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Completeness and regularity of generalized fuzzy graphs
 Sovan Samanta^{1},
 Biswajit Sarkar^{1}Email author,
 Dongmin Shin^{1} and
 Madhumangal Pal^{2}
 Received: 6 April 2016
 Accepted: 14 October 2016
 Published: 15 November 2016
Abstract
Fuzzy graphs are the backbone of many real systems like networks, image, scheduling, etc. But, due to some restriction on edges, fuzzy graphs are limited to represent for some systems. Generalized fuzzy graphs are appropriate to avoid such restrictions. In this study generalized fuzzy graphs are introduced. In this study, matrix representation of generalized fuzzy graphs is described. Completeness and regularity are two important parameters of graph theory. Here, regular and complete generalized fuzzy graphs are introduced. Some properties of them are discussed. After that, effective regular graphs are exemplified.
Keywords
 Generalized fuzzy graphs
 Effective edge
 Regular graphs
 Complete graphs
 Matrix
Background
Nowadays, graphs do not represent all the systems like networks, routes, schedules, images, etc. properly due to the uncertainty or haziness of the parameters of systems. For example, a social network may be represented as a graph, where vertices represent an account (person, institution, etc.) and edges represent the relation between those accounts. If the relations among accounts are measured as either good or bad according to the frequency of contacts among those accounts, then fuzzyness can be added for such representations. This and many other problems lead to define fuzzy graphs. The first definition of a fuzzy graph was introduced by Kauffman (1973). But, Rosenfeld (1975) described fuzzy relations on fuzzy sets and developed some theory of fuzzy graphs. Using these concept of fuzzy graphs, Koczy (1992) discussed fuzzy graphs to evaluate and to optimize any networks. Samanta and Pal (2013) showed that fuzzy graphs can be used in competition in ecosystems. After that, they introduced some different types of fuzzy graphs (Samanta and Pal 2015; Samanta et al. 2014). Bhutani and Battou (2003) and Bhutani and Rosenfeld (2003) discussed different arcs in fuzzy graphs. For further details of fuzzy graphs, readers may look in Mathew (2009), Mordeson and Nair (2000), Pramanik et al. (2014, 2016) and Rashmanlou et al. (2015). Applications of fuzzy graph include data mining, image segmentation, clustering, image capturing, networking, communication, planning, scheduling, etc.
Sunitha (2001) introduced and discussed some properties of complete fuzzy graphs. The same technique as in crisp complete graph, is used. After that, Parvathi and Karunambigai (2006) extended the concept of fuzzy graphs to introduce intuitionistic fuzzy graphs. Here, membership values and nonmembership values are associated. Nagorgani and Radha (2012) defined regular fuzzy graphs in which the degree of vertices are assumed as same as in crisp graphs. Akram (2011) introduced interval valued fuzzy graphs. In that paper, vertex and edge membership values are taken as intervals but the same technique applied for edge restriction as in fuzzy graphs. After that, Akram and Dudek (2011) introduced bipolar fuzzy graphs. Here, positive and negative membership values of vertices and edges are taken. Kittur (2012) and Radha and Kumaravel (2014) described some important properties of complete and regular fuzzy graphs.
In all these fuzzy graphs, there is a common property that edge membership value is less than to the minimum of it’s end vertex membership values. Suppose, a social network is to be represented as fuzzy graphs. Here, all social units are taken as fuzzy nodes. The membership values of the vertices may depend on several parameters. Suppose, the membership values are measured according to the sources of knowledge and the relation between those units is represented by fuzzy edges. Thus, the membership value is measured according to the transfer of knowledge. But, transfer of knowledge may be greater than one of the social actors/units as more knowledgeable person informs less knowledgeable person. But, this concept cannot be represented in fuzzy graphs as edge membership value should be less than membership values of end vertices. Thus, all images/networks cannot be represented by fuzzy graphs. To remove the restriction, generalized fuzzy graphs are introduced here.
Matrix representation is another way of representations. Different authors established different properties of fuzzy graph matrices. Recently, Khansamy and Thangaraj (2015) discussed about some properties of matrices of fuzzy graphs. In this research, generalized fuzzy graphs are represented by appropriate matrices. Sunitha (2001) discussed different major properties of fuzzy graphs in her research work. She described fuzzy complete graphs, regular graphs and many more. Kittur (2012) provided some properties on complete fuzzy graphs. Nagorgani and Radha (2012), Nagorgani and Latha (2008) and Radha and Kumaravel (2014) described some of the properties of regularity and irregularity of fuzzy graphs. These studies are regular extension of crisp graphs. In this paper, regularity is defined in more generalized way.
