Extended Graph of the Fuzzy Topographic Topological Mapping Model

: 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 , FTTM 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.


Introduction
Fuzzy topographic topological mapping (FTTM) [1] was introduced to solve the neuro magnetic inverse problem, particularly with regards to the sources of electroencephalography (EEG) signals recorded from epileptic patients. Originally, the model was a 4-tuple of topological spaces and mappings. The topological spaces are the magnetic plane (MC), base magnetic plane (BM), fuzzy magnetic field (FM) and topographic magnetic field (TM). The third component of FTTM, FM, is a set of three tuples with the membership function of its potential reading obtained from a recorded EEG. FTTM is defined formally as follows (see Figure 1).

Introduction
Fuzzy topographic topological mapping (FTTM) [1] was introduced to s neuro magnetic inverse problem, particularly with regards to the sources of el cephalography (EEG) signals recorded from epileptic patients. Originally, the mo a 4-tuple of topological spaces and mappings. The topological spaces are the m plane (MC), base magnetic plane (BM), fuzzy magnetic field (FM) and topograph netic field (TM). The third component of FTTM, FM, is a set of three tuples with th bership function of its potential reading obtained from a recorded EEG. FTTM is formally as follows (see Figure 1).
Furthermore, a sequence of FTTM, FTTM n , is an extension of FTTM and illustrated in Figure 2. It is arranged in a symmetrical form, since the model can accommodate magnetoencephalography (MEG) signals as well as image data due to its homeomorphism. ymmetry 2021, 13, x FOR PEER REVIEW Furthermore, a sequence of FTTM, , is an extension of FTTM a in Figure 2. It is arranged in a symmetrical form, since the model can accom netoencephalography (MEG) signals as well as image data due to its homeo

Generalized FTTM
Generally, the FTTM structure can also be expanded for any n numb nents.

Generalized FTTM
Generally, the FTTM structure can also be expanded for any n number of components.

Definition 2.
Ref. [2] A FTTM is defined as such that A 1 , A 2 , . . . , A n are the components of FTTM n The same generalization can be applied to any k number of FTTM versions as well, denoted as FTTM k n . Without the loss of generality, the collection of the k version of FTTM, in short FTTM k n , is now simply called as a sequence of FTTM unless otherwise stated.
where FTTM 1 n is the first version of FTTM n , the FTTM 2 n is the second version of FTTM n and so forth.
Obviously, a new FTTM can be generated from a combination of components from different versions of FTTM due to their homeomorphisms. Definition 4. Ref. [2] A new FTTM generated from * FTTM k n is defined as where 0 ≤ m 1 , m 2 , . . . , m n ≤ k and m i = m j for at least one i, j. A set of elements generated by * FTTM k n is denoted by G * FTTM k n . Mukaram et al. [2] showed that the number of FTTM can be determined from * FTTM k 4 using the geometrical features of its graph representation.
Theorem 1. Ref. [2] The number of generated FTTM that can be created from * FTTM k 4 is Theorem 1 is then extended to include n number of FTTM components.

Theorem 2.
Ref. [2] The number of generated FTTM that can be created from * FTTM k n is The following example is presented to illustrate Theorem 2.

Extended Generalization of FTTM
There are many studies on ordinary and fuzzy hypergraphs available in the literature such as [3,4]. However, * FTTM k n is an extended generalization of FTTM that is represented by a graph of a sequence of k number of polygons with n sides or vertices. The polygon is arranged from back to front where the first polygon represents FTTM 1 n , the second polygon represents FTTM 2 n and so forth. An edge is added to connect FTTM 1 n to the FTTM 2 n component wisely. A similar approach is taken for FTTM 2 n , FTTM 3 n and the rest ( Figure 3). ymmetry 2021, 13, x FOR PEER REVIEW Theorem 2. Ref. [2] The number of generated FTTM that can be created from * | ( * )| = − .
The following example is presented to illustrate Theorem 2.

