Unify Steiner Weiner Distance for Some Class of m-Polar Fuzzy Graphs

ABSTRACT Sometimes information in a network model is based on multi-agent, multi-attribute, multi-object, multi-polar information or uncertainty rather than a single bit. An m-polar fuzzy model is useful for such network models which gives more and more precision, flexibility, and comparability to the system as compared to the classical, fuzzy models. On the other, The Steiner tree problem in networks, and particularly in graphs, was formulated by Hakimi [25] and Levi [21] by definition minimal size connected tree sub graph that contains the vertices in S. Steiner trees have applications to multiprocessor computer networks. For example, it may be desired to connect a certain set of processors with a sub network that uses the fewest communication links. In this paper, we extend Steiner distance SWk (G) for m-polar fuzzy graphs and give this parameter for Join, composition and Cartesian product of two m-polar fuzzy graphs.


Introduction
A fuzzy set is an important mathematical structure to represent a collection of objects whose boundary is vague [1]. Fuzzy models are becoming useful because of their aim in reducing the differences between the traditional models used in engineering and science. Rosenfeld discussed the idea of fuzzy graph in 1975 [2]. Several concepts on fuzzy graphs were introduced by Mordeson [3]. Lately Ameri and his co-authors explained a new concepts of product interval-valued fuzzy graph [4]. Rashmanlou, Borzooei and their co-authors studied on the bipolar fuzzy graphs with categorical properties [5,6] and Talebi and Rashmanlou [7] studied the complement and isomorphism of bipolar fuzzy graphs. Juanjuan Chen [8] introduced the notion of the m-polar fuzzy set as a generalisation of bipolar fuzzy sets. Hayat and co-authors studied on bipolar Anti Fuzzy h-ideals in Hemi-rings [9]. Ghorai and Pal studied on some properties of m-polar fuzzy graphs as a bipolar fuzzy graphs [10,11]. Akram and Younas studied certain types of irregular m-polar fuzzy graphs in [12]. In 2016 Akram and Adeel studied on m-polar fuzzy labelling graphs [13] and Akram and Waseem introduced certain metrics in m-polar fuzzy graphs in [14]. Lately Prem Kumar studied on m-polar fuzzy graph representation of concept lattice [15][16][17][18]. On the other the Steiner distance of a set S of vertices in a connected graph G, is the number of edges in a smallest connected subgraph of G containing S, called a Steiner tree for S. If |S| = 2, then the Steiner distance of S is the distance between the two vertices of S. Steiner trees have applications to multiprocessor computer networks. For example, it may be desired to connect a certain set of processors with a subnetwork that uses the fewest communication links. A Steiner tree for the vertices that need to be connected corresponds to such a subnetwork. Lately, Gutman and co-authors introduced the concept of the Steiner Wiener index of a graph [19,20]. The Steiner k-Wiener index SW k (G) of G is defined by where the steiner distance d(S) of the vertices of S is the minimum size of a connected subgraph of G whose vertex set is S. In this paper, we expand this concept to m-polar fuzzy graphs and compute Steiner Weiner index for some class of m-polar fuzzy graphs.
Definition 1.1: [1] A fuzzy set μ in a universe X is a mapping σ : X → [0, 1]. A fuzzy relation on X is a fuzzy set μin X × X. Let σ be a fuzzy set in X and μ fuzzy relation on X. We call μ is a fuzzy relation on σ if μ(x, y) ≤ min{σ (x), σ (y)} ∀x, y ∈ X.

