topology-Enhanced Graph Convolutional Networks for Dynamic Graphs. Symmetry, 14.","DOI":"10.3390\/sym14102218"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Wang, G., Chen, L., and Xiong, Z. (2022). The l1-Embeddability of Hypertrees and Unicyclic Hypergraphs. Symmetry, 14.","DOI":"10.3390\/sym14112260"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"O\u2019Keeffe, M., and Treacy, M.M.J. (2022). The Symmetry and Topology of Finite and Periodic Graphs and Their Embeddings in Three-Dimensional Euclidean Space. Symmetry, 14.","DOI":"10.3390\/sym14040822"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2317","DOI":"10.1007\/s12190-021-01619-5","article-title":"Randic index of bipolar fuzzy graphs and its application in network systems","volume":"68","author":"Poulik","year":"2021","journal-title":"J. Appl. Math. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"4557","DOI":"10.1007\/s10462-021-10111-2","article-title":"Estimation of most affected cycles and busiest network route based on complexity function of graph in fuzzy environment","volume":"55","author":"Poulik","year":"2022","journal-title":"Artif. Intell. Rev."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"104267","DOI":"10.1016\/j.rinp.2021.104267","article-title":"A fuzzy graph approach analysis for COVID-19 outbreak","volume":"25","author":"Hassan","year":"2021","journal-title":"Results Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"4826","DOI":"10.1038\/s41598-022-08821-w","article-title":"Counterfeit fifty Ringgit Malaysian banknotes authentication using novel graph-based chemometrics method","volume":"12","author":"Hassan","year":"2022","journal-title":"Sci. Rep."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"18","DOI":"10.3844\/jmssp.2013.18.23","article-title":"Graph of finite sequence of fuzzy topographic topological mapping of order two","volume":"9","author":"Sayed","year":"2013","journal-title":"J. Math. Stat."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"151","DOI":"10.3844\/jmssp.2010.151.156","article-title":"Generalized finite sequence of fuzzy topographic topological mapping","volume":"6","author":"Jamaian","year":"2010","journal-title":"J. Math. Stat."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"012079","DOI":"10.1088\/1742-6596\/890\/1\/012079","article-title":"A Comparison of Ordinary Fuzzy and Intuitionistic Fuzzy Approaches in Visualizing the Image of Flat Electroencephalography","volume":"890","author":"Zenian","year":"2017","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_16","unstructured":"Rahman, W.E.Z.W.A., Ahmad, T., and Ahmad, R.S. (2002). Simulating the Neuronal Current Sources in the Brain. Proc. Biomed., 19\u201322."},{"key":"ref_17","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_18","doi-asserted-by":"crossref","unstructured":"Mukaram, M.Z., Ahmad, T., Alias, N., Shukor, N.A., and Mustapha, F. (2021). Extended Graph of Fuzzy Topographic Topological Mapping Model. Symmetry, 13.","DOI":"10.3390\/sym13112203"},{"key":"ref_19","unstructured":"Moura, L. (2022, October 01). Induction and Recursion. PowerPoint Presentation, University of Ottawa. Available online: https:\/\/www.site.uottawa.ca\/~lucia\/courses\/2101-12\/lecturenotes\/06Induction.pdf."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/12\/2645\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:41:35Z","timestamp":1760146895000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/12\/2645"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,15]]},"references-count":19,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["sym14122645"],"URL":"https:\/\/doi.org\/10.3390\/sym14122645","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,12,15]]}}}