Matrix Representations of Intuitionistic Fuzzy Graphs

Transcription

1 520 Matrix Representations of Intuitionistic Fuzzy Graphs M.G.Karunambigai,O.K.Kalaivani Department of Mathematics,Sri Vasavi College,Erode , Tamilnadu,India. May 23, 2016 Abstract Matrices play an important role in the broad area of science and engineering. However, the classical matrix theory sometimes fails to solve the problems involving uncertainties, occurring in an imprecise environment.sometimes it seems to be more natural to describe imprecise and uncertain opinions not only by membership functions and also by non membership function. In this paper, it is proved that (CONN µ(g) (v i, v j ), CONN ν(g) (v i, v j )) = (i, j) th entry of A + A A n 1, v i v j V, where A is the index matrix of the intuitionistic fuzzy graph G and A k is the k th power of an intuitionistic fuzzy matrix A and (CONN µ(g) (v i, v j ), CONN ν(g) (v i, v j )) is the strength of connectedness of v i and v j. Also, the properties of subdivision IFG, line IFG and power of an IFG are discussed Mathematics Subject Classification: 05C72, 03E72, 03F55. Index Terms: incidence intuitionistic fuzzy matrix, line intuitionistic fuzzy graph, Power of an intuitionistic fuzzy graph, subdivision intuitionistic fuzzy graph. I Introduction Graphs can be sometimes very complicated. So one needs to find more practical ways to represent them. Matrices are a very useful way of studying graphs, since they turn the picture into numbers. Networks can represent all sorts of systems in the real world. As computers are more adept at manipulating numbers than at recognizing pictures, it is standard practice to communicate the specification of a graph to a computer in matrix form. Matrices play an important role in the broad area of science and engineering. However, the classical matrix theory sometimes fails to solve the problems involving uncertainties, occurring in an imprecise environment. Sometimes it seems to be more natural to describe imprecise and uncertain opinions not only by membership functions and also by non membership function.so an Intuitionistic fuzzy matrix is the appropriate choice when exhibiting the membership degree and non- membership degree. In 1975, Rosenfeld [17] discussed the concept of fuzzy graphs whose basic idea was introduced by Kauffmann [12] in The fuzzy relations between fuzzy sets were also considered by Rosenfeld and he developed the structure of fuzzy graphs, obtaining analogs of several 1

2 521 2 graph theoretical concepts. The first definition of fuzzy graph was introduced by Kaufman[12] in 1973, based on Zadeh s fuzzy relations in 1971[20]. Atanassov[3][19] introduced the concept of intuitonistic fuzzy(if) relations and intuitionistic fuzzy graphs(ifgs). M.G.Karunambigai and R.Parvathi[10][14] introduced the concept of IFG elaborately and analysed its components. Atanassov introduced the index matrix reresentation of intuitionistic fuzzy graphs and discussed its operations in [5][4][6]. Akram et al. discussed the properties of strong intuitionistic fuzzy graphs and also the properties of intuitionisic fuzzy cycle and intuitionistic fuzzy trees in [1][2]. R.Parvathi et al.[16] discussed operations on intuitionistic fuzzy graphs using index matrices. Intuitionistic fuzzy matrix are extensively used for decision making problems, cluster analysis, pattern recognition, medical diagnosis and network problems. Intuitionistic fuzzy matrices can be used whenever uncertainity occurs in a problem. These application motivated us to consider intuitionistic fuzzy matrices and discuss its properties. The paper is organized as follows. In section 2, we review the basic definitions of intuitionistic fuzzy graph. Section 3 deals with the properties of the power of an intuitionistic fuzzy graph and given the relationship between the index matrix of an intuitionistic fuzzy graph and power of an intuitionistic fuzzy graph and section 4 concludes the paper. II Preliminaries In this section, the basic definitions and Theorems which are used to prove the forthcoming results are given. Definition 2.1 [8] A crisp graph G = (V, E) is an ordered triple (V (G ), E(G ), ψ G ) consisting of a nonempty V (G ) of vertices, a set E(G ), disjoint from V (G ), of edges and an incidence function ψ G that associates with each edge of G an unordered pair of vertices of G. Definition 2.2 [8]Let G = (V, E) be a crisp graph. A walk is a sequence of vertices and edges, where the endpoints of each edge are the preceding and following vertices in the sequence. A path is a walk without repeated vertices. If a walk (resp. trail, path) begins at u and ends at v then it is an u v walk. A walk is closed if it begins and ends at the same vertex. Definition 2.3 [8] Let G = (V, E) be a crisp graph. The length of a path P = v 1 v 2...v n+1 in G is n. Definition 2.4 [8] Let G = (V, E) be a crisp graph. The distance between the two vertices v i and v j in G is denoted by d G (v i, v j ) and is defined as the minimum length of the path connecting the vertices v i and v j. Definition 2.5 A matrix is a rectangular array of numbers arranged in rows and columns. The number of rows and columns that a matrix has, called its dimension or its order. That is, the dimension or order of a matrix with m rows and n columns is m n. The individual items in a matrix are called its elements or entries. Definition 2.6 Let A = [a ij ] and B = [b ij ] be two matrices. Then two matrices A and B are equal to each other, if they have the same dimensions m n and the same elements a ij = b ij for i = 1,..., n and j = 1,..., m. It is denoted by A = B Definition 2.7 [10] An Intuitionistic Fuzzy Graph (IFG) is of the form G = (V, E) said to be a minmax IFG if