Definition 1.2: [2]
A fuzzy graph is a pair G : (σ , μ) where σ is a fuzzy subset of a set S and μ is a fuzzy relation on σ . We assume that S is finite and nonempty, μ is reflexive and symmetric.  [14] Let σ be an m-polar fuzzy subset of a non-empty set V. An m-polar fuzzy relation on σ is an m-polar fuzzy subset μ of V × V defined by the mapping μ: denotes the ith degree of membership of the vertex x and pi • μ(xy) denotes the ith degree of membership of the edge xy. An m-polar fuzzy graph was introduced by Chen et al. [8] and modified by Akram and Waseem [14]. Definition 1.5: [14] An m-polar fuzzy graph is a pair G = (σ , μ), where σ : V → [0, 1] m is an m-polar fuzzy set in V and μ : V × V → [0, 1] m is an m-polar fuzzy relation on V such that pi • μ(xy) ≤ inf{pi • σ (x), pi • σ (y)} for all x, y ∈ V × V note that pi • μ(xy) = 0 for all xy ∈ V × V − E for all i = 1, 2, 3, ..., m. σ is called the m-polar fuzzy vertex set of G and μ is called the m-polar fuzzy edge set of G, respectively. An m-polar fuzzy relation μ on V is called symmetric if pi • μ(xy) = pi • μ(yx) for all x, y ∈ V. Definition 1.6: [16] Let G be a connected m-polar fuzzy graph. The length of an m-polar fuzzy path P : v 1 , v 2 , . . . , v n in G, is denoted by l(P) and defined as l(P) = ((l 1 (P), l 2 (P), . . . , l m (P)) where l i (P) = . . , m. If n = 0, define l i (P) = 0 and for n ≥ 1, l i (P) > 0.
Also if G is disconnected then l i (P)may be zero. Let G be a connected m-polar fuzzy graph.
For m-polar fuzzy graph G = (σ , μ), we defined μ−Unify as follows: Definition 1.7: If G is a m-polar fuzzy graph with |e(G)| = m, then μ−Unify is as follow Then, the composition of the graph The join of two simple graphs G 1 * = (V 1 , E 1 ) and G 2 * = (V 2 , E 2 ) is the simple graph with the vertex set V 1 ∪ V 2 and edge set E 1 ∪ E 2 ∪ E , where E is the set of all edges joining the nodes of V 1 and V 2 and assume that V 1 ∩ V 2 = ∅. The join G 1 * and G 2 * is denoted by G * = Ghorai and Pal extension above definition for m-polar fuzzy graph. Definition 1.9: [19] The Cartesian product G 1 × G 2 of two m-polar fuzzy graphs Definition 1.10: respectively, is defined as follows: for i = 1,2, . . . ,m tively, is defined as follows: where E is the set of all of the edges joining the nodes of V 1 and V 2 assuming that S-Steiner fuzzy tree is a subgraph T(V , E )of G with S ⊆ V . Such that the Steiner distance d(S) is the minimum size of edges of tree T(V , E )among all connected subgraphs whose vertex sets contain S (S ⊆ V ). Observe that d(S) = min{|e(T)||S ⊆ V(T)} for fuzzy graph G = (σ , μ), we defined k-Unify Steiner Wiener distance as follows Example 1.1: An example the graph G in Figure 1, which is the molecular graph of 1, 1, 3-trimethyl-cyclobutane. Its vertices are labelled by u 1 , u 2 , u 3 , u 4 , u 5 , u 6 , u 7 . The sequence 4, 3, 5, 6, 7 is a path of G. In this graph the Steiner Weiner for k = 2 are computed:

Main Result
Mao, Wang and Gutman [18] introduced Steiner Weiner index for join, composition and product of two graphs. Now we expand this concept to m-polar fuzzy graphs.

Theorem 2.1: Let G be a connected m-polar fuzzy graph with n vertices and let H be a connected m-polar fuzzy graph with n vertices (n ≥ n ) and k be an integer with
Then the k-Unify Steiner Weiner distance for join of two m-polar fuzzy graphs is as follow (2) If k ≤ n then where x kG is the number of the k-subsets of V(G), such that the fuzzy subgraph induced by each k-subset is connected, and x kH is the number of the k-subsets of V(H) such that the subgraph induced by each k-subset is connected.
is connected then G[S] contains nSP(G[S]) spanning fuzzy trees. If x kG is the number of k-subset of V(G) such that the fuzzy subgraph induced by each k-subset is connected, then this contribution is (k − 1)x kG μ UG , and if G[S] is not connected then spanning fuzzy tree has more than k edges. Assume that set = {g 1 , g 2 , . . . , g k }, clearly the fuzzy tree induced by the edges in {g i , h|1 ≤ i ≤ k} is one of the spanning fuzzy tree and has less than k edges, So d G+H (S) = k. Since x kG is the number of the k-subsets of V(G) such that the fuzzy subgraph induced by each k-subset is connected, it follows that this contri- For other case suppose that S ∩ V(G) = ∅ and ∩V(H) = ∅, clearly spanning fuzzy tree has k-1 edges, then this contribution is From the above argument, we get (3) Since n ≤ k ≤ n, it follows that for any , similarly to proof of 2, in this case the contribution is To consider S ∩ V(G) = ∅ and S ∩ V(H) = ∅, the spanning fuzzy tree has k-1 edge, then the portion is From the above argument it follows

Theorem 2.2: Let G be a connected m-polar fuzzy graph with n vertices, and let H be a connected m-polar fuzzy graph with n vertices. Let k be an integer with 2 ≤ k ≤ nn . Then the bounds of k-Unify Steiner Wiener distance for Cartesian product of two m-polar fuzzy graphs is
Proof: For any S ⊆ V(G × H) and |S| = k, all the vertices in S belong to some copies of G and some copies of H. Without loss of generality, let H(g 1 ), H(g 2 ), . . . , H(g l ) be all the copies of H such that S ∩ V(H(g i )) = ∅ (1 ≤ i ≤ l) and S ∩ V(H(g i )) = ∅, l + 1 ≤ i ≤ n) and letG(h 1 ), G(h 2 ), . . . , G(h l ) be all the copies of G such that ∩V(G(h j )) = ∅ (1 ≤ j ≤ l ) and On the other suppose that l ≤ k − 1. For each i (1 ≤ i ≤ l), there is an Steiner tree in H(g i ), say T i . Similarly, there is an Steiner tree in G(h 1 ), say T. Then the tree induced by the is an Steiner tree in ×H, and thus Suppose that l = k and l = k, all the vertices in S = {(g 1 , h 1 ), (g 2 , h 2 ), . . . , (g k , h k )} belong to different copies of G and different copies of H. For each i (1 ≤ i ≤ l − 1), there is an Steiner tree in H(g i ), say T i , Similarly, there is an Steiner tree in G(h 1 ), say T. Then the tree induced by the edges in E( is an Steiner tree in G × H, and therefore In the other case suppose that l = k and l ≤ k − 1. Since l = k it follows that each H(g i ) contains exactly one vertex of S. Since l ≤ k − 1. It follows that there exists some G(h i ) such that G(h i ) contains at least two vertices of S. For each i (1≤ i ≤ l − 1), there is an Steiner tree in H(g i ), say T i .
Similarly there is an Steiner tree in G(h 1 ), say T. Then the tree induced by the edges in is an Steiner tree in G × H, and thus Since there are Combining these results, we have and similarly, From now on, we assume that the vertices in S belong to at least two copies of H in G [H]. Without loss of generality, we assume that H(g 1 ), H(g 2 ), . . . , H(g {h 1 , h 2 , . . . , h l }). Since in G there exists a Steiner tree connecting {g 1 , g 2 , . . . , g l } of size d G ({g 1 , g 2

