An Investigation on M-Polar Fuzzy Saturation Graph and Its Application

The saturation graph is a well-deﬁned topic. But, saturation in a fuzzy graph was deﬁned recently and investigated many properties. In a fuzzy saturation graph, only one saturation is considered for every vertex. In a m -polar fuzzy graph ( m PFG), each vertex and edge has a m number of membership values. So, deﬁning saturation for m PFG is not easy and needs some new ideas. By considering m saturation for each component out of m components of membership values of a vertex, we deﬁned saturation in m PFG. α -saturation as well as β -saturation in m PFG is introduced here. Many interesting properties of it are also presented. α -vertex count and β vertex count in m PFG are also studied and the upper bound on some well known m PFG is also found here. Finally, a real-life application using saturation in m PFG is also presented.


Research background and Related works
sets. Next, Chen et al. [4] presented the thought of fuzzy sets in m-dimension as speculation of bipolar fuzzy sets. First, Kafmann [10] presented the fuzzy graph concept utilizing Zadeh's fuzzy relation. After that Rosenfeld [20] supplied the possibility of nodes, edges along with several hypothetical ideas like paths, connectedness, cycle, etc., in fuzziness. Different concepts and definitions are presented thereafter on fuzzy graphs [15,17,21]. Nair and Cheng presented fuzzy cliques in fuzzy graphs [18]. Mathew et al. also worked on different properties on fuzzy graphs [15,17,21]. Chen et al. [4] first presented a m-polar fuzzy graph (mPFG). Later on, Ghorai and Pal discussed several properties of mPFG [6,7,8,9,19]. Next, Mandal et al. studied different types of arcs on mPFG [13]. Akram and Adeel have also deeded on mPFGs and line graphs [1]. Akram et al. concentrated a few properties of edge on mPFG [2]. Next, Mahapatra and Pal presented fuzzy colouring of mPFG [11] and recently, Mahapatra et al. studied fuzzy fractional colouring on fuzzy graph [12].

Framework of this study
This paper is structured as follows: Section 2 describe some definitions which are useful in these manuscripts. In section 3, we have discussed the definitions of the strong vertex as well as SE count, α-vertex as well as α-edge count, β-vertex as well as β-edge count of mPFG and give the lower and upper bound of them in an mPFG. In section 3.1, we investigate vertex as well as edge counts of some well-known mPFG. In section 4, we introduced saturation in mPFG with the help of α-saturation and β-saturation. Section 5 describe algorithms to find α-saturated as well as β-saturated and saturation in mPFG. A real-life application based on the allocation problem has been solved using saturation in mPFG, given in Section 6. Lastly, the conclusion has been given in Sections 7.

Notations and symbols
In this portion, we reform some of the significant as well as useful notations which are used in the whole paper for development of the theories. The abbreviation form and their meanings are given in Table 1 2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 Here, we briefly call again some basic as well as useful definitions of graphs, mPFG and related terms connecting with it. Suppose G =( V, E) be a graph, where V (non-null set) is indicated as a node-set as well as E is indicated as edge-set. If a node is separate from all edges, then the node is said to be an isolated node. Otherwise, it is called a non-isolated node.
is a fuzzy set in V as well as f V 2 respectively and which obey the following rule . σ(a) as well as µ(a, d) indicates the node a and edge (a, d) MV.
Throughout this article [0, 1] m will be a partial order set having component wise order  , m 2 N and "  " stands for  Clearly, supp(A)=φ iff A = φ and supp(A) 6 = φ iff A 6 = φ. Therefore, A and supp(A) are equivalent in mPFS.
represents an mPFS of V as well asV ⇥ V respectively and which obey the rule such that 8i =1, 2, 3,...,m, Then G as well as G 0 are called isomorphic. We write it as G ⇠ = G 0 .
Definition 2.10. [13] Suppose G =(V, σ, µ) be an mPFG as well as P : a 1 ,a 2 ,...,a k be a path in G. Then S(P ) denotes the strength of the path P which is given by S(P )=( m i n 1i<jk p 1 µ(a i ,a j ), min 1i<jk p 2 µ(a i ,a j ),..., min 1i<jk p m µ(a i ,a j )) = (µ n 1 (a i ,a j ),µ n 2 (a i ,a j ),...,µ n m (a i ,a j )). The SC of the path in between a 1 and a k is given in the following way: In this article G ⇤ =(V, E) stands for the UCG of an mPFG G.

Vertex as well as edge saturation counts of mPFG
In this section, we introduced vertex as well as edge saturation counts in mPFG and discussed various useful properties of them. Vertex saturation count of an mPFG gives the dimension of the mean strong degree of mPFG and edge saturation count indicates the portion of SEs of mPFG. Here, we consider σ(u)=1 =(1, 1,...,1), for all u 2 V ,w h e r eG =(V, σ, µ).
Definition 3.2. Suppose G =( V, σ, µ) be an mPFG having UCG G ⇤ =( V, E). Then α-vertex count of G is indicated by α V (G) as well as given by 8i =1, 2,...,m and α-SE count of G is indicated by α E (G) as well as given by as well as given by ..,m and the β-SE count of G is indicated by β E (G) and given by

Vertex and edge counts of some well-known mPFG
In this portion, we talk over saturation counts of mPFG structures like mPF cycles, trees , and blocks in mPFG. Some necessary parts for these structures are also obtained.
Proof. Suppose G is an mPF tree. Then G and F are isomorphic. Therefore, Now, we consider another case. Suppose, G contains a cycle, say C. Then, it is not free from δ-SE. Let e be a δ-SE . If G e is a tree therefore G e as well as F are isomorphic. Therefore, If G e is not a tree, deleting the δ-SEs in G e in similar manner to obtain a MST F of G such that p i α V (G)=p i α V (F ), for each i =1, 2,...,m.