3 522 3 (1) V = {v 1,..., v n } such that µ i : V [0, 1] and ν i : V [0, 1], denotes the degree of membership and non-membership of an element v i V respectively and 0 µ i + ν i 1, for every v i V, (2) E V V where µ ij : V V [0, 1] and ν ij : V V [0, 1] are such that µ ij min(µ i, µ j ) ν ij max(ν i, ν j ), denotes the degree of membership and non-membership of an edge e ij = (v i, v j ) E respectively, where, 0 µ ij + ν ij 1, for every e ij = (v i, v j ) E. The degree of hesitance(hesitation degree) of the vertex v i V in G is Π i = 1 µ i ν i and the degree of hesitance(hesitation degree) of an edge e ij = (v i, v j ) E in G is Π ij = 1 µ ij ν ij. Definition 2.8 [10] Let G = (V, E) be an intuitionistic fuzzy graph.a walk is a sequence of vertices and edges, where the endpoints of each edge are the preceding and following vertices in the sequence, such that either one of the following conditions is satisfied. 1) µ ij > 0 & ν ij = 0 for some i & j. 2) µ ij > 0 & ν ij > 0 for some i & j. If a walk begins at v i and ends at v j then it is an v i v j walk. A walk is closed if it begins and ends at the same vertex. Definition 2.9 [10] Let G = (V, E) be an intuitionistic fuzzy graph. A path P in an intuitionistic fuzzy graph G is a sequence of distinct vertices v 1, v 2,..., v n such that either one of the following conditions is satisfied. 1) µ ij > 0 & ν ij = 0 for some i & j. 2) µ ij > 0 & ν ij > 0 for some i & j. Definition 2.10 [10] Let G = (V, E) be an intuitionistic fuzzy graph. The length of a path P = v 1 v 2...v n+1 (n > 0) in G is n. Definition 2.11 [10] An intuitionistic fuzzy graph G = (V, E) is connected if any two vertices are joined by a path. Definition 2.12 [10] Let G = (V, E) be an intuitionistic fuzzy graph. The µ strength of a path P = v 1 v 2...v n in an intuitionistic fuzzy graph G is denoted by S µ(g) (P ) and is defined as min{µ ij }, for all i, j = 1, 2,..., n Definition 2.13 [10] Let G = (V, E) be an intuitionistic fuzzy graph. The ν strength of a path P = v 1 v 2...v n in an intuitionistic fuzzy graph G is denoted by S ν(g) (P ) and is defined as max{ν ij }, for all i, j = 1, 2,..., n Definition 2.14 [10] If v i, v j V G, the µ strength of connectedness between the vertices v i and v j in G is CONN µ(g) (v i, v j ) = max{s µ(g) (P )} and ν strength of connectedness between the vertices v i and v j in G is CONN ν(g) (v i, v j ) = min{s ν(g) (P )} for all possible paths between v i and v j. Definition 2.15 [10] An intuitionistic fuzzy graph, G = (V, E) is said to be a strong intuitionistic fuzzy graph if µ ij = min(µ i, µ j ) and ν ij = max(ν i, ν j ), (v i, v j ) E. Definition 2.16 [10] An intuitionistic fuzzy graph, G = (V, E) is said to be a complete intuitionistic fuzzy graph if µ ij = min(µ i, µ j ) and ν ij = max(ν i, ν j ), v i, v j V.

4 523 4 Definition 2.17 [13] The order of an intuitionistic fuzzy graph G = (V, E) is defined as O(G) = (O µ (G), O ν (G)) where O µ (G) = µ i and O ν (G) = v i V Definition 2.18 [13] The size of an intuitionistic fuzzy graph is defined as S(G) = (S µ (G), S ν (G)) S µ (G) = e ij E v i V µ ij and S ν (G) = e ij E ν i ν ij Definition 2.19 [13] Let G = (V, E) be an intuitionistic fuzzy graph. The neighbourhood of a vertex v i V is denoted by N G (v i ) and is defined as N G (v i ) = {v j V (v i, v j ) E}. Definition 2.20 The function f : X Y is an one to one function if and only if for every element y Y there is exactly one element x X. In Symbol, f(x) = f(y) x = y, x, y X. Definition 2.21 The function f : X Y is an onto function if and only if for every element y Y there is at least one element x X. In Symbol, f(x) = y, y Y. Definition 2.22 A function f : X Y set X to a set Y. is a bijection if the function is both one-one and onto mapping of a Definition 2.23 [11] A homomorphism from a intuitionistic fuzzy graph G 1 = (V 1, E 1 ) to a intuitionistic fuzzy graph G 2 = (V 2, E 2 ), written f : G 1 G 2, is a mapping f : V 1 V 2 from the vertex set of G 1 to the vertex set of G 2 such that if any two vertices v i, v j V 1 are adjacent in G 1, then f(v i ), f(v j ) V 2 are adjacent in G 2 and µ(v i ) µ (f(v i )) and ν(v i ) ν (f(v i )), v i V 1 µ(v i, v j ) µ (f(v i ), f(v j )) and ν(v i, v j ) ν (f(v i ), f(v j )), v i, v j V 1. Definition 2.24 [11] Two intuitionistic fuzzy graphs G 1 = (V 1, E 1 ) and G 2 = (V 2, E 2 ) are said to be isomorphic if there is a bijections f : V 1 V 2 such that any two vertices v i, v j V 1 are adjacent in G 1 if an only if f(v i ), f(v j ) V 2 are adjacent in G 2 and µ(v i ) = µ (f(v i )) and ν(v i ) = ν (f(v i )), v i V 1 µ(v i, v j ) = µ (f(v i ), f(v j )) and ν(v i, v j ) = ν (f(v i ), f(v j )), v i, v j V 1. Definition 2.25 [11] Two intuitionistic fuzzy graphs G 1 = (V 1, E 1 ) and G 2 = (V 2, E 2 ) are said to be co-weak isomorphic if there is a bijections f : V 1 V 2 such that any two vertices v i, v j V 1 are adjacent in G 1 if an only if f(v i ), f(v j ) V 2 are adjacent in G 2 and µ(v i ) µ (f(v i )) and ν(v i ) ν (f(v i )), v i V 1 µ(v i, v j ) = µ (f(v i ), f(v j )) and ν(v i, v j ) = ν (f(v i ), f(v j )), v i, v j V 1