Extended Generalization of FTTM
There are many studies on ordinary and fuzzy hypergraphs available in such as [3,4]. However, * is an extended generalization of FTTM sented by a graph of a sequence of number of polygons with sides o polygon is arranged from back to front where the first polygon represen second polygon represents and so forth. An edge is added to conn the component wisely. A similar approach is taken for , rest ( Figure 3). When a new is obtained from * , it is then called a pse the generated and plotted on the skeleton of * . A generated pseudo-graph consists of vertices that signify the generated and edg nect the incidence components. Two samples of pseudo-graphs are illustrat When a new FTTM is obtained from * FTTM k n , it is then called a pseudo-graph of the generated FTTM and plotted on the skeleton of * FTTM k n . A generated element of a pseudo-graph consists of vertices that signify the generated FTTM and edges which connect the incidence components. Two samples of pseudo-graphs are illustrated in Figure 4. When a new is obtained from * , it is then called a pseudo-graph of the generated and plotted on the skeleton of * . A generated element of a pseudo-graph consists of vertices that signify the generated and edges which connect the incidence components. Two samples of pseudo-graphs are illustrated in Figure 4. Another concept related closely to the pseudo-graph is the pseudo degree. It is defined as the sum of the pseudo degree from each component of the . The pseudo Another concept related closely to the pseudo-graph is the pseudo degree. It is defined as the sum of the pseudo degree from each component of the FTTM. The pseudo degree of a component is the number of other components that are adjacent to that particular component.
Definition 6. Ref. [2] The deg p G : G * FTTM k n → Z defines the pseudo degree of the FTTM graph. Let F ∈ FTTM Definition 7. Ref. [2] The set of elements generated by * FTTM k n that have pseudo degree zero is #G 0 FTTM k n denotes the cardinality of the set G 0 FTTM k n .  Figure 5). (9)
2. # ( ) denotes the cardinality of the set ( ).  Previously, Elsafi proposed a conjecture in [5] related to the graph of pseudo degree.
In order to observe some patterns that may appear from the proposed conjecture, Mukaram et al. [2] have developed an algorithm to compute G 0 FTTM 3 n in order to prove the conjecture analytically. A flowchart on G 0 * FTTM n | ( )| = 4| ( )| + 6, ℎ In order to observe some patterns that may appear from the proposed co Mukaram et al. [2] have developed an algorithm to compute | ( )| in prove the conjecture analytically. A flowchart on | ( * )| is sampled in Figu   Figure 6. Flowchart for determining | ( * )|.
The researchers generated all combinations for 3 ≤ ≤ 4, 4 ≤ ≤ were able to isolate graphs with pseudo degree zero, which are listed below (Tab  The researchers generated all FTTM combinations for 3 ≤ k ≤ 4, 4 ≤ n ≤ 15 and were able to isolate graphs with pseudo degree zero, which are listed below (Table 1). The researchers then simulated G 0 FTTM k n for some values of k as well [2]. The number of graphs of pseudo degree zero for 2 ≤ k ≤ 8 and 2 ≤ n ≤ 10 are listed in Table 2.

Grid of FTTM
An alternative presentation of a sequence of FTTM, called an FTTM grid, is briefly overviewed. It provides a different perspective of the structure of FTTM. Instead of a polygon representation for each version of FTTM, a straight line is now used. The components of FTTM n are arranged on a horizontal line of vertices and the lines represent the homeomorphisms between the components of FTTM n . The only exception is the homeomorphism between the first and last components of FTTM n , A 1 and A n , respectively. Two open segments on the left of A 1 and on the right of A n are used to represent the homeomorphism between them. A vertical line is added to represent a homeomorphism between two components of different versions; hence, a grid is created (see Figure 7). The researchers then simulated | ( )| for some values of k as well [2]. Th number of graphs of pseudo degree zero for 2 ≤ ≤ 8 and 2 ≤ ≤ 10 are listed in Ta ble 2.

Grid of FTTM
An alternative presentation of a sequence of , called an grid, is briefl overviewed. It provides a different perspective of the structure of FTTM. Instead of a pol ygon representation for each version of , a straight line is now used. The compo nents of are arranged on a horizontal line of vertices and the lines represent th homeomorphisms between the components of . The only exception is the homeo morphism between the first and last components of , and , respectively Two open segments on the left of and on the right of are used to represent th homeomorphism between them. A vertical line is added to represent a homeomorphism between two components of different versions; hence, a grid is created (see Figure 7). There are four advantages when FTTM is represented as a grid instead of a sequenc of polygon.

•
It is represented in two dimensions; therefore, it reduces the complexity of the struc ture.

•
The process of adding a new component is easier than in a sequence of polygon.

•
It can take any number of components by adding the number of vertices at the en of the grid.

•
The homeomorphism between two components of the same version is presented a a horizontal edge, whereas the homeomorphism between two components of tw different versions is represented by a diagonal edge (see Figure 8). These arrange ments are necessary to produce the graph of pseudo degree zero. There are four advantages when FTTM is represented as a grid instead of a sequence of polygon.

•
It is represented in two dimensions; therefore, it reduces the complexity of the structure.

•
The process of adding a new component is easier than in a sequence of polygon. • It can take any number of components by adding the number of vertices at the end of the grid.

