N-structures Applied to Graphs

World Applied Sciences Journal 22 (Special Issue of Applied Math): 1-9, 2013 ISSN 1818-4952 c

IDOSI Publications, 2013 DOI: 10.5829/idosi.wasj.22.am.1.2013

N -structures Applied to Graphs M. Akram1 , Y.B. Jun2 , N.O. Alshehri3 and S. Sarwar1 1. Punjab University College of Information Technology, University of the Punjab, Lahore-54000, Pakistan [email protected], [email protected] 2. Department of Mathematics Education (and RINS), Gyeongsang National University, Chinju 660-701 Republic of Korea [email protected] 3. Department of Mathematics, Faculty of Sciences(Girls), King Abdulaziz University, Jeddah, Saudi Arabia [email protected] Abstract: In this paper, we introduce the notion of N -graphs and describe methods of their construction. We prove that the isomorphism between N -graphs is an equivalence relation (resp. partial order relation). We then introduce the concept of N -line graphs and discuss some of their fundamental properties. Key words: N -graphs, isomorphism, N -line graphs.

INTRODUCTION

to the right nor to the left of some reference object. In 1975, Rosenfeld [2] discussed the concept of fuzzy graphs whose basic idea was introduced by Kauffmann [3] in 1973. The fuzzy relations between fuzzy sets were also considered by Rosenfeld and he developed the structure of fuzzy graphs obtaining analogs of several graph theoretical concepts. Bhattacharya [4] gave some remarks on fuzzy graphs. Akram et al. introduced the concepts of bipolar fuzzy graphs and interval-valued fuzzy line graphs [5-9]. In this paper, we introduce the notion of N -graphs, describe methods of their construction. We prove that the isomorphism between N -graphs is an equivalence relation (resp. partial order relation). We then introduce the concept of N -line graphs and discuss some of their fundamental properties. We have used standard definitions and terminologies in this paper. For other notations, terminologies and applications not mentioned in the paper, the readers are referred to [10-14].

A (crisp) set A in a universe X can be defined in the form of its characteristic function µA : X → {0, 1} yielding the value 1 for elements belonging to the set A and the value 0 for elements excluded from the set A. The most of the generalization of the crisp set have been introduced on the unit interval [0, 1] and they are consistent with the asymmetry observation. In other words, the generalization of the crisp set to fuzzy sets relied on spreading positive information that fit the crisp point {1} into the interval [0, 1]. Because no negative meaning of information is suggested, we now feel a need to deal with negative information. To do so, we also feel a need to supply mathematical tool. To attain such object, Jun et al. [1] have introduced a new function which is called negative-valued function (briefly, N -function) to deal with negative information that fit the crisp point {−1} into the interval [−1, 0], and constructed N -structures. It is important to be able to deal with negative information. It is noted that positive information represents what is granted to be possible, while negative information represents what is considered to be impossible. As an example, let us consider the spatial relations. Human beings consider “left" and “right" as opposite directions. But this does not mean that one of them is the negation of the other. The semantics of “opposite" captures a notion of symmetry rather than a strict complementation. In particular, there may be positions which are considered neither

PRELIMINARIES Recall that a graph is an ordered pair G∗ = (V, E), where V is the set of vertices of G∗ and E is the set of edges of G∗ . Two vertices x and y in an undirected graph G∗ are said to be adjacent in G∗ if {x, y} = xy is an edge of G∗ . A simple graph is an undirected graph that has no loops and no more than one edge between any two different vertices. A subgraph of a graph G∗ = (V, E) is a graph H = (W, F ), where W ⊆ V and F ⊆ E. The complementary graph G∗ of a simple graph has the same 1

