Coordinated Transformation for Fuzzy Autocatalytic Set of Fuzzy Graph Type-3

Corresponding Author: Tahir Ahmad Department of Mathematical Science, Faculty of Science and Center for Sustainable Nanomaterials, Universiti Teknologi Malaysia, 81310 UTM, Skudai, Johor, Malaysia E-Mail: tahir@ibnusina.utm.my Abstract: Fuzzy Autocatalytic Set (FACS) of fuzzy graph Type-3 was established in the effort to describe the underlying phenomena of a clinical waste incineration process. Further research could be explored to scrutinize this concept especially in looking at its mathematical structure. In this paper, we highlighted the mathematical structures of FACS particularly on the algebraic structures of its matrix and graph. New definitions and theorems were generated through this investigation which leads to the visualization of the concept in Euclidean space. This visualization was accomplished through the transformation of non coordinated FACS to coordinated FACS in space. Eventually, the performance of coordinated FACS is then compared to the non-coordinated FACS in respect to the clinical waste incineration process.


Introduction
The emergence of fuzzy graph to autocatalytic sets has instigated a new concept named Fuzzy Autocatalytic Set (FACS). A clinical waste incineration process in Malacca was successfully modeled using this concept (Sabariah, 2005) whereby its theoretical foundation was first laid by Tahir et al. (2010a;2010b).

Definition 1.1
Let where E is a set of edges such that E{e 1 ,e 2 ,…,e n }. The fuzzy head of denotes as h(e i ) and the fuzzy tail t(e i ) are function of e i such that h: E → [0,1] and t:E → [0,1] for e i ∈E. Fuzzy edge connectivity is a tuple (t(e i ), h(e i )) and the set of all fuzzy edge connectivity is denoted as C = {(t(e i ),h(e i )):e i ∈E}. The membership value of fuzzy edge connectivity is denoted as µ(e i ) = min{t(e i ),h(e i )} (Fig. 1).
Naturally, the adjacency matrix of FACS is defined as follows:

Definition 1.2
The entries for adjacency matrix of FACS of fuzzy graph Type-3; Several new results of FACS which linked to Perron-Frobenius (PF) Theorem have been discussed in previous studies (Tahir et al., 2010a). In these studies, FACS was shown to provide better explanations of the real phenomena of the incineration process than crisp graph. Yet there are great opportunities and avenues for further exploration of this concept.
The main idea behind the definition of FACS is the merger of fuzzy graph Type-3 to autocatalytic set (Tahir et al., 2010b). The formal definition of FACS is given below.

Definition 1.3
Fuzzy Autocatalytic Set (FACS) is a subgraph where each nodes has at least one incoming link with membership value µ(e i )∈(0,1],∀e i ∈E. The transformation of any FACS of fuzzy graph Type-3 with no loop to a square matrix is then given by the following theorem (Sumarni et al., 2008).
is a bijective function. The above theorem allows FACS of fuzzy graph Type-3 to be mapped to a square matrix as shown in Fig. 2.

Materials and Methods
Some important features of FACS of fuzzy graph Type-3 can be explored through its corresponding adjacency matrix. Tahir et al. (2010b) have previously shown that if G is an FACS, the corresponding adjacency matrix contains a cycle and its Perron eigenvalue λ 1 >0. Subsequently, further exploration of non negative matrix with zero diagonal reveals additional features of the matrix which includes irreduciblity. These new features are presented in the following lemma and theorems.

Lemma 2.1
Adjacency matrix deduces from FACS of fuzzy graph Type-3 has a similar matrix.

Proof
Let B∈ M n be the FACS of fuzzy graph Type-3 such that: Since B∈ M n (R), the Real Schur Triangulation theorem (Horn and Johnson, 1985) guarantees that ∃P which is orthogonal (P T P = P T P = I) such that B = P T AP and A∈M n (R). By taking P T = P −1 , therefore B = P −1 AP. Hence, B is similar to A.

Proof
Let B be an adjacency matrix of FACS of fuzzy graph Type-3. Each b ii since the graph is with no loop. Thus B is a zero diagonal matrix.
Since B is similar by Lemma 2.1, then there ∃ matrix P, A and P −1 , ∋ B = P −1 AP, where A is a diagonal matrix. Furthermore and A k is also a diagonal matrix: This means it contradicts to similarity of B (i.e., B = P −1 AP). Therefore, B must be primitive.
Moreover, using theorem on irreducibility as given below.
The following result on FACS is attained.

