INTRODUCTION:

Fuzzy set [1] has developed a potential area of interdisciplinary research. The first definition of Fuzzy Graph was introduced 1973, and then it was developed in 1975 by Rosenfeld [2] and it has found applications in the analysis of cluster patterns. He also introduce the structure of fuzzy graph by the relation between the sets in fuzzy, also getting analogs of a few graph theoretical concepts. Followed by him Bhattacharya [3] gave some new ideas in fuzzy graph. Mordeson and Peng introduced some operation on fuzzy graph. The energy of bipolar fuzzy graph of adjacency matrix G is the sum of absolute value of eigenvalues. Many authors [4], [5], [6] and [7] discussed the existence of an epidemic threshold.

This paper is organised as follows. Section 2, consists of necessary definition. In section 3, we present the energy of a bipolar fuzzy graph. In section 4, compelling spreading rate and plague threshold idea were discussed. In section 5, we demonstrate these ideas with example. In section 6, result were discussed.

2. PRELIMINARIES:

Definition 2.1. Fuzzy set: [4] Let X be a nonempty set. A fuzzy set A in X is defined as

A = {(x,)/x X} which is characterized by a membership function:X→[0,1] and a fuzzy set satisfying the following condition, where is the non-membership function

Definition 2.2. [5] A fuzzy graph with V as the underlying set is a pair of functions G = (σ, μ) where σ : V→[0,1] is a fuzzy subset and μ : V x V→[0,1] is a symmetric fuzzy relation on the fuzzy subset σ for all u, v ∈ V such that μ (u,v)≤ σ(u) Λ σ(v). The underlying crisp graph of is denoted by G = (V, E) where E ⊆ V x V. A fuzzy relation can also be expressed by a matrix called fuzzy relation matrix where Throughout this paper, we suppose is undirected without loops and σ(u)=1 for each u∈ V.

Definition 2.3. An edge whose end points are the same is called a loop. A graph without loops and parallel edges is called a simple graph. Two vertices that are connected by an edge is called adjacent. The adjacency matrix for a graph G = (V, E) is a matrix with n rows and n columns, n = |V| and its entries defined by

Definition 2.4. [6] Let X be a nonempty set. A bipolar fuzzy set A in X is an object having the form

Where and are mappings.

We use the positive membership degree to denote the satisfaction degree of an element x to the property corresponding to a bipolar fuzzy set A and the negative membership degree to denote the satisfaction degree of an element x to some implicit counter property corresponding to a bipolar fuzzy set B. If and, it is the situation that x is regarded as having only positive satisfaction for A. If and, it is the situation that x does not satisfy the property of A but somewhat satisfies the counter property of A. It is possible for an element x to be such that and when the membership function of the property overlaps that of its counter property over some portion of X.

3. ENERGY OF BIPOLAR FUZZY GRAPH:

Definition 3.1. A bipolar fuzzy graph G=(V,A,B) is a nonempty set V together with a pair of functions and such that for all,

Definition 3.2. The bipolar fuzzy graph is defined as the adjacency matrix. That is for an bipolar fuzzy graph an bipolar fuzzy adjacency matrix is defined by A(BG)= where.

For the Figure 1 ,

Figure 1: Bipolar fuzzy graph

Definition 3.3. The Bipolar fuzzy graph of adjacency matrix can be split in to two matrices such as and , where denote membership of positive values , denote membership of negative values.

where

and

Definition 3.4. The adjacency matrix of bipolar fuzzy graph the eigen value is defined in two ways one is adjacency matrix of positive membership values and other is adjacency matrix of negative membership values . The eigen value of the matrix is called spectrum.

Definition 3.5: The energy of bipolar fuzzy graph is defined as the sum of absolute value of eigenvalues.

Example 3.1.

The Bipolar fuzzy graph for Fig: 1

The energy of bipolar fuzzy graph is [2.0204, 0.5556].

4. SPREADING OF MICROORGANISM IN BIPOLAR FUZZY SYSTEM

Let be the disease rate and be the healing rate of an bipolar fuzzy system. The ratio is an bipolar fuzzy compelling spreading rate and is the plague threshold of an bipolar fuzzy network denoted by , where is the largest eigen value of the adjacent matrix G. If the viable spreading rate , at that point infection proceed and a nonzero division of the hubs are contaminated, whereas , the pandemic vanishes. We characterize the accompanying definitions.

Definition 4.1. If , then the disease rate of an bipolar fuzzy system is characterized as and the healing rate of an bipolar fuzzy system is characterized as . The ratio is the compelling spreading rate and, where is the largest eigen value of the adjacent matrix G.

In the event that the quantity of guests are most extreme in a bipolar fuzzy system, at that point the spreading rate of infection will be greatest. Else we can say the framework or system is unstable.

Definition 4.2. If, then the disease rate of an bipolar fuzzy system is characterized as and the healing rate of an bipolar fuzzy system is characterized as. The ratio is the compelling spreading rate and , where is the largest eigen value of the adjacent matrix G.

In the event that the quantity of guests are least in bipolar fuzzy system, at that point the spreading rate of infection will be least. Else we can say the framework or system is steady.

5. NUMERICAL EXAMPLES:

Example 5.1.

In example 3.1, then,

The compelling spreading rate.

The plague threshold.

If the viable spreading rate , at that point infection proceed and a nonzero division of the hubs are contaminated. Hence the spreading rate of infection will be maximal.

Example 5.2.

Figure 2: Bipolar fuzzy graph

The energy of bipolar fuzzy graph is [0.7656, 1.677].

Here then,

The compelling spreading rate.

The plague threshold.

Therefore, the pandemic vanishes. Hence the spreading rate of infection will be minimal.

6. CONCLUSION:

In this paper, we examined the spreading of microorganism in bipolar fuzzy system and we defined compelling spreading rate and the sharp plague threshold of the microorganism spreading on bipolar fuzzy system and we demonstrate these ideas with example.

7. REFERENCES:

1. Zadeh, L.A., Fuzzy sets, Information and Control, (8) (1965), 338-353.

2. A. Rosenfeld, Fuzzy graphs, in: L.A. Zadeh, K.S. Fu, M. Shimura (Eds.), Fuzzy Sets and their Applications, Academic Press, New York, 1975, pp. 77–95.

3. P. Bhattacharya, Some remarks on fuzzy graphs, Pattern Recognition Letter 6 (1987) 297–302.

4. Bailey. N. T. J, The Mathematical Theory of Infectious Diseases and its applications, 2^nd ed. London, U.K, Charlin Grifﬁn, 1975.

5. Daley. D.J, and Gani. J, Epidemic Modelling: An Introduction. Cambridge University Press, Cambridge, 1999.

6. Pastor - Satorras. R and Vespignani. A, Epidemic Spreading in Scale, free networks, Phys. Rev. Lett, Vol. 86, no.14, pp. 3200–3203, Apr. 2001.

7. Van Mieghem. P, Member, IEEE, Jasmina Omic and Robert Kooij, Virus Spread in Networks, IEEE/ACM Transaction on Networking, Vol. 17, No. 1, 2009.

Received on 20.06.2019 Modified on 21.07.2019

Research J. Pharm. and Tech. 2020; 13(1):224-226.

DOI: 10.5958/0974-360X.2020.00045.1