Extended Graph of the Fuzzy Topographic Topological Mapping Model

Abstract

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.