MAX-MIN INTUITIONISTIC FUZZY MATRIX OF AN INTUITIONISTIC FUZZY GRAPH

: In this paper we introduce the Max-Min intuitionistic fuzzy matrix M ( G ) of an intuitionistic fuzzy graph. And the extreme energy of M ( G ) is deﬁned. And also we give the explicit expression for the coeﬃcients of the characteristic polynomial of M ( G ) . These concepts are illustrated with real time example.


Introduction
The foundation for graph theory was laid in 1735 by Leonhard Euler when he solved the 'Konigsberg bridges' problem.Many real life problems can be represented by graph.In computer science, graphs are used to represent net-works of communications, data organization, computational devices, the flow of computation, etc.The link structure of a website could be represented by a directed graph in which the vertices are the web pages available at the website and a directed edge from page A to page B exists if and only if A contains a link to B (see [14]).A similar approach can be taken to problems in travel, biology, computer chip design and many other fields.Hence graph theory is widely used in solving real time problems.But when the system is large and complex it is difficult to extract the exact information about the system using the classical graph theory.In such cases fuzzy graph is used to analyze the system.In 1973, the definition of fuzzy graph was introduced by Kafmann (see [10]) from the Zadeh's fuzzy relations.In 1975, Rosenfeld (see [16]) introduced another detailed definition including fuzzy vertex and fuzzy edges and several fuzzy analogs of graph theoretic concepts such as paths, cycles, connectedness and etc.In 1999, the first definition of intuitionistic fuzzy graph was proposed by Atanassov (see [2]).In 1978, Gutman (see [8]) introduced the energy of a graph as the sum of the absolute values of the eigen values of the adjacency matrix of a graph.The lower and upper bound for the energy of a graph are discussed in (see [4], [12], [9]).The energy of fuzzy graph and its bounds are discussed in (see [1]).The energy of an intuitionistic fuzzy graph and its bounds are discussed in (see [15]).In (see [15]), the adjacency matrix of an intuitionistic fuzzy graph is defined.Using this adjacency matrix we defined the energy of an intuitionistic fuzzy graph and the lower and upper bounds for the energy of an intuitionistic fuzzy graph are obtained.These concepts are illustrated with real time example.The link structure of a website could be represented by an intuitionistic fuzzy directed graph.The links are considered as vertices and the path between the links are considered as edges.The weightage of the each edge are considered as the number of visitors (membership value), the number of non visitors (non membership value) and drop off (intuitionistic fuzzy index) among that link structure.In (see [7]), The authors discussed the virus spread in an intuitionistic fuzzy network.Each node (vertex) in this network is either infected or healthy.An infected node can infect its neighbours with an infection rate β, and it is cured with curing rate δ .The ratio τ = β δ is called the effective spreading rate of the virus spread in an intuitionistic fuzzy network.Also the sharp epidemic threshold τ c of the virus spread in an intuitionistic fuzzy network is defined.Many authors (see [17], [3], [6], [11], [13]) discussed the existence of an epidemic threshold τ c .If the effective spreading rate τ = β δ > τ c , then virus continue and a nonzero fraction of the nodes are infected, whereas τ ≤ τ c , the epidemic dies out.The relationship between the number of visitors and the energy of an intuitionistic fuzzy graph is analyzed which is used to find the spreading rate of virus.This paper is organized as follows.In Section 2, we give the required definitions.In Section 3, we define the Max-Min intuitionistic fuzzy matrix M (G) of an intuitionistic fuzzy graph.And the extreme energy of M (G) is defined.And we give the explicit expression for the coefficients of the characteristic polynomial of M (G).In Section 4, we illustrate these concepts with real time example.In Section 5, we give the conclusion.

Intuitionistic Fuzzy Graph
Definition 1. (see [15]) An intuitionistic fuzzy graph is defined as G = (V, E, µ, γ) where V is the set of vertices and E is the set of edges.µ is a fuzzy membership function defined on V × V and γ is a fuzzy non -membership function defined on V × V .We denote µ (v i , v j ) by µ ij and γ (v i , v j ) by γ ij such that: Hence (V × V, µ, γ) is an intuitionistic fuzzy set.Definition 2. (see [15]) An intuitionistic fuzzy adjacency matrix of an intuitionistic fuzzy graph is defined as the adjacency matrix of the corresponding intuitionistic fuzzy graph.That is for an intuitionistic fuzzy graph G = (V, E, µ, γ), an intuitionistic fuzzy adjacency matrix is defined by where a ij = (µ ij , γ ij ) .Note that µ ij represents the strength of the relationship between v i and v j and γ ij represents the strength of the non-relationship between v i and v j .The adjacency matrix of an intuitionistic fuzzy graph can be written as two matrices one containing the entries as membership values and the other containing the entries as non-membership values.i.e.A (IG) = ((µ ij ) , (γ ij )) .

Energy of an Intuitionistic Fuzzy Graph
Definition 3. (see [15]) The energy of an intuitionistic fuzzy graph G = (V, E, µ, γ) is defined as defined as an energy of the membership matrix denoted by E (µ ij (G)) and δ i ∈Y |δ i | is defined as an energy of the non -membership matrix denoted by E (γ ij (G)).Theorem 4. (see [15]) Let G be an intuitionistic fuzzy directed graph (without loops) with |V | = n and |E| = m and A(IG) = ((µ ij ), (γ ij )) be an intuitionistic fuzzy adjacency matrix of G then: where |A| is the determinant of A(µ ij ), and (ii) where |B| is the determinant of A(γ ij ).

