Connectivity Index of M-polar Fuzzy Graph with Application

This paper brings in connectivity index of a m -polar fuzzy graph ( m PFG) with its boundedness. We investigate how the connectivity index of a m PFG changes when a vertex or an edge is deleted. Some special types of vertices include m -polar fuzzy connectivity enhancing node( m PFCEN), m -polar fuzzy connectivity neutral node ( m PFCNN), m -polar fuzzy connectivity reducing node ( m PFCRN) with their properties are discussed.


Introduction
Graph theory plays a major part, to connect some values with respect to some parameters by the use of an appropriate connection in many disciplines like IT, operation research,the network of wireless sensors, network routing, engineering and medical science etc. In 1965, Zadeh [26] replace the classical set by fuzzy set which gives better exactness in both theory and application. In 1975, Rosenfeld [21] initiate the concept of fuzzy graph and in various field it has manifold application. In 1994, Zhang [27] established the notion of bipolar fuzzy sets. Pal et al. [16] introduced more new concept on fuzzy graph on their book. In various connective network fields the bipolar fuzzy graph offers more exact results if there are a positive thinking and negative thinking side, such as: effect and side effect, forward and backward, cooperation and competition, loss and gain etc. The concepts of m-polar fuzzy (mPF) sets was established by Chen et al. [1] in 2014. Based on this concept, Ghorai and Pal [13] defined m-polar fuzzy graph(mPFG) and Presented several new result. Several new results, theories and applications on mPFGs are studied by Ghorai and Pal [14,10,15,12]. Binu M. [5] introduced the connectivity index in fuzzy graphs. The idea of strong arc in fuzzy graph was given by Bhutani and Rosenfeld [7] and types of arc in fuzzy graph was given by Mathew and Sunitha [20]. The notion of bridge, trees, cycles, cut node, end node were introduced by Rosenfeld [21]. The concepts of strength of connectedness in mPFG, mPF tree, mPF cut node are established by Mandal et al. [18].
In this paper, we will describe the connectivity index for mPFG. The upper and lower boundary of connectivity index for mPFG are discussed. If we delete an edge from a mPFG then the effects of the connectivity index in mPFG are given. The average connectivity index in mPFG are described.

Preliminaries
Firstly, we define mPFGs and other related terms.
In this paper, for a natural number m, m-power of [0,1] or [0, 1] m is considered as a poset with point-wise order ≤. "≤" is defined by x ′ ≤ y ′ ⇔ p i (x ′ ) ≤ p i (y ′ ) for each i = 1, 2, . . . , m, where x ′ , y ′ ∈ [0, 1] m and p i : [0, 1] m → [0, 1] be the ith projection mapping.  Definition 2.4. [18] The strength of the mPF path P : is the strength of connectedness between u ′ and v ′ and is defined as Definition 2.5. [18] An mPFG is said to be mPF connected graph if p i • B(a ′ , b ′ ) ∞ > 0, for at least one i = 1, 2, 3, . . . , m . 3 Connectivity index of mPFG In a network system, connectivity is a major factor. Here, we have presented a mPFG connectivity index.
Definition 3.1. The connectivity index of mPFG is characterised by CI mP F (G) ,defined as  In the above example, we calculated the connectivity index of a 3mPFG. Depended on this, we established the following Theorems. Different property on connectivity index of a mPFG is discussed below.
Again CON N G ′ (x, y) is connectedness between s and t in G ′ where G ′ is the partial mPF An mPFSG is a partial mPFG. Then the Theorem 3.3 holds for a mPFSG that is given in the next note.
Proof. Suppose n is the total number of nodes in Gi.e. |V | = n and s ∈ V . Then,