Saturation in m-polar fuzzy graph
Here, saturation in terms of a node as well as edge counts is presented. In this section, we also studied some of the interesting facts of it. We also studied saturated blocks in mPFG.     3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62 63 64 To show G 0 is saturated, we have to show that each node is connected with at least one α-SE as well as β-SEs. Let w 0 2 V 0 .T h e nt h e r em u s tb ean o d e ,s a yw,i nG for which φ(w)=w 0 . Since G is saturated, therefore w is an incident with at least one α-SEs as well as β-SEs. Since, G and G 0 are isomorphic to each other. Therefore, w 0 is also incident to at least one α-SEs and β-SEs. Hence, G 0 is also saturated.
Let G be an mPFG having UCG G ⇤ =( V, E)w h e r e|V | = k.W ed e fi n eafi n i t es e q u e n c e α S (G)=(n 1 ,n 2 ,...,n k ) called α-strong sequence where, n j = count of α-SEs connect at node v j .W ed e fi n eafi n i t es e q u e n c eβ S (G)=( n 1 ,n 2 ,...,n k ) called β-strong sequence where, n j = count of β-SEs connect at node v j . Since, the count of SEs of G = (the count of α-SEs of G + the count of β-SEs of G) therefore, Proof. Suppose G be an α-saturated mPFG. Therefore, at least one α-SEs incident with each vertex of G.T h u s Therefore, G is α-saturated mPFG.
Proof. Similar to the above theorem.
In the Fig. 4, we see that all the edges having membership value (0.7, 0.7, 0.7) are α-strong and the edges having membership value (0.5, 0.5, 0.5) are β-strong. Therefore, Fig. 4 is unsaturated as the vertex a 1 connected with both the β-SEs.
One simple observation of the above discussion is that Fig. 3 has an even number of vertices while in Fig. 4 has an odd number of vertices. Thus we have the following theorem.
Theorem 4.6. Suppose C n be an mPF cycle. It is saturated iff the following two hold: (i) n =2t, t is a positive integer.
(ii) α-SE as well as β-SEs occur alternatively on C n .
Proof. Suppose C n is an mPF cycle. Therefore, it is free from δ-SEs. All arcs that occur on C n are α-SE or β-SE. Let us assume that C n be saturated. Therefore, each node is connected with at least one α-SE and one β-SEs. Hence, count of α-SEs = t = count of β-SEs. Therefore, n =2 t. Again, every node connected with both α-SE as well as β-SEs happen if they occur alternatively on C n .
Conversely, let C n is a fuzzy cycle with an even number of nodes in which each node is connected with both α-SE as well as β-SEs alternatively. Therefore, each node is connected with precisely one α-SE as well as β-SEs. Hence, C n be a saturated fuzzy cycle.
Theorem 4.7. Suppose G =(V, σ, µ) be an mPF cycle. If G is saturated, it must be a block.
Theorem 4.8. Let G =(V, σ, µ) is mPF cycle. If G be an mPF blocks, then it is either saturated or β-saturated.
Proof. Let a block be G. We demand that G is free from δ-SEs. If possible, let e be a δ-SE. Then the remaining edges must be α-SE, and therefore G contains n 2 fuzzy cut nodes, an irrelevance. So, G has no δ-SEs. Thus, G is free from δ-SEs.
If G has only α-SE as well as β-SEs, they appear alternatively; else, the block shape will not be found. If count of α-SE = count of β-SEs = n 2 ,t h e nG be α-saturated as well as β-saturated and therefore it is saturated. If the count of α-SEs is less than the count of β-SEs, then G must be only β-saturated. For another case, when the count of α-SEs is greater than the count of β-SEs, it will not be true as it does not form a block. If every arc is β-strong of G,i tm u s tb e β-saturated. Therefore, the theorem.
Conversely, suppose (a, c) is an α-SE. Then, we have (a, c) is the one and only one strongest path in between a and c and removal of (a, c) will decrease the SC of a and b. Therefore, (a, c) is a bridge.
Theorem 4.11. A complete mPFG has at most one α-SEs.
Proof. We know that complete mPFG have at most one mPF bridge. Again, from Theorem 4.10, we have an arc (a, b) be an mPF bridge iff it is α-SE. Hence, a complete mPFG has at most one α-SEs.
Proposition 3. Every complete mPFG has at most n 2 or n 2 -1 β-SEs. Theorem 4.12. If G is a complete mPFG having n vertices, the following inequalities hold.

Application
The mPFG is an essential mathematical structure representing the facts in real-life connected through graphical systems, in which nodes and edges lie in an m-polar fuzzy information. In this section, by using saturation in mPFG, we solve one particular allocation problem.

Model construction
In modern days, education is an essential topic for every person. Under the rule of the Right to Education (RTE) in 2005, everybody has the opportunity to read and write. In the education system, IIT(Indian Institute of Technology) is one of the most important institutions for higher studies in India. Therefore, establishing an IIT in a town among some towns is not an easy task for any Government.

Decision making
Since a 6 is the only saturation node in the model 3PFG G, we can say that the town a 6 is the most suitable place to establish the IIT (Indian Institute of Technology) among all other towns considered in this proposed model.
We know that saturation in mPFG plays an essential role in this type of allocation problem through the above discussion. Moreover, we also recognize that saturation in mPFG is more applicable than saturation in FG in allocation problems.