5 524 5 Definition 2.26 [9] Let G 1 = (V 1, E 1 ) and G 2 = (V 2, E 2 ) be two IFGs with V 1 V 2 ϕ. Then the union of G 1 and G 2 is an IFG, denoted by G 1 G 2 = (V 1 V 2, E 1 E 2 ) and is defined as ((µ µ )(v i ), (ν ν )(v i )) = (µ i, ν i), if v i V 1 V 2, (µ i, ν i), if v i V 2 V 1, (max(µ i, µ i), min(ν i, ν i)) if v i V 1 V 2. µ ij if e ij E 1 E 2, µ ij if e ij E 2 E 1, max(µ ij, µ ij) if e ij E 1 E 2, (µ µ )(v i, v j ) = min((µ i µ i), max(µ j, µ j)) if v i V 1 V 2, v j V 1 V 2 and e ij E 1 E 2 or e ij E 2 E 1, (0, 1) otherwise. ν ij if e ij E 1 E 2, ν ij if e ij E 2 E 1, min((ν i ν i), (ν j ν j)) if e ij E 1 E 2, (ν ν )(v i, v j) = max((ν i ν i), min(ν j, ν j)) if v i V 1 V 2, v j V 1 V 2 and e ij E 1 E 2 or e ij E 2 E 1, (0, 1) otherwise. Definition 2.27 [15] An intuitionistic fuzzy matrix(ifm) is a matrix of order m n and is defined as A = {< a µij, a νij >} m n, where a µij [0, 1], a νij [0, 1] such that 0 a µij + a νij 1, 1 i m

6 525 6 and 1 j n. It can also be represented in the matrix form, A = {< a µij, a νij >} m n = < a µ11, a ν11 > < a µ12, a ν12 >... < a µ1n, a ν1n > < a µ21, a ν21 > < a µ22, a ν22 >... < a µ2n, a ν2n >... < a µm1, a νm1 > < a µm2, a νm2 >... < a µmn, a νmn > Definition 2.28 The number of rows and columns that IF matrix has called its dimension or its order. That is, the dimension or order of IF matrix with m rows and n columns is m n. The individual items in an IF matrix are called its elements or entries. Definition 2.29 [15] Let A = {< a µij, a νij >} m n be a intuitionistic fuzzy matrix. The transpose of the matrix A is denoted by A T and is defined as A T = {< a µji, a νji >} n m. Definition 2.30 Let A = {< a µij, a νij >} m n and B = {< b µij, b νij >} m n be two intuitionistic fuzzy matrices. Then two IF matrices A and B are equal to each other, if they have the same dimensions m n and the same elements a µij = b µij, a νij = b νij for i = 1,..., m and j = 1,..., n. It is denoted by A = B Definition 2.31 Let A = {< a µij, a νij >} m n and B = {< b µij, b νij >} m n be two intuitionistic fuzzy matrices, then the sum of A and B is denoted by A + max min B, is defined as A + max min B = {< c µij, c νij >} m n = [< max(a µij, b µij ), min(a νij, b νij ) >], 1 i m, 1 j n. Notation 2.1 Throughout this paper, we denote + max min as +. Theorem 2.2 [8] If a crisp graph G contains a u v walk of length l, then G contains a u v path of length l. Theorem 2.3 [8] Let G be a crisp graph. Then G is connected if and only if every pair of vertices in G is connected. III Matrix Representations of Intuitionistic Fuzzy Graphs In this section, the properties of the power of an intuitionistic fuzzy graph and the relationship between the index matrix of an intuitionistic fuzzy graph and power of an intuitionistic fuzzy graph have been analysed. Definition 3.32 Let A = {< a µij, a νij >} m n and B = {< b µij, b νij >} n p be two intuitionistic fuzzy matrices. Then the two types of product of A and B are defined as 1) max min product of IF matrices : A max min B = {< c µij, c νij >} m p = [< max(min(a µij, b µjk )), min(max(a νij, b νjk )) >], 1 i m, 1 j p, 1 k n and 2) min max product of IF matrices: A min max B = {< c µij, c νij >} m p = [< min(max(a µij, b µjk )), max(min(a νij, b νjk )) >], 1 i m, 1 j p, 1 k n.