After introductory section, generalized fuzzy graphs of type 1 and type 2 (GFG1, GFG2) are described with suitable examples. After that, GFG1 and GFG2 are represented by matrices. Complete GFG1, GFG2 are introduced. Then, regular and effective regular GFG1, GFG2 are introduced and several properties are established. At last, conclusions are given.
Problem definition of this work
The direction of this work is to generalize the fuzzy graphs by removing the edge restriction. The relation between vertices and edges are to be established. Two properties, completeness and regularity, are to be discussed.
Generalized fuzzy graphs
A fuzzy graph \(\xi =(V,\sigma ,\mu )\) is a nonvoid set V with a pair of functions \(\sigma :V\rightarrow [0, 1]\) and \(\mu :V \times V \rightarrow [0,1]\) such that for each \(x, y\in V\), \(\mu (x,y) \le \sigma (x)\wedge \sigma (y)\), where \(\sigma (x)\) and \(\mu (x,y)\) represent the membership values of the vertex x and edge (x, y) in \(\xi\) respectively.
Authors contributions towards generalized fuzzy graphs
Authors  Year  Contributions 

Kauffman (1973)  1973  Introduction of fuzzy graphs 
Rosenfeld (1975)  1975  Modification of the concept of fuzzy graphs given by Kauffman (1973). He added that edge membership value is less than minimum of vertex membership values 
Sunitha (2001)  2001  Introduction and discussion of properties of complete fuzzy graphs 
Parvathi and Karunambigai (2006)  2006  Introduction of intuitionistic fuzzy graphs 
Nagorgani and Radha (2012)  2008  Introduction of regular fuzzy graphs 
Akram (2011)  2011  Introduction of interval valued fuzzy graphs 
Akram and Dudek (2011)  2011  Introduction of bipolar fuzzy graphs 
Kittur (2012)  2012  Some properties of complete fuzzy graphs 
Radha and Kumaravel (2014)  2014  Introduction of edge regular fuzzy graphs 
Khansamy and Thangaraj (2015)  2015  Introduction of vertexedge matrix of fuzzy graphs 
This paper  –  Introduction of generalized fuzzy graphs Matrix representation of generalized fuzzy graphs Introduction of regular and complete generalized fuzzy graphs 
Here, two types of relations are considered. In the following, generalized fuzzy graph of first kind is defined. Here, vertex membership values are considered first. Then, depending on vertex membership values, edge membership values are considered.
Definition 1
Let V be a nonvoid set. Two functions are considered as follows: \(\rho :V\rightarrow [0,1]\) and \(\omega : V\times V\rightarrow [0,1]\). We suppose \(A=\{(\rho (x),\rho (y))\omega (x,y)> 0\}\). The triad \((V,\rho ,\omega )\) is defined to be generalized fuzzy graph of first kind (GFG1) if there exists a function \(\phi :A\rightarrow (0,1]\) such that \(\omega (x,y)=\phi ((\rho (x),\rho (y)))\) where \(x,y\in V\). Here \(\rho (x),x\in V\) is the membership value of the vertex x and \(\omega (x,y),x,y\in V\) is the membership value of the edge (x, y).
Example 1
Now, generalized fuzzy graphs of second kind is defined. Here, the membership values of edges are considered first. Then, depending on edge membership values, vertex membership values of vertices are assigned.
Definition 2
Let V be a nonvoid set. Two functions are considered as follows: \(\rho :V\rightarrow [0,1]\) and \(\omega : V\times V\rightarrow [0,1]\) and let B be the range set of \(\omega\). The triad \((V,\rho ,\omega )\) is defined to be generalized fuzzy graph of second kind (GFG2) if there exists a function \(\psi :B\rightarrow (0,1]\) such that for every \(x\in V\), \(\rho (x)=\psi (\omega (e_x))\), where \(e_x=(x,y)\) such that \(y\in V\). Here, \(\rho (x),x\in V\) is the membership values of the vertex x and \(\omega (x,y)\) is the generalized membership value of the edge (x, y).
