Explicit Closed-Form Formulas for the Cardinality of Generated FTTMs With Graphs of Pseudo Degree Zero

Anam, Amirul Aizad Ahmad Fuad, Yuhani Yusof, Rustam and Mujahid Abdullahi

International Journal of Mathematics and Mathematical Sciences, 2026, vol. 2026, 1-13

Abstract: Fuzzy topographic topological mapping FTTM is a mathematical model of homeomorphic topological spaces formulated to solve the neuromagnetic inverse problem. Homeomorphisms between these spaces generate new FTTMs, which can be represented as pseudo-graphs. A graph of pseudo degree zero is the pseudo-graph associated with a generated FTTM of order k and pseudo degree zero. The set of all such generated FTTMs is denoted by G0k∗FTTMnk, and its cardinality is denoted by G0k∗FTTMnk. The cardinalities G0k∗FTTMnk for k=3 and 4 have previously been expressed only recursively and thus require preceding terms for their computation. This paper focuses on the case k=3. Encoding each of the three versions of FTTM as a distinct labeled color on the vertices of the cycle graph Cn translates the pseudo degree zero condition into a proper coloring of Cn, and the order-three condition into the requirement that all three colors occur on its vertices. This reduces the enumeration of G03∗FTTMn3 to counting surjective proper three-colorings of Cn. The resulting explicit closed-form formulas are G03∗FTTMn3=2n−2 for odd n≥3 and G03∗FTTMn3=2n−4 for even n≥4. These formulas determine the cardinalities directly, without requiring prior terms, and reveal both exponential growth and a parity distinction. The formulas are also derived from the previously established recurrence relation, and their equivalence to the earlier recurrence formulas is proved.

Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:4684761

DOI: 10.1155/ijmm/4684761

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