Theorem 2.4
If matrix B∈M n×n of FACS of fuzzy graph Type-3 is primitive, then it is irreducible.

Proof
Let B∈M n×n be a matrix of FACS of fuzzy graph Type-3 as defined in Def. 1.2, therefore B is a nonnegative matrix such that b ij ≥0 for all ij. Since B is primitive, choose k>0 such that B k >0. Let D = (I+B) n-1 B, therefore D = B n + C n-1 B n-1 + …+ C 1 B Consequently, D>0⇒(I + B) n-1 B>0⇒ (I + B) n-1 >0 since B k >0 for some k∈N. Hence B is irreducible by Theorem 2.3. Perez et al. (2001) described length of edge for an undirected, simple and connected graph G(V,E) by introducing a positive function l to possess the ordered property (Tahir et al., 2010b). He deduced that if it is an undirected, simple and connected graph with vertex set V = {v 1 ,v 2 ,…,v n } and edge set E = {e ij , e pq ,…e mn }, then for every edge e ij = (v i ,v j )∈E, there exist a positive number l(e ij ) = l((v i ,v j )) that represents the length of the edge.
The idea of Perez et al. (2001) as described above has initiated the following definition to represent length of an edge of FACS.
Definition 2.1 leads to the following definition on length for directed path of FACS. This definition is necessary to establish a concept of aperiodicity of FACS.

Definition 2.2
Suppose G FT3 (V,E) is a no-loop Fuzzy Autocatalytic set of fuzzy graph Type-3. The length of directed path that start and end at v i with respect to r route is denoted as ( ) The condition of aperiodicity is natural in the study for FACS of fuzzy graph Type-3 (Ninove, 2008). This is due to the fact that a particular route is different to the other route in the graph by its length or by vertices that it passes through. Precisely, aperiodicity denotes that a route is not periodic; which means that each route is not a repetition of the other route and thus length of each route in the graph is unique. This motivates the formation of the following definition which inspires from Euclid's algorithm for any two real numbers.

Definition 2.3
K be lengths of two routes of FACS atv i , then the greatest common divisor (gcd) between these two routes is defined as: if g k m Z gk K and gm K K K g if g k m Z gk K and gm K 3 is a gateway to a permissible adoption and adaptation of some of the results in theordinary concept of greatest common divisor for integers.

Corollary 2.6
Therefore using similar argument as in Theorem 2.5, there existinteger a and b such that ( ) But then da≤ 1 and and d∈Z + . If that the case Corollary 2.7: Theorem 2.8: The definition of aperiodicity given by (Ninove, 2008) is as follows.

Definition 2.4
A strongly connected graph is aperiodic if, for every node i∈N, the greatest common divisor of the length of all paths from i to i is equal to one.
Consequently, the above results instigated the following theorem on the aperiodicity of FACS of fuzzygraph Type-3.

Theorem 2.9
Any graph of FACS of fuzzy graph Type-3 is aperiodic.

Proof
Let G FT3 (V,E) be a graph of FACS. By Theorem 2.4, the adjacency matrix of FACS is irreduciblewhich imply that the graph is strongly connected. By using Definition 2.4 and Theorem 2.5, it isshown that any FACS of fuzzy graph Type-3 is aperiodic.
Similarly, this investigation on the algebraic structure of adjacency matrix can be extended to itstransition matrix.
Naturally, the formulation for the transition matrix for FACS of fuzzy graph Type-3 is based on itsfuzzy membership value (Sumarni et al., 2010).

Definition 2.5
The entries for transition matrix of FACS of fuzzy graph Type-3 is * ij P such that: The definition for stochastic matrix (Stewart, 1994;Norris, 1998) is as follows.

Definition 2.6
A non-negative matrix is stochastic matrix if row sums equal to one.
FACS of fuzzy graph Type-3 can be shown to be stochastic by the following theorem.

Theorem 2.10
Transition matrix for no loop FACS of fuzzy graph Type-3 is stochastic.

