# Graph of Fuzzy Topographic Topological Mapping in relation tok-Fibonacci Sequence

Authors: Noorsufia Abd Shukor; Tahir Ahmad; Amidora Idris; Siti Rahmah Awang; Amirul Aizad Ahmad Fuad
Journal: Journal of Mathematics
 (2021)

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## Abstract

A generatedn-sequence of fuzzy topographic topological mapping, FTTMn, 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 FTTMn and establish their relations to the Hamiltonian polygonal paths. Finally, the relation between the Hamiltonian polygonal paths induced from FTTMn to the k-Fibonacci sequence is established and their upper and lower bounds’ number of paths is determined.


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## Body

## 1. Introduction

The fuzzy topographic topological mapping (FTTM) model is built to solve the neuromagnetic inverse problem proposed in 1999 [1]. It consists of four topological spaces, namely, magnetic contour plane (MC), base magnetic plane (BM), fuzzy magnetic field (FM), and topographic magnetic field (TM). The FTTM is developed to determine the location of a simulated neuromagnetic current source [2] as shown in Figure 1.Figure 1
FTTM model.Later, Ahmad et al. [3] proved that the components of FTTM, namely, MC, BM, FM, and TM, were homeomorphic. The FTTM’s structures and proofs of their homeomorphisms were outlined in [4].Furthermore, FTTM can also be viewed as a sequence. The idea is possible when FTTM version 2 was successfully constructed by Rahman et al. [5] as shown in Figure 2. It was specially designed to solve the multiple current sources.Figure 2
FTTM1 and FTTM2.In 2006, Yun and Ahmad [4] noticed that if there are two elements of FTTM (see Figure 2), they will generate(1)21×21×21×21−2=14new elements of FTTM.These 14 elements of FTTM are (MC, BM, FM, TMI), (MC, BM, FMI, TM), (MC, BMI, FM, TM), (MI, BM, FM, TM), (MC, BM, FMI, TMI), (MC, BMI, FMI, TM), (MI, BMI, FM, TM), (MI, BM, FM, TMI), (MI, BM, FMI, TM), (MC, BMI, FM, TMI), (MC, BMI, FMI, TMI), (MI, BM, FMI, TMI), (MI, BMI, FM, TMI), and (MI, BMI, FMI, TM).Further, Yun [6] conjectured the following.Conjecture 1.
If there existn elements of FTTM that are homeomorphic to each other componentwise, the number of new elements of FTTM that can be generated from these n elements is(2)n1×n1×n1×n1−n=n!1!n−1!×n!1!n−1!×n!1!n−1!×n!1!n−1!−n=n×n×n×n−n=n4−n.In order to prove the conjecture, Jamaian et al. [7] introduced the concept of sequence of FTTM as stated below.Definition 1 (see [7]).
LetFTTMi=MCi,BMi,FMi,TMi such that MCi,BMi,FMi,TMi are topological spaces with MCi≅BMi≅FMi≅TMi. Set of FTTMi is denoted by FTTM=FTTMi:i=1,2,3,…,n. Sequence of nFTTMi of FTTM is FTTM1,FTTM2,FTTM3,FTTM4,…,FTTMn such that MCi≅MCi+1,BMi≅BMi+1,FMi≅FMi+1 and TMi≅TMi+1.
A sequence ofnFTTMi, without loss of generality, abbreviated as FTTMn, is illustrated in Figure 3.
Finally, the conjecture was proven [7], and surprisingly the FTTMn is related to the Pascal triangle.
Elsafi [8] then brought the concept of sequence of FTTM to another level. The researcher viewed and furnished sequence of FTTM as a graph. The details of the concept are presented in the following section.Figure 3
The sequence ofFTTMn.