Virus Spread in an Intuitionistic Fuzzy Network
Definition 5. (see [7]) If the number of visitors are maximum in an intuitionistic fuzzy network then the spreading rate of virus will be maximum.If (E (µ ij (G))) > (E (γ ij (G))) then the infection rate of an intuitionistic fuzzy network is defined as β = max i,j µ ij and the curing rate of an intuitionistic fuzzy network is defined as δ = min i,j γ ij .The ratio τ = β δ (δ = 0) is the effective spreading rate and τ c = 1 λmaxA(µ ij ) where λ max A(µ ij ) is the largest eigen value of the adjacency matrix A (µ ij ) of an intuitionistic fuzzy network.Definition 6. (see [7]) If the number of visitors are minimum in an intuitionistic fuzzy network then the spreading rate of virus will be minimum.
) then the infection rate of an intuitionistic fuzzy network is defined as β = min i,j µ ij and the curing rate of an intuitionistic fuzzy network is defined as δ = max i,j γ ij .The ratio τ = β δ (δ = 0) is the effective spreading rate and τ c = 1 λmaxA(γ ij ) where λ max A(γ ij ) is the largest eigen value of the adjacency matrix A (γ ij ) of an intuitionistic fuzzy network.

Max-Min Intuitionistic Fuzzy Matrix of an Intuitionistic Fuzzy Graph
In this section we define the Max-Min intuitionistic fuzzy matrix M (G) of an intuitionistic fuzzy graph.And the extreme energy of M (G) is defined.And we give the explicit expression for the coefficients of the characteristic polynomial of M (G) .
Definition 7. Let G = (V, E, µ, γ) be an intuitionistic fuzzy graph.∀ node i, define α j = max i µ ij and σ j = min i γ ij .Definition 8. Let G = (V, E, µ, γ) be an intuitionistic fuzzy graph.The Max-Min intuitionistic fuzzy matrix of an intuitionistic fuzzy graph is defined as M (G) = ((r ij ) , (s ij )), where and 0 (0.9, 0.1) 0 (0.9, 0.1) (0.9, 0.1) 0 (0.8, 0.1) 0 (0.9, 0.1) (0.8, 0.1) 0 (0.6, 0.1) (0.9, 0.1) (0.8, 0.1) (0.6, 0.1) 0 0 0.9 0 0.9 0.9 0 0.8 0 0.9 0.8 0 0.6 0.9 0.8 0.6 0 Let M (G) = ((r ij ) , (s ij )) be the Max-Min intuitionistic fuzzy matrix of an intuitionistic fuzzy graph G .Let R = (r ij ) and S = (s ij ) .The characteristic polynomial of R and S is an equation of order n if G has n nodes.For our further discussion we confine our self the polynomial of order 4. The Characteristic polynomial of R of order 4 is trR) + 2tr R 3 and c 4 = det (R) .We now give the explicit expression for the coefficients of c 2 and c 3 .Definition 10.Mutually adjacent.Let G be the given intuitionistic fuzzy graph.Two vertices v i and v j are said to be mutually adjacent if there is an edge from v i to v j and there is an edge from v j to v i .Definition 11.Cyclic.Let G be the given intuitionistic fuzzy graph.Three vertices v i , v j and v k are cyclic if there is an edge from v i to v j , v j to v k and v k to v i .
v j are mutually adjacent, then r ij = r ji otherwise any one of r ij or r ji will be zero.Therefore (1) becomes c 2 = − 1≤i<j≤n (r ij ) 2 if v i and v j are mutually adjacent.
Lemma 13.In the characteristic polynomial of R, c 3 = r ij r jk r ki where the summation is taken over all i , j and k such that v i , v j and v k are cyclic in G .
Proof.In general we have, r ii r ik r ji r jj r jk r ki r kj r kk = r 12 r 23 r 31 + r 12 r 24 r 41 + r 13 r 34 r 41 + r 13 r 32 r 21 + r 14 r 43 r 31 + r 14 r 42 r 21 + r 23 r 34 r 42 + r 24 r 43 r 32 .
Note that if v i , v j and v k are cyclic then c 3 = r ij r jk r ki otherwise any one of r ij or r jk or r ki will be zero.Therefore (2) becomes c 3 = r ij r jk r ki where the summation is taken over all i , j and k such that v i , v j and v k are cyclic in G .
Theorem 14. (see [5]) If θ 1 , θ 2 , ..., θ n are the eigen values of R, then Proof.We know that  s ij s jk s ki where the summation is taken over all i , j and k such that v i , v j and v k are cyclic in G. i.e.Extreme energy of an intuitionistic fuzzy graph is (E 1 , E 2 ) = (3.9243,0.4828).

Numerical Examples
In this section we give the extreme energy of an intuitionistic fuzzy graph through real time example.We have taken the website http://www.pantechsolutions.net/.This website is modeled as an intuitionistic fuzzy graph by considering the navigation of the customer.An intuitionistic fuzzy graph of this site for four different time periods is taken.For each of these periods the Max-Min intuitionistic fuzzy matrix is constructed and the extreme energy is calculated.These results are represented in the table .
Example 19.In the same website (mentioned above in example 17), we consider the same four links for the period September 16, 2013 to October 15, 2013.

The following table represents the comparison of energy of A (G) and the Extreme energy of M (G).
From the above table we can observe that the energy of µ of A (G) for all the time periods is < then the Extreme energy of R in M (G).Similarly the energy of γ of A (G) > then the Extreme energy of S in M (G) .

Conclusion
In this paper we introduced the Max-Min intuitionistic fuzzy matrix M (G) of an intuitionistic fuzzy graph G.And the extreme energy of M (G) is defined.And also we give the explicit expression for the coefficients of the characteristic polynomial of M (G).These concepts are illustrated through real time example.These results are compared with the energy of an intuitionistic fuzzy graph.