World Appl. Sci. J., 22 (Special Issue of Applied Math): 1-9, 2013

vertices as G∗ . Two vertices are adjacent in G∗ if and only if they are not adjacent in G∗ . Consider the Cartesian product G∗ = G∗1 × G∗2 = (V, E) of graphs G∗1 and G∗2 . Then V = V1 × V2 and E = {(x, x2 )(x, y2 )|x1 ∈ V1 , x2 y2 ∈ E2 } ∪{(x1 , z)(y1 , z)|z ∈ V2 , x1 y1 ∈ E1 }. Let G∗1 = (V1 , E1 ) and G∗2 = (V2 , E2 ) be two simple graphs. Then, the composition of graph G∗1 with G∗2 is denoted by G∗1 [G∗2 ] = (V1 × V2 , E 0 ), where E 0 = E ∪ {(x1 , x2 )(y1 , y2 )|x1 y1 ∈ E1 , x2 6= y2 } and E is defined in G∗1 × G∗2 . Note that G∗1 [G∗2 ] 6= G∗2 [G∗1 ]. The union of graphs G∗1 and G∗2 is defined as G∗1 ∪ G∗2 = (V1 ∪ V2 , E1 ∪ E2 ). The join of G∗1 and G∗2 is the simple graph G∗1 + G∗2 = (V1 ∪ V2 , E1 ∪ E2 ∪ E ′ ), where E ′ is the set of all edges joining the nodes of V1 and V2 . In this construction it is assumed that V1 ∩ V2 6= ∅. An isomorphism of the graphs G∗1 and G∗2 is a bijection between the vertex sets of G∗1 and G∗2 such that any two vertices v1 and v2 of G∗1 are adjacent in G∗1 if and only if f (v1 ) and f (v2 ) are adjacent in G∗2 . If an isomorphism exists between two graphs, then the graphs are called isomorphic and we write G∗1 ≃ G∗2 . An automorphism of a graph is a graph isomorphism with itself, i.e., a mapping from the vertices of the given graph G∗ back to vertices of G∗ such that the resulting graph G∗ is isomorphic with G∗ . By an intersection graph of a graph G∗ = (V, E), we mean, a pair P (S) = (S, T ) where S = {S1 , S2 , . . . , Sn } is a family of distinct nonempty subsets of V and T = {Si Sj | Si , Sj ∈ S, Si ∩ Sj 6= ∅, i 6= j}. It is well know that every graph is an intersection graph. By a line graph of a graph G∗ = (V, E), we mean, a pair L(G∗ ) = (Z, W ) where Z = {{x} ∪ {ux , vx } | x ∈ E, ux , vx ∈ V, x = ux vx } and W = {Sx Sy | Sx ∩ Sy 6= ∅, x, y ∈ E, x 6= y}, and Sx = {x} ∪ {ux , vx }, x ∈ E. It is reported in the literature that the line graph is an intersection graph. Denote by F (X, [−1, 0]) the collection of functions from a nonempty set X to [−1, 0]. We say that an element of F (X, [−1, 0]) is a negative-valued function from X to [−1, 0] (briefly, N -function on X). By an N -structure we mean an ordered pair (X, µ) of X and an N -function µ on X. By an N -relation on X we mean an N -function ν on X × X satisfying the following inequality: (∀x, y ∈ X)(ν(x, y) ≥ max{µ(x), µ(y)}),

in V and ν is an N -function in E ⊆ V × V such that ν({x, y}) ≥ max(µ(x), µ(y)) for all {x, y} ∈ E. We call µ the N -vertex function of V, ν the N -edge function of E, respectively. Note that ν is a symmetric N -relation on µ. We use the notation xy for an element {x, y} of E. Thus, G = (µ, ν) is an N -graph of G∗ = (V, E) if ν(xy) ≥ max(µ(x), µ(y)) for all xy ∈ E. Definition 2. Let G = (µ, ν) be an N -graph. P The order of an N -graph is defined by O(G) = x∈V µ(x). The degree of a vertex x in G is defined by deg(x) = P xy∈E ν(xy).

Example 3. Consider a graph G∗ = (V, E) such that V = {x, y, z}, E = {xy, yz, zx}. Let µ be an N function of V and let ν be an N -function of E ⊆ V × V defined by µ

x -0.7

y -0.5

z -0.7

xy -0.3

ν

y

yz -0.2

zx -0.4

z −0.2

−0.5

−0.7

−0.3

−0.4

−0.7

x G (i) By routine computations, it is easy to see that G = (µ, ν) is an N -graph of G∗ . (ii) Order of an N -graph= O(G)=-1.9. (iii) Degree of each vertex in G is deg(x) = −0.7, deg(y) = −0.5, deg(z) = −0.6.

