Graph theory is a well-established mathematical concept that is widely used in numerous applications such as in biology, chemistry and network analysis. The advancement in the theory of graph has led to the development of a concept called autocatalytic set. In this paper, a mathematical modeling technique namely graph-based dynamic modeling of palm oil refining process is introduced. The system parameters are identified in detail in the beginning of the paper. The parameters involved are the chemical compounds used or produced during the refining process. These identified parameters are then modeled as the vertices and edges of the graph. The dynamicity of the system is then simulated and analyzed. The system is simulated using MATLAB software programing. The two final products produced by the refining process agreed with results obtained from other published methods. Hence, the effectiveness and simplicity of the model are established.

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, $FTT{M}_n$, 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 $FTTM$s 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.

Fuzzy topological topographic mapping ($FTTM$) is a mathematical model that consists of a set of homeomorphic topological spaces designed to solve the neuro magnetic inverse problem. The key to the model is its topological structure that can accommodate electrical or magnetic recorded brain signal. A sequence of FTTM, $FTT{M}_n$, is an extension of FTTM whereby its form can be arranged in a symmetrical form, i.e., polygon. The special characteristic of $FTTM$, namely, the homeomorphisms between its components, allows the generation of new $FTTM$. The generated $FTTM$s can be represented as pseudo graphs. A pseudo-graph consists of vertices that signify the generated $FTTM$ and edges that connect their incidence components. A graph of pseudo degree zero, $G_0\left(FTT{M}_n^k\right),$ however, is a special type of graph where each of the $FTTM$ components differs from its adjacent. A researcher posted a conjecture on $G_0^3\left(FTT{M}_n^3\right)$ in 2014, and it was finally proven in 2021 by researchers who used their novel grid-based method. In this paper, the extended $G_0^3\left(FTT{M}_n^3\right)$, namely, the conjecture on $G_0^4\left(FTT{M}_n^4\right)$ that was posed in 2018, is narrated and proven using simple mathematical induction.

A generated n-sequence of fuzzy topographic topological mapping, , is a combination of n number of FTTM’s graphs. An assembly graph is a graph whereby its vertices have valency of one or four. A Hamiltonian path is a path that visits every vertex of the graph exactly once. In this paper, we prove that assembly graphs exist in and establish their relations to the Hamiltonian polygonal paths. Finally, the relation between the Hamiltonian polygonal paths induced from to the k-Fibonacci sequence is established and their upper and lower bounds’ number of paths is determined.

Fuzzy Topological Topographic Mapping (FTTM) is a mathematical model that consists of a set of homeomorphic topological spaces designed to solve the neuro magnetic inverse problem. A sequence of FTTM, denoted as $FTT{M}_n$, 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. Later, the $FTT{M}_n$ can also be viewed as a graph. Previously, a group of researchers defined an assembly graph and utilized it to model a DNA recombination process. Some researchers then used this to introduce the concept of tangled cords for assembly graphs. In this paper, the tangled cord for $FTT{M}_4$ is used to calculate the Eulerian paths. Furthermore, it is utilized to determine the least upper bound of the Hamiltonian paths of its assembly graph. Hence, this study verifies the conjecture made by Burns et al.