Boundedness of connectivity index of m Polar fuzzy graph
In this section we described boundedness implies lower and upper boundary of connectivity index of a mPFG. Next, we introduced some Theorems on connectivity index of mPFG.
Proof. If G has only one vertex. Then also G ′ has one vertex. Then, CON N G (s, t) = 0 = Let G contain more than one vertex. Then the complete mPFG From the above relation, we say that proof. Let u 1 be an node in G. Here  = 0.997. if we remove an edge from G, it could decrease or stay the same connectivity index. In this section we will present the connectivity index of edge deleted mPFG.
Next, we find out the connectivity index of edge deleted mPFSG of mPFG using the following example. Here, = (7.5, 6.5, 4.5).
= (7.5, 6.5, 4.5).  So, Here we see that proof. Let (s, t) be a mPFB of G. Using the concept of mPFB, there exist one edge Conversely, suppose p i • CI mP F (G) > p i • CI mP F (G ′ ) for all i. Next we suppose that, the edge (s, t) is not a mPF bridge. Then for each pair of vertices (s 1 , t 1 ) in G, we have, is a mPFB of G.
proof. Let G be mPFG. Since 0 ≤ r 1 ≤ r 2 ≤ 1, G r 1 and G r 2 are the r 1 and r 2 cuts of G respectively. So G r 1 is a partial mPFSG of G r 2 . Then CON N G r 1 (s, t) ≥ CON N G r 2 (s, t). Then by Theorem , we have CI mP F (G r 1 ) ≥ CI mP F (G r 2 ).

Average connectivity index of a mPFG
In this section, we explain average connectivity index of mPFG. In Theorem , we see that, if a node is removed from a connected mPFG, then its connectivity index reduces. But if a vertex is exclude from a connected mPFG then its average connectivity index may reduces or increases or same. Here certain types nodes mPFCRN, mPFCEN,mPFCNN are introduced and some properties on those nodes has been established.
Definition 6.1. The average connectivity index of mPFG G denoted by ACI mP F (G) is defined CON N G (s, t)), . . . , Here, p i • ACI mP F (G) is the i th component of average mPF Connectivity index and |V | = n.
Similarly, (ii) and (iii) can be proved.
Definition 6.7. Let G be a mPFG.
(i) If G has at least one mPFCEN, then G is called Connectivity enhancing m-polar fuzzy graph (CEmPFG).
(ii) If G has no mPFCRN, then G is called Connectivity reducing m-polar fuzzy graph (CRmPFG).
(iii) If every nodes are of G mPFCNN, then G is called neutral m-polar fuzzy graph (NmPFG).
Example 6.8. For the mPFG, G in Example 6.4,we show that s 5 is an mPFCEN of G. Hence G is a CEmPFG and G is not contain any mPFCRN vertex then G is also called CRmPFG but G is not a NmPFG.  CON N G (s, t)). Then Proof. Since G is a mPFT and x is not an mPFEN of G, G − s is disconnected. Again, as G 1 and G 2 are connected components of G − s, So G 1 G 2 and G 1 G 2 = φ. First assume that the node x is a mPFCEN.
The converse part is shown when all these steps are reversed. Similarly, (ii) and (iii) can be proved.

Application
We 1. If you take expert advise, they will warn you that staying in a mobile tower within 50m is like staying in a microwave oven the whole day. Electromagnetic radiations are carcinogenic components that are unquestionably harmful to one's health.
2. Although cancer is the worst hazard to those who live near a mobile phone tower, also other health problems can occur. Sleeping difficulties, memory loss, tiredness,skin problems, headache, depression, hearing issues, joint aches and even cardiovascular illnesses are experienced by those in the region.
3. The birds have been most noticeably affected by mobile tower radiation. The number of birds such as sparrows has begun decreasing from the residential areas where mobile towers have been installed. The mobile towers are known to emit microwaves that harm birds' eggs and embryos, as they thin the skulls of chicks and eggshells.
4. Children have thinner skulls and are thus known to be impacted more by cell tower radiation. It is the same for an underdeveloped child bearing pregnant women. It is simpler to penetrate radiation and the impact might be extremely catastrophic if not controlled.
Children these days are known to have less concentration power -one impact of putting cell towers close to schools and hospitals in residential areas.
So, mobile tower is very harmful for our environment. For this reason, we want to reduce the number of tower. Here we can give an idea for reducing the tower from a city (or country).
Using the concept of Average connectivity index, we reduce the number of mobile tower.  Here we see that ACI mP F G (G−v 4 ) < ACI mP F G (G). From here it is clear that, if we eliminate the mobile network that is present at the v 4 location, the average network speed in each device in the city remains the same. In this way we reduce the mobile tower in a city(or country).

Conclusion
In this paper, we have shown how to calculate the connection index in mPFG.