## 2. Graph ofFTTMn

Sayed and Ahmad introduced for the first time the representation of FTTM as a graph in [5]. Further, they defined the notion of order with respect to sequence of FTTM as follows.Definition 2 (see [5]).
LetFTTMn=FTTM1,FTTM2,FTTM3,…,FTTMn be a sequence of n-FTTM (see Figure 3); then,(1)
Ci,jFTTMn are cubes of order two that can be produced from the combination of FTTMi and FTTMj in FTTMn for 1≤i,j≤n:(3)i=1,2,3,…,n−1,j=2,3,…,n.(2)
Ci,jFTTMn1≤i<j≤n represent the number of cubes of order two that can be produced from the combination of FTTMi and FTTMj in FTTMn, such that i∈I,j∈J,∀i<j≤n.
Figure4 shows the sequence of three terms of FTTM3 such that FTTM3=M1,B1,F1,T1,M2,B2,F2,T2,M3,B3,F3,T3 and(4)Ci,jFTTM3=C1,2FTTM3,C2,3FTTM3,C1,3FTTM3,Ci,jFTTM31≤i<J≤3=3.
Figure5(a) shows that M1,B1,F3,T3 is an element of order two since its components appear in two terms of FTTM, namely, in FTTM1 and FTTM3. By replacing B1 with B2, then M1,B2,F3,T3 is an element of order three since its components appear in FTTM1,FTTM2, and FTTM3 as presented in Figure 5(b).
Later, Ahmad et al. [9] established the relation of sequence of FTTMn to k-Fibonacci sequence.Figure 4
FTTM3.Figure 5
Example ofFTTM3 with elements of different orders: (a)M1,B1,F3,T3; (b)M1,B2,F3,T3.
(a)(b)Theorem 1 (see [9]).
The number of cubes produced by the combination of any three terms inFTTMn; FTTM3/n can be presented as(5)FTTM3/n=∑i=3nn+3−ii−n+2−ii+1=nn−1n−23!,forn≥3.For examples,FTTM3/1=0 for FTTM1 (see Figure 1), FTTM3/2=0 since FTTM2 is made of two terms FTTM only and FTTM3/3=1 (see Figure 3). The numbers of FTTM3/n for n=1,2,3,…,10 are summarized in Table 1.Table 1
FTTM3/n for n=1,2,3,…,10
nFTTM3/nn = 10n = 20n = 31n = 44n = 510n = 620n = 735n = 856n = 984n = 10120


## 3. Assembly Graph and Hamiltonian Path

The concept of an assembly graph was first introduced by Angeleska et al. [10] for DNA structure through recombination process. The formal definition of an assembly graph is as follows.Definition 3 (see [10]).
An assembly graph is a finite connected graph, where all vertices are rigid vertices of valency 1 or 4. A vertex of valency 1 is called an end point. LetΓ=V,E be a finite graph with a set of vertices V and a set of edges E. The number of 4-valent vertices in Γ is denoted with Γ. The assembly graph is called trivial if Γ=0 (see Figure 6).
Angeleska et al. [10] also defined isomorphism between two assembly graphs. Basically, their isomorphism is a special case of the ordinary graph isomorphism.Figure 6
Examples of assembly graph [10].Definition 4 (see [10]).
Two assembly graphsΓ1=V1,E1 and Γ2=V2,E2 are isomorphic if there is a graph isomorphism Φ that preserves the cyclic order of each rigid vertex. More specifically, for a graph isomorphism Φ=Φv,Φe:Γ1⟶Γ2 with Φv=V1⟶V2 and Φe=E1⟶E2, for every rigid vertexv,e1,e2,e3,e4cyc in Γ1, we have(6)Φvv,Φve1,Φve2,Φve3,Φve4cyc=Φvv,EcycΦvv.
Angeleska et al. [10] then defined a composition operator for two assembly graphs. In particular, the initial vertex of Γ1∘Γ2 is the initial vertex of Γ1 and the terminal vertex of Γ1∘Γ2 is the terminal vertex of Γ2.Definition 5 (see [10]).
A compositionΓ1∘Γ2 of two (directed simple) assembly graphs Γ1 and Γ2 is the directed simple assembly graph, obtained by identifying the terminal vertex of Γ1 with the initial vertex of Γ2.
Furthermore, the following definitions yield some immediate properties for graphFTTMn.Definition 6 (see [10]).
LetΓ be an assembly graph. An open path in Γ is a homeomorphic image of the open interval (0, 1) in Γ. An open path is also represented by a sequence:(7)e1\v0,v1,e2,v2,e3,…,vm−1,em,vm,em+1\vm+1,where vi′s are vertices in Γ for i∈1,2,…,m such that vi≠vj when i≠j and ei′s are edges in Γ for i∈1,2,…,m with endpoints vi−1 and vi, respectively, such that the initial vertex of e1 (and possibly part of e1) and the terminal vertex of em+1 (and possibly part of em+1) are not included.
An open path is a cycle ife1=em+1.Definition 7 (see [10]).
A set of pairwise disjoint open pathsγ1,…,γk in Γ is called Hamiltonian if their union contains all 4-valent vertices of Γ. An open path γ is called Hamiltonian if the set {γ} is Hamiltonian.Definition 8 (see [10]).
LetΓ be an assembly graph. The assembly number of Γ, denoted by An(Γ), is defined by An(Γ) = min {k | there exists a Hamiltonian set of polygonal paths γ1,…,γk in Γ}.Definition 9 (see [10]).
For a positive integerm, we define the minimal realization number for m to be Rminmmin=Γ:AnΓ=m where Γis the number of 4-valent vertices in Γ. A graph Γ such that Rminm=Γ is called a realization of Rminm.
A Hamiltonian cycle is a cycle which passes through all vertices and the path ends at the initial vertex, and a Hamiltonian path is a path that visits every vertex of the graph exactly once.
A theorem that relates the number of Hamiltonian polygonal paths in an assembly graph is as follows.Theorem 2 (see [11]).
IfΓ is a simple assembly graph with Γ=k and C is the collection of all Hamiltonian polygonal paths of Γ, then(8)C≤F2k+1−1,where Fk is the kth Fibonacci number.