(1)

Definition 4. Let µ1 and µ2 be N -functions of V1 and V2 and let ν1 and ν2 be N -functions of E1 and E2 , respectively. The Cartesian product of two N -graphs G1 and G2 of the graphs G∗1 and G∗2 is denoted by G1 × G2 = (µ1 × µ2 , ν1 × ν2 ) and is defined as follows:

where µ ∈ F (X, [−1, 0]). Throughout this paper, G∗ will be a crisp graph, and G a N -graph. N -STRUCTURES APPLIED TO GRAPHS

• (µ1 × µ2 )(x1 , x2 ) = max(µ1 (x1 ), µ2 (x2 )) for all (x1 , x2 ) ∈ V ,

Definition 1. An N -graph with an underlying set V is defined to be a pair G = (µ, ν) where µ is an N -function 2

World Appl. Sci. J., 22 (Special Issue of Applied Math): 1-9, 2013

• (ν1 × ν2 )((x, x2 )(x, y2 )) = max(µ1 (x), ν2 (x2 y2 )) for all x ∈ V1 , for all x2 y2 ∈ E2 ,

(x1 , x2 )

• (ν1 × ν2 )((x1 , z)(y1 , z)) = max(ν1 (x1 y1 ), µ2 (z)) for all z ∈ V2 , for all x1 y1 ∈ E1 .

−0.2

−0.2

Definition 5. Let G1 and G2 be two N - graphs. The degree of a vertex in G1 × G2 can be defined as follows: for any (x1 , x2 ) ∈ V1 × V2 , X (ν1 × ν2 )(x1 , x2 )(y1 , y2 ) dG1 ×G2 (x1 , x2 ) = X

max(µ1 (x), ν2 (x2 y2 ))

−0.2

G1 × G2

−0.1

−0.1

−0.3

(x1 ,x2 )(y1 ,y2 )∈E

=

(x1 , y2 )

−0.3 −0.2

(y1 , x2 )

(y1 , y2 )

x1 =y1 =x,x2 y2 ∈E2

G1 × G2 is an N - graph of G∗1 × G∗2 . max(µ2 (z), ν1 (x1 y1Clearly, )). (ii) Routine computations give degree of each vertex in x2 =y2 =z,x1 y1 ∈E1 G1 × G2 as Example 6. Let G∗1 = (V1 , E1 ) and G∗2 = (V2 , E2 ) be two graphs, where V1 = {x1 , y1 } and V2 = {x2 , y2 } dG1 ×G2 (x1 , x2 ) = −0.3, dG1 ×G2 (x1 , y2 ) = −0.3, are underlying sets. Let µ1 and µ2 be N -functions of V1 and V2 and let ν1 and ν2 be N -functions of E1 and E2 , dG1 ×G2 (y1 , x2 ) = −0.3, dG1 ×G2 (y1 , y2 ) = −0.3. respectively. We define µ1 : V1 → [−1, 0], µ2 : V2 → Definition 7. Let µ1 and µ2 be N -functions of V1 and [−1, 0], ν1 : E1 → [−1, 0] and ν2 : E2 → [−1, 0] by V2 and let ν1 and ν2 be N -functions of E1 and E2 , reµ1 (x1 ) = −0.2, µ1 (y1 ) = −0.3, spectively. The composition of two N -graphs G1 and G2 of the graphs G∗1 and G∗2 is denoted by G1 [G2 ] = µ2 (x2 ) = −0.35, µ2 (y2 ) = −0.4, (µ1 ◦ µ2 , ν1 ◦ ν2 ) and is defined as follows: ν1 (x1 y1 ) = −0.1, ν2 (x2 y2 ) = −0.2. • (µ1 ◦ µ2 )(x1 , x2 ) = max(µ1 (x1 ), µ2 (x2 )) for all (i) It is easy to see that G1 = (µ1 , ν1 ) and G2 = (µ2 , ν2 ) (x1 , x2 ) ∈ V , are N -graphs of G∗1 and G∗2 , respectively. Routine computations give • (ν1 ◦ ν2 )((x, x2 )(x, y2 )) = max(µ1 (x), ν2 (x2 y2 )) for all x ∈ V1 , for all x2 y2 ∈ E2 , (µ1 × µ2 )(x1 , x2 ) = −0.2, (µ1 × µ2 )(x1 , y2 ) = −0.2, • (ν1 ◦ ν2 )((x1 , z)(y1 , z)) = max(ν1 (x1 y1 ), µ2 (z)) (µ1 × µ2 )(y1 , x2 ) = −0.3, (µ1 × µ2 )(y1 , y2 ) = −0.3, for all z ∈ V2 , for all x1 y1 ∈ E1 . (ν1 × ν2 )((x1 , x2 )(x1 , y2 )) = −0.2, • (ν1 ◦ν2 )((x1 , x2 )(y1 , y2 ))=max(µ2 (x2 ), µ2 (y2 ), ν1 (x1 y1 )) (ν1 × ν2 )((x1 , x2 )(y1 , x2 )) = −0.1, for all z ∈ V2 , for all (x1 , x2 )(y1 , y2 ) ∈ E 0 − E. (ν1 × ν2 )((y1 , x2 )(y1 , y2 )) = −0.2, Note that µ1 ◦ µ2 = µ1 × µ2 on V and ν1 ◦ ν2 = ν1 × ν2 (ν1 × ν2 )((x1 , y2 )(y1 , y2 )) = −0.1. on E. x1 x2 Definition 8. Let G1 and G2 be two N -graphs. The degree of a vertex in G1 [G2 ] can be defined as follows: −0.2 −0.35 for any (x1 , x2 ) ∈ V1 × V2 , X (ν1 ◦ ν2 )(x1 , x2 )(y1 , y2 ) dG1 [G2 ] (x1 , x2 ) = +