Note 1
In GFG2, the codomain set of \(\psi\) excludes the number 0, as the membership values of vertices are always positive.
Example 2
Let us consider a generalized fuzzy graph of second kind (GFG2) shown in Fig. 2. Here, \(V=\{a,b,c,d,e\}\) be a nonvoid set and {((a, b), 0.4), ((a, c), 0.5), ((a, e), 0.6), ((b, c), 0.8), ((b, e), 0.3), ((c, e), 0.7), ((d, e), 0.7), ((c, e), 1)} is the fuzzy edge set. Also let, \(\rho (x)=\frac{\sum _{y\in V}\omega (x,y)}{n}\) in which n is the whole number of existing edges adjacent to x. Now, \(\rho (a)=\frac{0.4+0.5+0.6}{3}=0.5\). The vertex set is {a(0.5), b(0.5), c(0.75), d(0.85), e(0.575)}.
Matrix representation of generalized fuzzy graphs
Matrix representation of GFG1
\(\phi\)  \(v_1(\rho (v_1))\)  \(v_2(\rho (v_2))\)  \(\ldots\)  \(v_n(\rho (v_n))\) 

\(v_1(\rho (v_1))\)  \(\rho (v_1)\)  \(\phi (\rho (v_1),\rho (v_2))\)  \(\ldots\)  \(\phi (\rho (v_1),\rho (v_n))\) 
\(v_2(\rho (v_2))\)  \(\phi (\rho (v_2),\rho (v_1))\)  \(\rho (v_2)\)  \(\ldots\)  \(\phi (\rho (v_2),\rho (v_n))\) 
\(\ldots\)  \(\ldots\)  \(\ldots\)  \(\ldots\)  \(\ldots\) 
\(v_n(\rho (v_n))\)  \(\phi (\rho (v_n),\rho (v_1))\)  \(\phi (\rho (v_n),\rho (v_2))\)  \(\ldots\)  \(\rho (v_n)\) 
Matrix representation of GFG1
Note 2
 1.
The total information of GFG1 can be interpreted perfectly from the matrix representation.
 2.
The function \(\phi\) is put in (1, 1)position. Entries of the matrix are calculated from the function \(\phi\) and vertex membership values which are put in 1st row and 1st column.
 3.
As matrices are considered for undirected graphs, edges (u, v) and (v, u) are same and hence the matrices are symmetric.
 4.
Number of rows = number of columns = V.
 5.
If a row (column) has all its entries to be zero then that vertex is an isolated vertex.
Example 3
An example of matrix representation of Fig. 2
\(\phi (x,y)=\max \{x,y\}\)  x(0.5)  y(0.9)  z(0.3)  t(0.8) 

x(0.5)  0.5  0.9  0.5  0.8 
y(0.9)  0.9  0.9  0  0.9 
z(0.3)  0.5  0  0.3  0 
t(0.8)  0.8  0.9  0  0.8 
Theorem 1
Let \(M_{G_1}\) be the matrix representation of GFG1. Then \(D(x_k)=\sum _{j=1,j\ne k}^{n}a(k+1,j+1)\), \(x_k\in V\) or \(D(x_p)=\sum _{i=1,i\ne p}^{n}a(i+1,p+1)\), \(x_p\in V\).
Proof
Matrix representation of GFG2
Note 3
 1.
The total information of GFG2 can be interpreted perfectly from the matrix representation.
 2.
The function \(\psi\) is put in (1, 1)position. Entities of last column of the matrix are calculated from the function \(\psi\) and edge membership values which are put in corresponding positions of 1st row.
 3.
If a row (column) has all its entries to be zero, then that vertex is an isolated vertex.
Example 4
Matrix representation of GFG2
\(\psi\)  \(e_1(\omega (e_1))\)  \(e_2(\omega (e_2))\)  \(\ldots\)  \(e_m(\omega (e_m))\)  \(\rho (v)\) 

\(v_1\)  0  1  …  0  \(\psi (\omega (e_1))\) 
\(v_2\)  1  1  …  1  \(\psi (\omega (e_1),\omega (e_2),\ldots ,\omega (e_1))\) 
…  …  …  …  …  … 
\(v_n\)  1  0  …  1  \(\psi (\omega (e_1),\omega (e_m))\) 
First row of the matrix indicates the existence of vertices among the edges. Here, the vertex ‘a’ is incident to the edges \(e_1,e_2,e_3\). Hence, ‘1’ is put in the corresponding columns and ‘0’ in the remaining columns. The membership value of the vertices will be calculated from the function \(\psi (x_1,x_2,\ldots ,x_k)=\frac{\sum _{i=1}^kx_i}{k}\), where \(x_i\), \(i=1,2,\ldots ,k\) are the edge membership values.