## 4. Assembly Graph ofFTTMn

A graph ofFTTMn as described above contains many subgraphs including assembly graphs. A new concept called maximal assembly graph for assembly subgraphs of FTTMn is introduced.Definition 10.
LetG1,G2,G3,…,Gn be subgraphs of G (V, E) whereby each Giis an assembly graph. A maximal assembly subgraph of Gi is defined asΓGi=maxΓG1,ΓG2,…,ΓGn.
Table2 lists all assembly subgraphs for FTTM3.
LetΓ1 be the assembly subgraph as in Table 2 (5) and Γ2 be the assembly subgraph as in Table 2 (7); then Γ1∘Γ2=Γ1+Γ2=2+2=AnΓ1+AnΓ2.
AnFTTM4 produced 23 assembly subgraphs [12].
Then, considerΓ1=V1,E1 and Γ2=V2,E2 as assembly graphs as depicted in Figures 7(a) and 7(b) whereby V1=B1,B2,M2,F2,B3,M3,F3,B5, E1=B1,B2,B2,M2,B2,F2,B2,B3,B3,M3,B3,F3,B3,B4 and V2=T1,T2,M2,F2,T3,M3,F3,T4, E2=T1,T2,T2,F2,T2,M2,T2,T3,T3,M3,T3,F3,T3,T4, respectively.
Now, defineΦ=ΦV,ΦE:Γ1⟶Γ2such that(9)ΦvB1=T1,ΦEB1,B2=T1,T2,ΦvB2=T2,ΦEB2,M2=T2,F2,ΦvM2=M2,ΦEB2,F2=T2,M2,ΦvF2=F2,ΦEB2,B3=T2,T3,ΦvB3=T3,ΦEB3,M3=T3,M3,ΦvM3=M3,ΦEB3,F3=T3,F3,ΦvF3=F3,ΦEB3,B4=T3,T4,ΦvB4=T4.
Hence,Γ1≅Γ2.
Clearly, a maximal assembly subgraph forFTTMn is the resultant graph with edges for the first and the last terms of FTTM, in particular,FTTM1 and FTTMn are neglected. The formal definition of a maximal assembly graph of FTTMn is stated.Table 2
Assembly subgraphs ofFTTM3
—No.Geometrical featuresThe number of 4-valent vertices1ΓFTTM3=M2=12ΓFTTM3=B2=13ΓFTTM3=F2=14ΓFTTM3=T2=15ΓFTTM3=M2,B2=26ΓFTTM3=B2,F2=27ΓFTTM3=F2,T2=28ΓFTTM3=M2,T2=29ΓFTTM3=M2,B2,F2,T2=4Figure 7
Example ofFTTM4 with different 4-valent vertices. (a) ΓFTTM4=B2,B3=2. (b) ΓFTTM4=T2,T3=2.
(a)(b)Definition 11.
The maximal assembly graph ofFTTMn is(10)ΓFTTMn=FTTMn−EFTTM1∪EFTTMn,forn≥3,and ΓFTTMn is the number of its 4-valent vertices.
From now on, the maximal assembly subgraph ofFTTMn is referred to as an assembly graph of FTTMn until mentioned otherwise. Some properties on assembly graph of FTTMn for n=3 and 4 are summarized as follows.Theorem 3 (see [13]).
TheFTTM3 consists of an assembly subgraph.Theorem 4 (see [13]).
TheFTTM4 consists of an assembly subgraph.The previous two results can be generalized to anyFTTMn.Theorem 5 (see [13]).
Every sequence ofFTTMn contains an assembly subgraph for n≥3.Furthermore, Ahmad et al. [13] proved that the number of 4-valent vertices of the maximal assembly graph, ΓFTTMn, is as follows.Theorem 6 (see [13]).
ΓFTTMn+2=4+n−14,forn∈N.
The following theorems are immediate.Theorem 7.
Every sequence ofFTTMn yields minimal realization, Rminm number, for n≥3.Proof.
Theorem5 guarantees that every sequence of FTTMn contains an assembly subgraph for n≥3. An assembly graph for sequence of FTTMn is a maximal assembly graph by Definition 10 whereby ΓFTTMn is the number of its 4-valent vertices. By Definition 8, every sequence of FTTMn yields minimal realization, Rminm number, for n≥3.Theorem 8.
RminFTTMn+2=4n for sequence of FTTMn and n∈N.Proof.
Theorem7 guarantees that every sequence of FTTMn yields minimal realization, RminFTTMn number, for n≥3. Theorem 6 states that ΓFTTMn+2=4+n−14 for n∈N. Hence, RminFTTMn+2=4+n−14=4+4n−4=4n for sequence of FTTMn and n∈N.
In fact,ΓFTTMn is a realization of RminFTTMn since RminFTTMn=ΓFTTMn for n≥3. Consequently, the following theorem is deduced.Theorem 9.
RminFTTMn<RminFTTMn+1 for sequence of FTTMn and n≥3.Proof.
(11)RminFTTMn=4n,By Theorem8<4n+1<RminFTTMn+1for sequence ofFTTMnandn≥3.