7 526 7 Definition 3.33 Let A = {< a µij, a νij >} m n be intuitionistic fuzzy matrix and k is a positive integer. Then the k th power of an intuitionistic fuzzy matrix is denoted by A k and is defined as max min product of k copies of an intuitionistic fuzzy matrix A. Definition 3.34 [5] Let G = (V, E) be an intuitionistic fuzzy graph. The index matrix representation of intuitionistic fuzzy graph(imifg) is of the form [V, E V V ] where V = {v 1, v 2,..., v n } and v 1 v 2... v n v 1 < µ 11, ν 11 > < µ 12, ν 12 >... < µ 1n, ν 1n > v 2 < µ 21, ν 21 > < µ 22, ν 22 >... < µ 2n, ν 2n > E = {< µ ij, ν ij >} m n =.... v n < µ n1, ν n1 > < µ n2, ν n2 >... < µ nn, ν nn > where < µ ij, ν ij > [0, 1] [0, 1](1 i, j n), the edge between two vertices v i and v j is indexed by < µ ij, ν ij >. Note 3.4 Index matrix representation of any intuitionistic fuzzy graph is an intuitionistic fuzzy matrix. Definition 3.35 Let G = (V, E) be an intuitionistic fuzzy graph where V = {v 1, v 2,..., v n }. The incidence matrix of an intuitionistic fuzzy graph G is B = {< b µij, b νij >} n m, where n and m represents the number of vertices and number of edges of G respectively, whose entries of B are as follows: < µ(e j ), ν(e j ) >, if an edge e j is incident on the vertex v i B = {< b µij, b νij >} n m = < 0, 1 >, otherwise It can also be represented in the matrix form, e 1 e 2... e n v 1 < µ(e 1 ), ν(e 1 ) > < µ(e 2 ), ν(e 2 ) >... < µ(e n ), ν(e n ) > v 2 < µ(e 1 ), ν(e 1 ) > < µ(e 2 ), ν(e 2 ) >... < µ(e n ), ν(e n ) > B = {< b µij, b νij >} n m =.... v n < µ(e 1 ), ν(e 1 ) > < µ(e 2 ), ν(e 2 ) >... < µ(e n ), ν(e n ) >

8 527 8 where < µ(e j ), ν(e j ) > [0, 1] [0, 1]. Definition 3.36 Let G = (V, E) be an intuitionistic fuzzy graph, where V = {v 1, v 2,..., v n } and E = {e 1, e 2,..., e k }. Then the line intuitionistic fuzzy graph is denoted by G L = (V L, E L ), where the vertices of G L are in one-one correspondence with the edges of G and there exist an edge between the vertices of G L if and if only if the corresponding edges of G are adjacent. The membership and non-membership value of V L and E L are defined as follows: µ L (v i ) = µ(e i ) and ν L (v i ) = ν(e i ), e i E. min(µ L (v i ), µ L (v j )), if e i µ L (v i, v j ) = (0, 1), otherwise max(ν L (v i ), ν L (v j )), if e i ν L (v i, v j ) = (0, 1), otherwise and e j are adjacent in G and e j are adjacent in G Example 3.1 Consider an intuitionistic fuzzy graph, G = (V, E), such that V = {v 1, v 2, v 3, v 4 }, E = {(v 1, v 2 ), (v 1, v 3 ), (v 2, v 3 ), (v 3, v 4 ), (v 4, v 1 )} and V L = {v 12, v 23, v 34, v 14, v 13 } and E L = {(v 12, v 23 ), (v 23, v 34 ), (v 34, v 14 ), (v 14, v 12 ), (v 12, v 13 ), (v 14, v 13 ), (v 13, v 23 ), (v 13, v 34 )} Figure 1: G and G L Definition 3.37 Let G = (V, E) be an intuitionistic fuzzy graph with the underlying crisp graph G = (V, E). Then the subdivision of an intuitionistic fuzzy graph G is denoted by G sd = (V sd, E sd ) and is obtained by adding a new vertex u k into every edge e ij = (v i, v j ) E of G such that the membership and the non-membership of the vertex v k and the edges v i u k and u k v j are defined as follows: µ sd (u k ) = µ ij and ν sd (u k ) = ν ij, u k V sd µ sd (v i, u k ) min(µ sd (v i ), µ sd (u k )) and ν sd (v i, u k ) max(ν sd (v i ), ν sd (u k )) µ sd (u k, v j ) min(µ sd (u k ), µ sd (v j )) and ν sd (u k, v j ) max(ν sd (u k ), ν sd (v j )), v i, v j, u k V sd.

9 528 9 Example 3.2 Consider an intuitionistic fuzzy graph, G = (V, E), such that V = {v 1, v 2, v 3, v 4, v 5 }, E = {(v 1, v 2 ), (v 1, v 5 ), (v 2, v 3 ), (v 2, v 4 ), (v 3, v 4 ), (v 4, v 5 )} and V sd = {v 1, v 2, v 3, v 4, v 5, u 1, u 2, u 3, u 4, u 5 } and E sd = {(v 1, u 1, (u 1, v 2 ), (v 2, u 2 ), (u 1, v 3 ), (v 3, u 3 ), (u 3, v 4 ), (v 4, u 6 ), (u 6, v 2 ), (v 4, u 4 ), (u 4, v 5 ), (v 5, u 5 ), (u 5, v 1 ) Figure 2: G and G sd Definition 3.38 Let G = (V, E) be an intuitionistic fuzzy graph with the underlying crisp graph G = (V, E) where V = {v 1, v 2,..., v n }. Then the power of an intuitionistic fuzzy graph G is denoted by, G k = (V k, E k ), where V k = V and the vertices v i and v j are adjacent in G k if and only if d G (v i, v j ) k (Refer Definition 1.4). The membership and non -membership values of the edges of G k are defined as follows: (min(µ i, µ j ), max(ν i, ν j )), if d G (v i, v j ) k (µ k (v i, v j ), ν k (v i, v j )) = (0, 1), otherwise Example 3.3 Consider an intuitionistic fuzzy graph, G = (V, E), such that V = {v 1, v 2, v 3, v 4, v 5 }, E = {(v 1, v 2 ), (v 2, v 3 ), (v 3, v 4 ), (v 4, v 5 )}, E 2 = {(v 1, v 2 ), (v 2, v 3 ), (v 3, v 4 ), (v 4, v 5 ), (v 1, v 3 ), (v 3, v 5 ), (v 2, v 4 )}, E 3 = {(v 1, v 2 ), (v 2, v 3 ), (v 3, v 4 ), (v 4, v 5 ), (v 1, v 3 ), (v 3, v 5 ), (v 2, v 4 ), (v 1, v 4 ), (v 2, v 5 )} and E 4 = {(v 1, v 2 ), (v 2, v 3 ), (v 3, v 4 ), (v 4, v 5 ), (v 1, v 3 ), (v 3, v 5 ), (v 2, v 4 ), (v 1, v 4 ), (v 2, v 5 ), (v 1, v 5 )} Figure 3: G and G 2