Proof
Let G FT3 (V,E) be a graph of FACS of fuzzy graph Type-3. Using Definition 2.5 of transition matrix of FACS and P ii = 0 due to the non-existence of loop in the graph, therefore: Hence P * is stochastic. The resultant matrix is also irreducible and is presented in the following theorem.

Theorem 2.11
Stochastic matrix for no loop FACS of fuzzy graph Type-3 is irreducible.

Proof
Let P * ∈M n (R) be a transition matrix for FACS of fuzzy graph Type-3 such that Since P∈M n (R), Real Schur Triangulation theorem (Horn and Johnson, 1985) guarantees that ∃P which is orthogonal (P T P= PP T = I) such that P * = P T AP. By taking P T = P −1 , therefore P * = P −1 AP. Hence, P * is similar to A where A is a diagonal matrix. Furthermore, and k A is also a diagonal matrix. By contradiction, > / for all k. Consequently, 1 1 0 P A P − > / . This means it contradicts to similarity of P * (i.e., P −1 AP). Therefore, P * must be primitive. Next, choose k>0 such that (P * ) k >0 and let for some N k ∈ . Hence P * is irreducible by Theorem 2.3 (Horn and Johnson, 1985).
The eigenvalues of Laplacian are related to structural properties of graph (Mohar, 1991). Chung defined thedirected Laplacian for aperiodic strongly connected directed graph by using transition probability matrix P u,v which is the probability of moving from vertex u to v as is the out-degree of u and w(u,v) is the weight of edge from u to v (Chung, 2005). However, in this work, edges of FACS are constructed based on fuzzy edge connectivity as stated in Definition 1.1. This situation leads to the formulation of the following new definition for directed Laplacian ofFACS.

Definition 2.7
Suppose P * is the transition matrix for no loop FACS of fuzzy graph Type-3 with P * u,v represents fuzzy value of moving from u to v. Let 1denotes the all 1s vector such that P * 1 = 1. If the graph is strongly connected and a periodic, then PF Theorem guarantees that there exist a unique (row) vector, namely ϕ for which ϕP * = ϕ with ϕ(v)>0 for all v and where, by (P * ) t denotes the transpose of P*. The combinatorial Laplacian of FACS is defined as: Some of the properties for the directed Laplacian of FACS have been formulated in the previous study (Tahir et al., 2010a;2010b) in which they are summarized in the following theorem.

Theorem 2.12
Suppose G FT3 (V,E) be a no loop of FACS of fuzzy graph of Type-3 which is strongly connected, then the combinatorial Laplacian n×n matrix L has the following properties: • The matrix is real and symmetric • The matrix is Hermitian • The matrix is positive semi-definite and have n nonnegative real eigenvalues which can beordered sequentially in an ascending order as 0 = λ 0 ≤λ 1 ≤…≤λ n-1 . • The smallest eigenvalue, λ 0 = 0 has multiplicity 1 where it is equal to the number of connected components in the graph The procedure to transform the graph of FACS into 2D-Euclidean space is adopted from Carmel et al. (2002) which is based on Laplacian matrix and solving a unique one dimensional optimization problem in order to determine their coordinates. The general overview of the transformation is depicted in the following Fig. 3.
In order to transform the graph of FACS to coordinated graph, definition of balanced vector and optimal arrangement of directed graph (Carmel et al., 2002) is adopted. Thus, balanced vector and optimal arrangement with regard to FACS can be presented as follows. Consequently, y = (y 1 ,…,y n ) is determined by using concept of minimization of hierarchy energy function where it is equivalent to solving an optimal arrangement (Carmel et al., 2002). Similarly, the same conceptcan be incorporated to FACS and it is redefined as follows.

Definition 2.9
Let G FT3 (V,W * ) be a graph of FACS with Laplacian, L and balance, b. Its optimal arrangement, y * , is the solution for Ly = b, subject to the constraint y t .1 n = 0.
The optimal arrangement y * is the vector for ycoordinates of FACS and is solved by using Conjugate Gradient method.
On the other hand, x = (x 1 ,…,x n ) t representing the xcoordinates of FACS is solved using minimization of edge squared lengths via Tutte-Hall energy function, According to (Carmel et al., 2002), minimization of this energy function is the Fiedler vector, which is the eigenvector of the Laplacian associated with the smallest positive eigenvalue. In this case, the Fiedler vector of combinatorial directed Laplacian is obtained by solving its eigenvalue problem.
From the properties deduced for combinatorial Laplacian matrix of FACS of fuzzy graph Type-3 and method of finding the (x, y) coordinate of FACS as described earlier, the procedure for the transformation can be presented formally by the following theorem.