## 5. Hamiltonian Paths in an Assembly Graph ofFTTMn

In previous section, we proved the existence of an assembly graph in any sequence ofFTTMn for n≥3. Hamiltonian polygonal paths exist in any assembly graph FTTMn as well.Theorem 10 (see [13]).
ΓFTTM3 consists of a set of Hamiltonian polygonal paths.Theorem 11 (see [13]).
ΓFTTM4 consists of a set of Hamiltonian polygonal paths.In fact, the existence of Hamiltonian paths in any sequence ofFTTMn for n≥3 is generalized in the following theorem.Theorem 12 (see [13]).
ΓFTTMn consists of a set of Hamiltonian paths, for n≥3.A coded program in [14] is modified to calculate the number of all Hamiltonian polygonal paths in an assembly graph of FTTMn. Table 3 summarizes the number of Hamiltonian polygonal paths in assembly graphs of FTTMn for n=3,4,5,…,10.Table 3
Hamiltonian polygonal paths in assembly graphs ofFTTMn for n=3,4,5,…,10
FTTMnNo. of verticesNo. of 4-valent verticesHamiltonian polygonal pathsFTTM31248FTTM4168144FTTM520121,168FTTM624168,032FTTM7282049,312FTTM83224281,248FTTM936281,523,920FTTM1040327,953,408