Theorem 2
Let \(M_{G_2}\) be the \((n+1)\) by \((m+2)\) matrix of GFG2, \(\xi\) . Also let, \(i+1\) th row has ‘1’ as entries in \(a(i+1,m_1),a(i+1,m_2),\ldots , a(i+1,m_p)\) positions where \(i=1,2,\ldots , n\) . Then, \(a(i+1,m+2)=\phi (e_{m_1},e_{m_2},\ldots ,e_{m_p})\).
Proof
Complete generalized fuzzy graphs
In general, if the membership value of an edge is greater than half of its maximum of membership values of its end vertices, the edge is called to be an effective edge. Suppose, someone informs another person about any news. If the transfer of knowledge is greater than to a certain amount (may be assumed half of the source knowledge), then the second person can be informed effectively. Thus, transfer of knowledge (which indicates the membership value of an edge) helps a person (a vertex with lower membership value) to be informed from a source (a vertex with greater membership value). Definition of an effective edge is given below.
Definition 3
Let \(\xi =(V,\rho ,\omega )\) be a GFG1 (or GFG2). An edge (x, y) is defined to be effective edge if \(\omega (x,y)\ge \frac{1}{2}\max \{\rho (x),\rho (y)\}\). A generalised fuzzy graph GFG1 (or GFG2) is defined to be effective, if for all \(x,y\in V\), \(\omega (x,y)\ge \frac{1}{2}\max \{\rho (x),\rho (y)\}\).
Note 4
It is obvious that for GFG1, \(\omega (x,y)=\phi (x,y)\). Now, if an edge of a generalized fuzzy graph is effective, then others edge may not be effective. The following is the analytic description of this statement. Let \(\xi\) be a GFG1 and it has vertex set \(\{a(0.8),b(0.2),c(0.3)\}\) and edge set \(\{(a,b),(b,c)\}\) and \(\phi (x,y)=\min \{x,y\}\). Then, it can be found \(\phi (a,b)=0.2,\phi (b,c)=0.2\). The edge (b, c) is effective as \(\phi (b,c)=0.2>\frac{1}{2}max\{b,c\}=1.5\). Thus, the edge (a, b) is not effective.
A simple graph G is defined to be complete if every vertex in G is connected with every other vertex, i.e., if G contains only one edge between each pair of distinct vertices. Now, we have to check the completeness of generalized fuzzy graphs. Now, it is easy to consider that an edge is called to be noneffective if it is not effective. In that case, the edge may be ignored for some representation. Thus to consider that a generalized fuzzy graph is complete, effectiveness of edges are important. The definition of complete generalized fuzzy graph is defined below.
Definition 4
Let \(\xi =(V,\rho ,\omega )\) be a GFG1 (or GFG2). The graph \(\xi\) is defined to be complete if all pairs of vertices are connected by effective edges. Otherwise, the graph is defined to be incomplete generalized fuzzy graph.
Example 5
Note 5
If we change the function \(\phi\) of Fig. 4 and redefine the function \(\phi (x,y)=xy\), then the graph is not complete as some of its edges are noneffective (see Fig. 4b).
Theorem 3
Let \(\xi =(V,\rho ,\omega )\) be a complete GFG1 or GFG2. Then \(D(x)\ge \frac{V1}{2}\rho (a)\) for all \(x\in V\).