10 Figure 4: G 3 and G 4 It can also be represented in the matrix form, v 1 v 2 v 3 v 4 v 5 v 1 < 0, 1 > <.6,.3 > < 0, 1 > < 0, 1 > < 0, 1 > v 2 <.6,.3 > < 0, 1 > <.1,.4 > < 0, 1 > < 0, 1 > (i, j) th entry of A = v 3 < 0, 1 > <.1,.4 > < 0, 1 > <.1,.6 > < 0, 1 > v 4 < 0, 1 > < 0, 1 > <.1,.6 > < 0, 1 > <.3, 5 > v 5 < 0, 1 > < 0, 1 > < 0, 1 > <.3,.5 > < 0, 1 > v 1 v 2 v 3 v 4 v 5 v 1 <.6,.3 > < 0, 1 > <.1,.4 > < 0, 1 > < 0, 1 > v 2 < 0, 1 > <.6,.3 > < 0, 1 > <.1,.6 > < 0, 1 > (i, j) th entry of A 2 = v 3 <.1,.4 > < 0, 1 > <.1,.4 > < 0, 1 > <.1,.6 > v 4 < 0, 1 > <.1,.6 > < 0, 1 > <.3,.5 > < 0, 1 > v 5 < 0, 1 > < 0, 1 > <.1,.6 > < 0, 1 > <.3,.5 > Theorem 3.5 Let G = (V, E) be a intuitionistic fuzzy graph. If an intuitionistic fuzzy graph G contains a u v walk of length l, then G contains a u v path of length l. Proof. Proof follows from the Definition 1.8 and Theorem 2.2. Theorem 3.6 Let G = (V, E) be a strong intuitionistic fuzzy graph and G sd = (V sd, E sd ) be the subdivision

11 of an intuitionistic fuzzy graph G. Then S µ (G sd ) 2S µ (G) and S ν (G sd ) 2S ν (G), where S µ and S ν are size of G. Proof. Let G = (V, E) be a strong intuitionistic fuzzy graph and G sd = (V sd, E sd ) be the subdivision of an intuitionistic fuzzy graph G. The size of G is S(G) = (S µ (G), S ν (G)), where Consider, S µ (G sd ) = e ik, e kj E sd (µ sd (e ik ) + µ sd (e kj )), µ sd (v k ) + µ sd (v k ), since G is strong µ ij + µ ij, since by Definition µ ij 2S µ (G). and S ν (G sd ) = (ν sd (e ik ) + ν sd (e kj )) e ik, e kj E sd ν sd (v k ) + ν sd (v k ), since G is strong ν ij + ν ij, since by Definition ν ij 2S ν (G). Theorem 3.7 Let G 1 = (V 1, E 1 ) and G 2 = (V 2, E 2 ) be two strong IFGs. Then A(G 1 G 2 ) = A(G 1 ) + A(G 2 ) if and only if V 1 = V 2. Proof. Let us assume that A(G 1 G 2 ) = A(G 1 ) + A(G 2 ). Suppose that G 1 and G 2 are having different vertex set and let V 1 = {v 1, v 2,..., v m } and V 2 = {u 1, u 2,..., u n }, where v i u j, i = 1, 2,..., m, j = 1, 2,..., n, i and j Case(i):Let m n. Then V (G 1 G 2 ) = {v 1, v 2,..., v m, u 1, u 2,..., u n } and the order of the IF matrix A(G 1 G 2 ) = m + n m + n. But the order of the IF matrix A(G 1 ) = m m and the order of the IF matrix A(G 2 ) = n n. Therefore A(G 1 ) + A(G 2 ) is not possible, since by Definition 1.31, the order of the matrices A(G 1 ) and A(G 2 ) are not equal. Hence by Definition 1.30, A(G 1 G 2 ) A(G 1 ) + A(G 2 ), which is a contradiction to our assumption that A(G 1 G 2 ) = A(G 1 ) + A(G 2 ). Hence V 1 = V 2. Case(ii): Let m = n. Then V (G 1 G 2 ) = V 1 V 2 = {v 1, v 2,..., v m, u 1, u 2,..., u m } and the order of the matrix A(G 1 G 2 ) = 2m 2m. But the order of the IF matrix A(G 1 ) = m m, A(G 2 ) = m m. Therefore the order of the IF matrix A(G 1 ) + A(G 2 ) = m m A(G 1 G 2 ), which is contradiction to our assumption that A(G 1 G 2 ) = A(G 1 ) + A(G 2 ). Hence V 1 = V 2. Conversely, Let us assume that V 1 = V 2. and let V 1 = {v 1, v 2,..., v m } and V 2 = {v 1, v 2,..., v m }. Then the order of IF matrix A(G 1 G 2 ) = m m = A(G 1 ) + A(G 2 ). Next, in order to prove the entries of the IF matrix A(G 1 G 2 ) = the entries of the IF matrix A(G 1 ) + A(G 2 ), we need to consider the following three subcases: Subcase(i): Let (v i, v j ) E 1 E 2