Theorem 2.13
Any FACS of fuzzy graph of Type-3 can be induced to coordinated FACS of fuzzy graph of Type-3.

Proof
Suppose G FT3 (V,E) is an FACS of fuzzy graph Type-3 with n vertices and fulfill all the conditions described in Theorem 2.1, Theorem 3.3, Theorem 3.5 and Corollary 3.6. Next, define transition matrix for FACS of fuzzy graph type-3, P * as in Definition 4.1 which fulfill all the conditions described in Theorem 4.1 and Theorem 4.2. Next, find left Perron vector, ϕ, for P * using standard matrix multiplication (P * )2 n−1 = (P * ) n (P * ) n−1 or by using Theorem 1 in (Anton and Rorres, 1977). Next, construct combinatorial Laplacian for FACS, L, as defined in Definition 2.7 and is satisfying Theorem 2.12. Let balance vector for G FT3 (V,E), is b, as in Definition 2.8 and define optimal arrangement, y * as in Definition 2.9. Thenycoordinates of the nodes of G FT3 (V,E) is given by y*. Herewith, solve eigenvalue problem of L toget the Fiedler vector, x = (x 1 ,…,x n ) T which represents xcoordinates of the nodes. Positioned the nodes in 2D-Euclidean space using (x, y) coordinate obtained earlier and draw its corresponding edges. Finally FACS of fuzzy graph Type-3 is induced to coordinated FACS of fuzzy graph Type-3.
The following corollaries are immediate cases of Theorem 2.13.

Proof
From Theorem 5.2 of (Tahir et al., 2010a), the graph for clinical waste incineration process in (Sabariah, 2005) can be represented as h(e i )} which is FACS of fuzzy graph Type-3. By using Theorem 2.10 above, immediately it can be induced to coordinated FACS of fuzzy graph Type 3. Now by letting i:V→V F the following corollary can be obtained.

Corollary 2.15
FACS of fuzzy graph with fuzzy vertices and edges can be induced to coordinated fuzzy graph.

Proof
Consider G FT3 (V F , E F ), then each vertices in V F has fuzzy membership value; i.e., (V i , µ(V i ))∀i. By replacing V i with (V i ,µ(V i )) in Theorem 2.10, any FACS with fuzzy vertices and edges can thereforebe induced to coordinated graph.

Results and Discussion
A clinical waste incineration process was modeled by FACS of fuzzy graph Type-3 (Sabariah, 2005;Tahir et al., 2010b) as in Fig. 4.
In this present work, FACS representing the clinical waste incineration process can be transformed into 2D-Euclidean space stated in Collorary 2.14 by using Theorem 2.13 as in Fig. 5.
Every edge of the graph in Fig. 4 is associated to a membership value for fuzzy edge connectivity whereby no information on location of each node in this graph. In contrast, every edge of the graph in Fig.  5 provides not only information on membership value, it also gives the information of its length. Here, every node has its own coordinate and the location of every node is given in Table 1.   From the coordinates given in Table 1, the edge's length between two nodes is calculated as shown in Table 2.
The graph dynamic procedure for non coordinated FACS of fuzzy graph Type-3 was discussed in (Tahir et al., 2010b). The work was extended to coordinated FACS and the comparison between the two is given in Table 3. Table 3 shows that coordinated FACS provides not only the information on the sequence of depleted variables and the information on the by-product, but it also gives the set of most important variables in the incineration process.

Conclusion
Mathematical structures of FACS of fuzzy graph Type-3 has been investigated through its adjacency matrix, transition matrix and Directed Laplacian matrix. This mathematical structure provides a platform to transform FACS to coordinated FACS. Comparison between the results of graph dynamic procedure for noncoordinated and coordinated FACS has shown that coordinated FACS is superior to non-coordinated FACS with respect to information for sequence of depleted, important and detail of by-product of variables in the clinical waste incineration process.