Proof
Here, \(\xi =(V,\rho ,\omega )\) be a complete GFG1 or GFG2. From the definition of complete GFG1 or GFG2, every vertex is connected with all remaining vertices, i.e. \((V1)\) vertices by effective edges. Now, effective edges have a property that \(2\omega (x,y)\ge \max \{\rho (x),\rho (y)\}\). Hence, \(D(x)=\) sum of membership values of all adjacent edges of x which is obviously greater than \(\frac{V1}{2}\rho (a)\). Hence, it proves Theorem 3. \(\square\)
Regularity of generalized fuzzy graphs
An example of matrix of Fig. 3
\(\psi\)  \(e_1(0.6)\)  \(e_2(0.4)\)  \(e_3(0.5)\)  \(e_4(0.8)\)  \(e_5(1)\)  \(e_6(0.7)\)  \(e_7(0.3)\)  \(e_8(0.7)\)  \(\rho (x)\) 

a  1  1  1  0  0  0  0  0  0.5 
b  0  1  0  1  0  0  1  0  0.5 
c  0  0  1  1  1  0  0  1  0.75 
d  0  0  0  0  1  1  0  0  0.85 
e  1  0  0  0  0  1  1  1  0.575 
Definition 5
Let \(\xi =(V,\rho ,\omega )\) be a GFG1 (or GFG2). Now, degree of a vertex x in \(\xi\) is denoted as D(x) and is defined as \(D(x)=\sum _{y\in V}\omega (x,y))\). Also, let \(k=\frac{\sum _{x\in V}{D(x)}}{n}\), where n is the whole number of existing elements of V. Now \(\xi\) is defined to be regular if for every \(x\in V\), \(D(x)k\le \epsilon\), where \(\epsilon\) is very small number and \(0\le \epsilon \le 1\).
In this study, \(\epsilon\) is assumed as the upper bound of \(D(x)k\). This value may vary for different networks and it is decided by decision makers. This graph can be called \(\epsilon\)regular generalized fuzzy graphs.
Note 6
If \(\epsilon =0\), then \(D(x)=k\) for all \(x\in V\). Hence, it is called regular GFGs i.e. 0regular GFGs are regular GFGs.
Example 6
Theorem 4
Let \(\xi\) be \(\epsilon\) regular GFG1. Also, let the corresponding crisp graph be a cycle. Suppose, the length of the crisp cycle is odd. Then, \(\sup A\inf A\le 2\epsilon\) , where \(A=\{\phi (x,y): x,y\in V\}\).
Proof
Note 7
The converse of the Theorem 4 is not necessarily true. If the condition of edge of any cycle is true, the cycle need not be regular.
Theorem 5
Let \(\xi\) be \(\epsilon\) regular GFG1. Also, let the corresponding crisp graph be a cycle. Suppose the length of the cycle is even. Then, \(\sup A\inf A\le 2\epsilon\) where \(A=\{\phi (x,y): x,y\in V\}\) or \(\sup B\inf B\le 2\epsilon\) , where \(B=\{\phi (e)\) : for all alternating edges e of the cycle \(\xi \}\).
Proof
Let \(\xi =(V,\rho , \omega )\) be a \(\epsilon\)regular fuzzy graphs. Thus, \(D(x)k\le \epsilon\) for all \(x\in V\) and \(A=\{\phi (x,y): x,y\in V\}\). Now, let \(e_1,e_2,\ldots ,e_{2n}\) be the edges of the graph (see Fig. 7). As, the supremum and infimum are to be found out for this proof, the maximum/minimum value of the range set of \(\phi\) are considered without loss of generality. Let \(\phi (e_1)=k_1\) then, \(\phi (e_2)=kk_1\pm \epsilon\), \(\phi (e_3)=k(kk_1\pm \epsilon\) \(=k_1\pm \epsilon\), thus, it is found \(\phi (e_{2n})=kk_1\pm \epsilon\). If u is the connecting vertex of \(e_1\) and \(e_{2n}\), \(D(u)=\phi (e_1)+\phi (e_{2n})\) \(=k\pm \epsilon\). Thus, proceeding the concept of Theorem 4, it is found \(\sup A\inf A\le 2\epsilon\), where \(A=\{\phi (x,y): x,y\in V\}\). Besides, for odd edges maximum/minimum value of \(\phi (e)=k_1\pm \epsilon\) and for even edges \(\phi (e)=kk_1\pm \epsilon\). Hence, \(\sup B\inf B\le 2\epsilon\), where \(B=\{\phi (e): \text { for all alternating edges e of the cycle } \xi \}\). \(\square\)
Note 8
Effective edges have significant values in every systems. Thus, only these edges are counted for degree of a vertex and the definition of effective degree is given below (Fig. 8).