Application
Fuzzy graphs of the 1-polar type are nothing more than the most familiar fuzzy graphs and have many applications for cluster analysis and solving fuzzy intersection equations, database theory, problems concerning group structure, and so on. The further possible applications of m-polar fuzzy graphs in real world problems can be viewed in the case of bipolar fuzzy graphs, i.e. 2 polar fuzzy graphs. Bipolar fuzzy graphs have many applications in social networks, engineering, computer science, database theory, expert systems, neural networks, artificial intelligence, signal processing, pattern recognition, robotics, computer networks, and medical diagnosis and so on. Additionally, m-polar fuzzy graphs (m > 2) are very useful in many decision making situations. [22] Nowadays fuzzy sets are playing a substantial role in chemistry. For k = 2, the above defined Steiner Wiener index coincides with the ordinary Wiener index W(G) = u,v∈V (G) d (u, v). [23,24] We define one of the applications of Steiner Wiener index for k = 2 by definition of Wiener index. Harold Wiener reported the existence of correlations between the new index and a large number of physico-chemical  properties of alkanes. Wiener defined W only for alkans as the number of carbon carbon bonds between all pairs of carbon atoms. There is a relation between W and the distances in the molecular graph. In particular, pointed out that W is equal to the half of the sum of all elements of the distance matrix of the respective molecular graph. By this, the concept of Wiener number could be extended to cyclic molecules also. The applications of graphs in chemistry are because there are relation between structural formula and a graph. The first application of the Wiener number was for predicting the boiling points of alkanes on the formula: b:p = αw + βw (3) where, α and β are empirical constants and w(3) is the so called path number, namely, the number of pairs of vertices whose distance is equal to 3. Wiener index used to estimate boiling points, molar volumes, refractive indices, heats of isomerisation and heats of vaporisation of alkanes. In Figure 3 and Table 1, it was shown some relation between Steiner wiener index for k = 2 (Wiener index) and chemical parameters. The Wiener number found several noteworthy applications in polymer chemistry. The melting points and other physical properties of polymers were predicted on the basis of their W -values. The average electron energies and the energy gaps in conjugated polymers were also shown to depend on W. Of the newest applications of the Wiener number, we may mention its use in the rationalisation of the mechanism of electro education of chlorobenzene derivatives and for distinguishing between fullerene isomers, as well as its role in the recent approaches towards the quantitation of molecular similarity. The fact that W is correlated with so many physico-chemical properties of non-polar organic substances leads to the conclusion that it must be a rough measure of the intermolecular forces. Curiously, however, in spite of the extensive research on the Wiener number in the last years, this fundamental feature was directly tested only quite recently.

Conclusions
Graph theory is an extremely useful tool for solving combinatorial problems in different areas, including algebra, number theory, geometry, topology, operations research, optimisation and computer science. Because research on or modelling of real world problems often involve multi-agent, multi-attribute, multi-object, multi-index, multi-polar information, uncertainty, and/or process limits, m-polar fuzzy graphs are very useful. The m-polar fuzzy models give increasing precision, flexibility, and comparability to the system compared to the classical, fuzzy and bipolar fuzzy models. On the other hand, the recent synthesis of macromolecules with highly branched skeletons (e.g. dendrimers), increase the need for estimating Wiener index via pertinent approximate formulae. So given that for k = 2, the above defined Steiner Wiener index coincides with the ordinary Wiener index, therefore, in this paper, we introduced and studied k-Unify Steiner Weiner for m-polar fuzzy graphs and obtained this title for Join, Composition and Cartesian Product of two m-polar fuzzy graphs that can be a molecule graph. We plan to extend our research work on m-polar fuzzy graphs and its applications in support system ( Figure 4).

Disclosure statement
No potential conflict of interest was reported by the author(s).

Notes on contributors
Asefeh Karbasioun is a Ph.D student in Department of Mathe mathematics, Payame Noor University, Tehran Iran. She is working in fuzzy graphs and its Applications.
Reza Ameri is a professor in Department of Mathematics, School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, Iran. He received his M.S. in 1993 in Fuzzy Algebra, and