## 6. Graph ofFTTMn in Association to k-Fibonacci Sequence

The following theorem is the highlight of this paper. It links the work of Sayed and Ahmad [15] and Ahmad et al. [9], i.e., the relation of graph of FTTM and Fibonacci number (see Figure 8).Figure 8
Three mathematical concepts with respect to FTTM.Theorem 13 (see [12]).
LetFTTMn be a sequence of n-FTTM with ΓFTTMn=k and C be the collection of all sets of Hamiltonian polygonal paths of FTTMn; then,(12)C≤F2k+1−1,where Fk is the kth Fibonacci number.Proof.
LetFTTMn be a sequence of n-FTTM. By Theorem 5, FTTMn consists of assembly graphs, namely, ΓFTTMn. Then, Theorem 12 guarantees that ΓFTTMn consists of a set of Hamiltonian polygonal paths, say C. Using Theorem 2, for FTTMn, C≤F2k+1−1 as required whereby Fk is the kth Fibonacci number.
Thus, the connections illustrated in Figure8 are completed. A refinement of Theorem 13 is given in the following corollary.Corollary 14 (see [12]).
LetFTTMn be a sequence of n-FTTM for n≥3 and C be the set of all Hamiltonian polygonal paths of FTTMn; then,(13)C≤F8n+1−1.Proof.
LetFTTMn be a sequence of n-FTTM for n≥3. By Theorem 5, FTTMn consists of assembly graphs, namely, ΓFTTMn. Further, Theorem 6 reveals that ΓFTTMn+2=4+n−14,forn∈N. Theorem 12 guarantees that ΓFTTMn consists of a set C, that is, all its Hamiltonian polygonal paths. By replacing k=4+n−14,forn∈N, in Theorem 13,(14)C≤F2k+1−1=F24+n−14+1−1,replacek=4+n−14=F24+4n−4+1−1=F24n+1−1=F8n+1−1.
Table4 lists Hamiltonian polygonal paths of FTTMn in relation to k-Fibonacci numbers for n = 3 to 10.
The following theorem highlights the lower and upper bounds for Hamiltonian polygonal paths ofFTTMn.Table 4
Hamiltonian polygonal paths andF2k+1 for n = 3 to 10.
ΓFTTMn=kF2k+1CC≤F2k+1−1ΓFTTM3=4F24+1=F9=3488 ≤ 33ΓFTTM4=8F28+1=F17=1,597144144 ≤ 1,596ΓFTTM5=12F212+1=F25=75,0251,1681,168 ≤ 75,024ΓFTTM6=16F216+1=F33=3,524,5788,0328,032 ≤ 3,524,577ΓFTTM7=20F220+1=F41=156,580,14149,31249,312 ≤ 165,580,140ΓFTTM8=24F224+1=F49=7,778,742,049281,248281,248 ≤ 7,778,742,048ΓFTTM9=28F228+1=F57=365,435,296,1621,523,9201,523,920 ≤ 365,435,296,161ΓFTTM10=32F232+1=F65=17,167,680,177,5657,953,4087,953,408 ≤ 17,167,680,177,564Theorem 15.
LetFTTMn be a sequence of n-FTTM for n≥3 and C be the set of all Hamiltonian polygonal paths of FTTMn; then,(15)nn−1n−23!≤C≤F8n+1−1.Proof.
Definition11 states that ΓFTTMn is the number of 4-valent vertices. These 4-valent vertices can only exist only for (at least) three terms of FTTM, i.e., FTTM3/n. But Theorem 1 guarantees that the number of cubes produced by the combination of any three terms in FTTMn; FTTM3/n is nn−1n−2/3!. In other words, the lower bound of Hamiltonian polygonal paths of FTTMn, C, is obtained as C≥nn−1n−2/3!. Corollary 14 states C≤F8n+1−1 for n≥3. Hence, nn−1n−2/3!≤C≤F8n+1−1 as required.
The following table (see Table5) lists the lower and upper bounds for Hamiltonian polygonal paths in FTTMn for n = 3 to 10.Table 5
Lower and upper bounds for Hamiltonian polygonal paths ofFTTMn for n = 3 to 10.
FTTMnFTTM3/nCF8n+1−1FTTM31833FTTM441441,596FTTM5101,16875,024FTTM6208,0323,524,577FTTM73549,312165,580,140FTTM856281,2487,778,742,048FTTM9841,523,920365,435,296,161FTTM101207,953,40817,167,680,177,564


## 7. Conclusions

The aim of this paper is to prove that there exists a relationship betweenFTTMn (sequence of n-FTTM) and k-Fibonacci sequence. We have established the lower and upper bounds for the established relation.



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