12 By Definition 1.26, (µ µ )(v i, v j ) = max(µ ij, µ ij) (ν ν )(v i, v j ) = min((ν i ν i), (ν j ν j)) = min(min(ν i, ν i), min(ν j, ν j)) = min(min(ν i, ν j), min(ν i, ν j)), Since G 1 &G 2 are strong IFGs (ν ν )(v i, v j ) = min(ν ij, ν ij) Therefore, (i, j) th entry of A(G 1 G 2 ) = (max(µ ij, µ ij), min(ν ij, ν ij)) (1) By Definition 1.31 and 1.34, (i, j) th entry of A(G 1 ) + A(G 2 ) = (max(µ ij, µ ij), min(ν ij, ν ij)) (2) From Equation (1) and (2), (i, j) th entry of A(G 1 G 2 ) = (i, j) th entry of A(G 1 ) + A(G 2 ) Subcase(ii): Let e ij E 1 E 2. Then by Definition 1.26, (µ µ )(v i, v j ) = µ ij and (ν ν )(v i, v j ) = ν ij. Therefore, (i, j) th entry of A(G 1 G 2 ) = (µ ij, ν ij ) (3) If e ij E 1, then (µ ij, ν ij ) (0, 1) and if e ij E 2, then (µ ij, ν ij ) = (0, 1). Therefore, (i, j) th entry of A(G 1 ) + A(G 2 ) = (max(µ ij, µ ij), min(ν ij, ν ij)) = (µ ij, ν ij ) (4) From Equation (3) and (4), (i, j) th entry of A(G 1 G 2 ) = (i, j) th entry of A(G 1 ) + A(G 2 ). Subcase(iii): Let e ij E 2 E 1. Then proof follows from the Subcase(ii) by replacing E 1 by E 2 and E 2 by E 1. Hence, from the Subcases (i) (iii), it follows that (i, j) th entry of A(G 1 G 2 ) = (i, j) th entry of A(G 1 ) + A(G 2 ). Theorem 3.8 Let G = (V, E) be an IFG and let A = {< µ ij, ν ij >} be the index matrix of G. Then for each positive integer k, the (i, j) th entry of A k = strength of connectedness of v i v j walks of length k. (5) Proof. Let G = (V, E) be an IFG and let A = {< µ ij, ν ij >} be the index matrix of G and the vertex set V = {v 1, v 2,..., v n }. Let A k = {< b µij, b νij >} be the k th power of the IF matrix A. Let us prove the Equation (1) by mathematical induction method on the power of A. Initial Step: Let k = 1, then A k = A. Then A = {< µ ij, ν ij >}, where µ ij is µ strength of connectedness of (v i, v j ) walk of length 1 and ν ij is ν strength of connectedness of (v i, v j ) walk of length 1 = (CONN µ(g) (v i, v j ), CONN ν(g) (v i, v j )), where (v i, v j ) is v i v j walk of length 1. Inductive Step: Assume that the result is true for k. By the inductive hypothesis, (i, j) th entry of A k = (CONN µ(g) (v i, v j ), CONN ν(g) (v i, v j )), where (v i, v j ) is v i v j walk of length k. Next we have to prove the result for k + 1. (i, j) th entry of A k+1 = (i, j) th entry

13 of A k max min (i, j) th entry of A = ( n l=1 (a µ bµlj il ), n l=1 (a ν bνlj il ) = (max(min(conn µ(g) (v i, v l ), CONN µ(g) (v l, v j ))), min(max(conn ν(g) (v i, v l ), CONN ν(g) (v l, v j )))), where (v i, v l ) is v i v l walk of length k and (v l, v j ) is v l v j walk of length 1 = (CONN µ(g) (v i, v j ), CONN ν(g) (v i, v j )), where (v i, v j ) is v i v j walks of length k + 1, v l V. Hence the result is true for every k. That is, (i, j) th entry of A k = strength of connectedness of v i v j walks of length k. Theorem 3.9 Let G = (V, E) be an intuitionistic fuzzy graph, where V = {v 1, v 2,..., v n } be the vertices of G. Let A = {< µ ij, ν ij >} be the index matrix of G. Then (CONN µ(g) (v i, v j ), CONN ν(g) (v i, v j )) = (i, j) th entry of A + A A n 1, v i v j V and (CONN µ(g) (v i, v i ), CONN ν(g) (v i, v i )) = (i, i) th entry of A + A A n, v i V Proof. Let G be an IFG, where V = {v 1, v 2,...v n } be the vertices of G. Let A be the index matrix of G and A k be the power of IF matrix A. By Theorem 3.8, (i, j) th entry of A k = (CONN µ(g) (v i, v j ), CONN ν(g) (v i, v j )), where (v i, v j ) is v i v j walks of length k. Case(i): Let v i v j. By Theorem 2.2, v i v j walk of length k contains v i v j path of length k. Since the vertex set V has n vertices, the v i v j path passes through at most n vertices. Therefore (v i, v j ) is path of length less than or equal to n 1 Hence, (i, j) th entry of A + A A n 1 = (max(conn µ(g) (v i, v j )), min(conn ν(g) (v i, v j ))) where (v i, v j ) is v i v j path of length less than or equal to n 1 = (max(max(s µ(g) (v i, v j )), min(min(s ν(g) (v i, v j ))) = (CONN µ(g) (v i, v j ), CONN ν(g) (v i, v j )) Therefore, (CONN µ(g) (v i, v j ), CONN ν(g) (v i, v j )) = (i, j) th entry of A + A A n 1, v i v j V. Case(ii): Let v i = v j V. Since the vertex set V has n vertices, the closed v i v i path passes through at most n vertices. Therefore (v i, v i ) is v i v i path of length less than or equal to n. Hence, (i, i) th entry of A + A A n = (max(conn µ(g) (v i, v i )), min(conn ν(g) (v i, v i ))) = (max(max(s µ(g) (v i, v i ))), min(min((s ν)g) (v i, v i ))))) = (CONN µ(g) (v i, v i ), CONN ν(g) (v i, v i )) where (v i, v i ) is v i v j paths of length less than or equal to n (CONN µ(g) (v i, v i ), CONN ν(g) (v i, v i )) = (i, i) th entry of A + A A n, v i V. Theorem 3.10 Let G = (V, E) be a strong intuitionistic fuzzy graph and A be the index matrix of G. Let C k = {< c µij, c νij >} = A + A A k and C k 1 = {< c µ ij, c ν ij >} = A + A A k 1. Then G is connected and G k is complete if and only if {< c µij, c νij >} =< 0, 1 >, for every i and j {< c µ ij, c ν ij >} =< 0, 1 >, for some i and j

