Combinatorial structure of cube of fuzzy topographic topological mapping and K-fibonacci sequence

Abstract

Fuzzy Topographic Topological Mapping (FTTM) is a model for solving neuromagnetic inverse problem. FTTM consists of four topological spaces that are homeomorphic to each other. A sequence of FTTMn is a combination of n terms of FTTM. In previous studies, FTTM are linked with three mathematical concepts namely; FTTM with Pascal’s Triangle, FTTM as a graph and FTTM in relation to k-Fibonacci sequence. In this research, the relationship between graph of FTTMn and k-Fibonacci is established via Hamiltonian polygonal paths in an assembly graph of FTTMn. The assembly graph is a graph with all vertices have valency of one or four. The Hamiltonian path is a path that visits every vertex of a graph exactly once. The structure of assembly graph of FTTMn including maximal assembly graph of FTTMn is introduced and its properties are investigated. The existence of Hamiltonian polygonal path in maximal assembly graph of FTTMn is proven. Several new definitions and theorems for the assembly graph of FTTMn and Hamiltonian polygonal path in maximal assembly graph of FTTMn are stated and proven, respectively. Finally, a theorem that highlight the relation between graph of FTTMn to k-Fibonacci sequence is proven.