Explicit Closed‐Form Formulas for the Cardinality of Generated F T T M s With Graphs of Pseudo Degree Zero

Source abstract

Fuzzy topographic topological mapping ( F T T M ) is a mathematical model of homeomorphic topological spaces formulated to solve the neuromagnetic inverse problem. Homeomorphisms between these spaces generate new F T T M s , which can be represented as pseudo‐graphs. A graph of pseudo degree zero is the pseudo‐graph associated with a generated F T T M of order k and pseudo degree zero. The set of all such generated F T T M s is denoted by , and its cardinality is denoted by . The cardinalities 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 F T T M as a distinct labeled color on the vertices of the cycle graph C n translates the pseudo degree zero condition into a proper coloring of C n , and the order‐three condition into the requirement that all three colors occur on its vertices. This reduces the enumeration of to counting surjective proper three‐colorings of C n . The resulting explicit closed‐form formulas are for odd n ≥ 3 and 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.