14 Proof. Let G = (V, E) be a strong intuitionistic fuzzy graph and A be the index matrix of G. Let A k be the k th power of the IF matrix A. Let G k be a complete intuitionistic fuzzy graph. Then by Theorem 3.8: (i, j) th entry of A = (CONN µ(g) (v i, v j ), CONN ν(g) (v i, v j )), where (v i, v j ) is v i v j walk of length 1 (i, j) th entry of A 2 = (CONN µ(g) (v i, v j ), CONN ν(g) (v i, v j )), where (v i, v j ) is v i v j walk of length 2... (i, j) th entry ofa k 1 = (CONN µ(g) (v i, v j ), CONN ν(g) (v i, v j )), where (v i, v j ) is v i v j walk of length k 1 (i, j) th entry of A k = (CONN µ(g) (v i, v j ), CONN ν(g) (v i, v j )), where (v i, v j ) Therefore, is v i v j walk of length k (i, j) th entry of A + A A k 1 = (max(conn µ(g) (v i, v j )), min(conn ν(g) (v i, v j ))), (i, j) th entry of A + A A k 1 = (CONN µ(g) (v i, v j ), CONN ν(g) (v i, v j ))) (6) where (v i, v j ) is v i v j walk of length less than or equal to k 1 and (i, j) th entry of A + A A k = (max(conn µ(g) (v i, v j )), min(conn ν(g) (v i, v j ))), (i, j) th entry of A + A A k = (CONN µ(g) (v i, v j ), CONN ν(g) (v i, v j ))) (7) where (v i, v j ) is v i v j walk of length k. From Equation (2), (i, j) th entry of A + A A k 1 is the strength of connectedness of (v i, v j ) walk of length 1, 2,..., k 1 except k. Therefore, {< c µ ij, c ν ij >} = (CONN µ(g) (v i, v j ), CONN ν(g) (v i, v j )) < 0, 1 >, (v i, v j ) is walk of length 1, 2,..., k 1 < 0, 1 >, (v i, v j )is walk of length k (8) Since G k is complete, the maximum shortest path of length in G k is k. Then there exist at least one path of length k in G k. Therefore from Equation (6), (7) and (8), (i, j) th entry of A + A A k = (CONN µ(g) (v i, v j ), CONN ν(g) (v i, v j ))) < 0, 1 >, (9) where (v i, v j ) is walk of length k, i, j. Hence from Equation (8) and (9), {< c µij, c νij >} =< 0, 1 >, for every i and j {< c µ ij, c ν ij >} =< 0, 1 >, for some i and j (10)

15 Conversely, suppose that Equation (10) is true, then for each distinct pair i, j we have C k 0. Therefore, there exist at least one walk of length less than n from v i to v j. This implies that v i is connected to v j, for every v i, v j V. Hence G is connected. Again, let us assume that Equation (10) is true, then there exists v i v j walk of length k and shortest path of length k in G. By the Definition 1.38 and Theorem 2.2, µ k ij = min(µ i, µ j ) and ν k ij = max(ν i, ν j ), (v i, v j ) Eand d G (v i, v j ) k (11) Since G is strong intuitionistic fuzzy graph and from the Equation(11), µ k ij = min(µ i, µ j ) and ν k ij = max(ν i, ν j ), v i, v j V Hence G k is complete. Corollary 3.11 Let G = (V, E) be an intuitionistic fuzzy graph and A be the index matrix of G. Let C k = {< c µij, c νij>} = A + A A k and C k 1 = {< c µ ij, c ν ij >} = A + A A k 1. Then G is connected and (G ) k complete if and only if {< c µij, c νij>} =< 0, 1 >, for every i and j {< c µ ij, c ν ij >} =< 0, 1 >, for some i and j Corollary 3.12 Let G = (V, E) be a strong directed intuitionistic fuzzy graph and A be the index matrix of G. Let C k = {< c µij, c νij >} = A + A A k and C k 1 = {< c µ ij, c ν ij >} = A + A A k 1. Then G is connected and G k is complete if and only if {< c µij, c νij>} =< 0, 1 >, for every i and j {< c µ ij, c ν ij >} =< 0, 1 >, for some i and j Theorem 3.13 Let G and H be two intuitionistic fuzzy graphs. Then G is co-weak isomorphic with H then G k is homomorphic with H k. Proof. Proof follows from the definition 1.23, 1.25 and Theorem 3.14 Let G = (V, E) be an intuitionistic fuzzy graph. Let A = {< µ ij, ν ij >} m m and B = {< b µij, b νij>} m n be the index matrix and incidence matrix of G respectively. Then the entries of B max min B T are {< b µij, b νij >} m m = (µ ij, ν ij ), if i j (max(µ ik ), min(ν ik )), if i = j, v k N G (v i )