•
The homeomorphism between two components of the same version is presented as a horizontal edge, whereas the homeomorphism between two components of two different versions is represented by a diagonal edge (see Figure 8). These arrangements are necessary to produce the graph of pseudo degree zero. Furthermore, Zilullah et al. [2] introduced some operations and spect to the FTTM grid. They are recalled, summarized and listed bel Then, we will move on to the next main section of the paper wherein C proven as a theorem.  Furthermore, Zilullah et al. [2] introduced some operations and properties with respect to the FTTM grid. They are recalled, summarized and listed below for convenience. Then, we will move on to the next main section of the paper wherein Conjecture 1 is finally proven as a theorem.
B G * FTTM k n is the set of FTTM blocks that can be generated from G * FTTM k n .
Definition 9. The function C j i is defined as C : G * FTTM k n → B G * FTTM k n for F ∈ G * FTTM k n , A generated FTTM is then divided into blocks of three components. A set of blocks is defined as follows.

Definition 12. A set of blocks B ijk is defined as
Since this study is concerned with graphs of pseudo degree zero, the sets that need to be taken into consideration are the ones with diagonal paths, namely, B 121 , B 121 , B 123 , B 131 ,  B 132 , B 212 , B 213 , B 232 , B 231 , B 321 , B 312 , B 323 n is the set of generated FTTMs with a diagonal path, then deg p G(F) = 2 or 0.
Corollary 1. The element of G 0 FTTM k n has a FTTM path with the following properties: 1.
All the edges connecting the path are diagonal.

2.
The starting and the end points of the path belong to different versions of FTTM.
, then all the paths for x are diagonals.

Lemma 2.
If F ∈ G * FTTM k n , then ∃x, y such that x ∈ G * FTTM k n−2 , y ∈ C n n−2 G * FTTM k n and F = x ⊕ y.

Lemma 3.
If F ∈ G * FTTM k n , then ∃ unique tuple (x, y) such that x ∈ G * FTTM k n−2 , y ∈ C n n−2 G * FTTM k n and F = x ⊕ y.
for any a, b ∈ Z and a = b.

The Theorem
All the materials laid down in previous sections are assembled to produce the analytical proof of Conjecture 1. The first step is to find G d * FTTM 3 n since G 0 * FTTM 3 n is a subset of G d * FTTM 3 n by Theorem 2.
By using Theorem 4, P(m + 1) = G 0 * FTTM 3 2k+3 = |K| such that By using Theorem 5, The set G d can be constructed from (x, y) where x ∈ G d m n−2 =1 * FTTM 3 2k+1 and y ∈ C n n−2 G d * Thus, Similarly, the same induction process can be used as proof for even parts.
The set G d * FTTM 3 n has only two possible subsets, namely G 0 * FTTM 3 n and H n = x ∈ G d * FTTM 3 n deg p x = 2 . To find G 0 * FTTM 3 n , the relation between G 0 * FTTM 3 n , G d * FTTM 3 n and H n must be investigated.
Proof of Theorem 9. Using Theorem 6, G 0 * FTTM 3 From Theorem 8 and Lemma 6, Hence by Theorem 7, such that G 0 * FTTM 3 Theorem 9 is another version of the earlier conjecture. A simple algebraic manipulation is needed to show their equivalence. We formally state and prove this as the final theorem.
The whole process of proving Conjecture 1 is summarized below in Figure 9. It shows that the equation in Theorem 9 is exactly the statement of the conjecture. I other words, the conjecture is proven by construction.□ The whole process of proving Conjecture 1 is summarized below in Figure 9. Outline of proving Conjecture 1 by construction. Figure 9. Outline of proving Conjecture 1 by construction.

Conclusions
The developed grid-based method of proof is new; some definitions and properties were introduced, whereas others were investigated along the way. The originality and advantages of this method can be summarized in the point forms below.

•
It provides a different perspective to the structure of FTTM. Instead of a polygon representation for each version of FTTM, a straight line is now used. The components of FTTM n are arranged on a horizontal line of vertices and the lines represent the homeomorphisms between the components of FTTM n . • A vertical line is added to represent a homeomorphism between two components of different versions; hence, a grid is created. • It is represented in two dimensions; therefore, it reduces the complexity of the structure.

•
The process of adding a new component is easier than in a sequence of polygon. • It can take any number of components by adding the number of vertices at the end of the grid.

•
The homeomorphism between two components of the same version is presented as a horizontal edge, whereas the homeomorphism between two components of two different versions is represented by a diagonal edge (see Figure 8).

•
This grid-based technique offers an edge in proving the conjecture; in particular, it enables one to visualize a given problem in a 2-dimensional space. • Finally, the conjecture that spells the number of the generated FTTM graph of pseudo degree zero with respect to n number of components and k number of versions is proven analytically for the first time using this method.
However, the lengthy computing time for simulation needs to be resolved for larger k and n, accordingly. This may be overcome by employing parallel computing, and the grid-based technique can be very handy for such enumerative combinatorics problems in the near future.