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

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.