Definition 6
Let \(\xi =(V,\rho ,\omega )\) be a GFG1 (or GFG2) and let E be the set of all effective edges of \(\xi\). Now, effective degree of a vertex x in \(\xi\) is denoted as \(\mathcal {D}(x)\) and is defined as \(\mathcal {D}(x)=\sum _{(x,y)\in E,y\in V}\phi (x,y)\).
If all the vertices of a GFG1 or GFG2 are of same effective degree or almost same effective degree, the graph is defined to be effective regular graph. The definition is given in the following:
Definition 7
Let \(\xi =(V,\rho ,\omega )\) be a GFG1 (or GFG2) and let E be the set of all effective edges of \(\xi\) and \(k=\frac{\sum _{(x,y)\in E}{\mathcal {D}(x)}}{n}\) in which n is the whole number of elements of V. Now \(\xi\) is defined to be \(\epsilon\)effective regular if for all \(x\in V\), \(\mathcal {D}(x)k\le \epsilon\).
Example 7
Let us consider a GFG1, \(\xi\) with vertex set \(\{a(0.7),b(0.1),c(0.7),d(0.2)\}\) and \(\phi (x,y)=\max \{x^2,y^2\}\). Thus, the edge membership values of (a, b), (b, c), (c, d), (d, a), (b, d) are 0.49, 0.49, 0.49, 0.49, 0.04 respectively. Now by calculation, it can be observed that (b, d) is not effective. Thus, effective degrees of the vertices are same and equal to 0.98. Thus, the graph is \(\epsilon\)effective regular GFG1.
Definition 8
Let \(\xi =(V,\rho ,\omega )\) be a GFG1 or GFG2. \(\xi\) is defined to be pregular effective generalized fuzzy graph if every vertex of \(\xi\) is incident to exactly p number of effective edges.
Example 8
Theorem 6
Let \(\xi =(V,\rho ,\omega )\) be a pregular effective GFG1 or GFG2, then \(D(x)\ge \frac{p1}{2}\rho (x)\).
Proof
Proof of this theorem is obvious, keeping the reference of the proof of Theorem 3. \(\square\)
Theorem 7
Every complete generalized fuzzy graph \(\xi =(V,\rho ,\omega )\) (GFG1 or GFG2) is \((V1)\)regular effective generalized fuzzy graph.
Proof
Let \(\xi =(V,\rho ,\omega )\) be a complete GFG1 or GFG2. Now, every vertex of a complete GFG1 or GFG2 is adjacent to remaining \((V1)\) vertices. By the definition of peffective regular fuzzy graphs, it is easy to verify that \(\xi\) is \((V1)\)regular effective generalized fuzzy graph. \(\square\)
Insights of this study

Fuzzy graphs are generalized with removal of the edge restriction. Thus any kinds of networks can be represented by GFGs.

Vertex and edge relation of GFGs are established.

Matrix representation of GFGs are given. This is the easier way to represent any GFGs.

Two major properties of GFGs, completeness and regularity, are provided. Some important results are proved.
Conclusions
This study described some major properties of generalized fuzzy graphs. In the literature, adjacent matrices and incident matrices are available. Here, GFG1 was represented by matrices, which was similar to adjacent matrices of fuzzy graphs. But, the difference is that each element is determined from the function \(\phi\). Again, GFG2 was represented by matrices similar to incident matrices. These representations are helpful to understand the generalized fuzzy graphs in easier way. This study also introduced two properties namely, completeness and regularity of GFG1 or GFG2. Effective regular and peffective regular GFG1 or GFG2 were described. Some results regarding the definitions were established. In near future, eigenvalues of matrices and their properties will be established (Gholmy and Hawary 2016). This study will develop the theory of fuzzy graphs along with some important algorithms and networking problems (Oghlan et al. 2016).
Declarations
Authors' contributions
All authors contributed equally to every part of this study. All authors read and approved the final manuscript.
Acknowledgements
The authors are overwhelmed to the Editor in Chief and Honorable reviewers of the journal “Springer Plus” for their suggestions to improve the quality and representation of the paper. A special thanks to the Honourable H.O.D., Department of Applied Mathematics with Oceanology and Computer programming, Vidyasagar University, India for his major support during this research.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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