16 Proof. Let G = (V, E) be an intuitionistic fuzzy graph. Let A = {< µ ij, ν ij >} and B = {< b µij, b νij >} be the index matrix and incidence matrix of G of order m m and m n respectively. Let B T = {< b µ ij, b ν ij >} n m be the transpose of the matrix B. Then the order of B max min B T is m m. Case(i): Let e k (v i, v j ), i j. By the Definition 1.35, the entries in B are as follows: {< b µij, b νij >} = (µ ij, ν ij ), e k (v i, v j ) (0, 1), otherwise The (i, j) th entries of B max min B T is given by (B max min B T ) ij = ( k (b µik b µ kj ), k (b νik b ν kj )) That is, (B max min B T ) ij = (µ ij, ν ij ), sincee k (v i, v j ) (12) Case(ii): Let (v i, v j ) E of G. Subcase(i): Let v i V is incident on e k E and v j V is not incident on e k E, then (b µik, b νik ) (0, 1) = (µ(e k ), ν(e k )), (b µjk, b νjk ) = (0, 1) and (b µ kj, b ν kj ) = (0, 1). Therefore, (B max min B T ) ij = ( k (b µik b µ kj ), k (b νik b ν kj )) = (0, 1) (13) Subcase (ii): Let v i V is not incident on e k E and v j V is incident on e k E, then (b µik, b νik ) = (0, 1) and (b µjk, b νjk ) (0, 1) = µ(e k ) and (b µ kj, b ν kj ) (0, 1) = µ(e k ). Therefore, (B max min B T ) ij = ( k (b µik b µ kj ), k (b νik b ν kj )) = (0, 1) (14) Subcase(iii): Let v i, v j V is not incident on e k E, then (b µik, b νik ) = (0, 1) and (b µjk, b νjk ) = (0, 1) and (b µ kj, b ν kj ) = (0, 1). Therefore, (B max min B T ) ij = ( k (b µik b µ kj ), (b νik b ν kj )) = (0, 1) Hence from the above three cases, (B max min B T ) ij = ( k (b µik b µ kj ), k (b νik b ν kj )) = (0, 1), (v i, v j ) E (15) Case(iii): Let v i = v j V in G. (B max min B T ) ij = ( k (b µik b µ ki ), k (b νik b ν ki )) = ( k (b µik ), k (b νik )) Hence from Equation (8), (9) and (10) and Case (iii), = ((b µi1 b µi2... b µin ), (b νi1 b νi2... b νin )) = ( k µ ik, k ν ik ), e k E is incident on v i V {< b µij, b νij >} m m = {< µ ij, ν ij >}, if i j (max(µ ik ), min(ν ik )), if i = j, v k N G (v i )

17 Theorem 3.15 Let G = (V, E) be an intuitionistic fuzzy graph and G L = (V L, E L ) be a line intuitionistic fuzzy graph. Let A = {< µ ij, ν ij >} and B = {< b µij, b νij>} be the index matrix and incidence matrix of G respectively. Then the entries of B T max min B are (µ L (v i, v j ), ν L (v i, v j ) >), if i j {< b µij, b νij>} n n (max(µ ik ), min(ν ik )), if i = j, v k N G (v i ) where (µ L (v i, v j ), ν L (v i, v j )) is the membership and non membership value of an edge e ij E L. Proof. Let G = (V, E) be an intuitionistic fuzzy graph. Let A = {< µ ij, ν ij >} and B = {< b µij, b νij>} be the index matrix and incidence matrix of G of order m m and m n respectively. Let B T = {< b µ ij, b ν ij>} n m be the transpose of the matrix B. Then the order of B T max min B is n n. Case(i): Let e i E, e j E are incident on v k V. Then ((b µki, b νki ) (0, 1) = (µ(e i ), ν(e i )) = (b µ ik, b ν ik ), (b µkj, b νkj ) (0, 1) = (µ(e j ), ν(e j )) = (b µ jk, b ν jk ). The (i, j) th entry of B T max min B are as follows: (B max min B T ) ij = ( k (b µ ik b µkj ), k (b ν ik b νkj )) = ( (µ(e i ) µ(e j )), ((ν(e i ) ν(e j )))) = ((µ(e i ) µ(e j )), (ν(e i ) ν(e j ))) = ((µ L (v i ) µ L (v j )), (ν L (v i ) ν L (v j ))), since by Definition 1.36 = (µ L (v i, v j ), ν L (v i, v j )), (v i, v j ) E Case(ii): Let e i E, e j E are not incident on v k V Then ((b µki, b νki ) = (0, 1) = (b µ ik, b ν ik ), ((b µkj, b νkj ) = (0, 1) = (b µ jk, b ν jk ). The (i, j) th entry of B T max min B are as follows: Case(iii): Let v i = v j V in G. (B T max min B) ij = ( k (b µ ik b µki ), k (b ν ik b νki )) = (0, 1) (B T max min B) ij = ( k (b µ ik b µki ), k (b ν ik b νki )) = ( k (b µik ), k (b νik )) = ((b µi1 b µi2... b µin ), (b νi1 b νi2... b νin )) = ( k µ ik, k ν ik ), e k E is incident with v i V Hence from the Cases (i) and (ii) and by Definition 1.36, (B T max min B) ij = (µ L (v i, v j ), ν L (v i, v j )), if i j Hence from the Case (iii), (B T max min B) ij = (max(µ ik ), min(ν ik )), if i = j, v k N G (v i ). IV Conclusion In this paper, we discussed the properties of the power of an intuitionistic fuzzy graph, subdivision intuitionistic fuzzy graph and line intuitionistic fuzzy graph. Intuitionistic fuzzy graph effectively expresses the approximate and interpolative reasoning used by humans when they employ linguistic propositions for deductive reasoning. The authors further extend this work so it can have application in decision making and network analysis.

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