X

−0.1

(x1 ,x2 )(y1 ,y2 )∈E

−0.2

=

X

max(µ1 (x), ν2 (x2 y2 ))

x1 =y1 =x,x2 y2 ∈E2 −0.3

+

−0.4

y1

y2

G1

G2

X

max(µ2 (z), ν1 (x1 y1 ))

x2 =y2 =z,x1 y1 ∈E1

+

X

x2 6=y2 ,x1 y1 ∈E1

3

max(µ2 (x2 ), ν1 (x1 y1 )).

World Appl. Sci. J., 22 (Special Issue of Applied Math): 1-9, 2013

Example 9. Let G∗1 = (V1 , E1 ) and G∗2 = (V2 , E2 ) be two graphs, where V1 = {x1 , y1 } and V2 = {x2 , y2 } be underlying sets. Let µ1 and µ2 be N -functions of V1 and V2 and let ν1 and ν2 be N -functions of E1 and E2 , respectively. We define µ1 : V1 → [−1, 0], µ2 : V2 → [−1, 0], ν1 : E1 → [−1, 0] and ν2 : E2 → [−1, 0] by

Clearly, G1 [G2 ] is an N - graph of G∗1 [G∗2 ]. (ii) Routine computations give degree of each vertex in G1 [G2 ] as dG1 [G2 ] (x1 , x2 ) = −0.5, dG1 [G2 ] (x1 , y2 ) = −0.5, dG1 [G2 ] (y1 , x2 ) = −0.5, dG1 [G2 ] (y1 , y2 ) = −0.5.

µ1 (x1 ) = −0.2, µ1 (y1 ) = −0.3,

Definition 10. Let µ1 and µ2 be N -functions of V1 and V2 and let ν1 and ν2 be N -functions of E1 and E2 , respectively. Then union of two N -graphs G1 and G2 of the graphs G∗1 and G∗2 is denoted by G1 ∪G2 = (µ1 ∪µ2 , ν1 ∪ ν2 ) and is defined as follows:

µ2 (x2 ) = −0.35, µ2 (y2 ) = −0.4, ν1 (x1 y1 ) = −0.35, ν2 (x2 , y2 ) = −0.5. (i) It is easy to see that G1 = (µ1 , ν1 ) and G2 = (µ2 , ν2 ) are N -graphs of G∗1 and G∗2 , respectively. Routine computations give

(A) (µ1 ∪ µ2 )(x) = µ1 (x) if x ∈ V1 ∩ V2 , (µ1 ∪ µ2 )(x) = µ2 (x) if x ∈ V2 ∩ V1 , (µ1 ∪ µ2 )(x) = max(µ1 (x), µ2 (x)) if x ∈ V1 ∩ V2 .

(µ1 ◦ µ2 )(x1 , x2 ) = −0.2, (µ1 ◦ µ2 )(x1 , y2 ) = −0.2, (µ1 ◦ µ2 )(y1 , x2 ) = −0.3, (µ1 ◦ µ2 )(y1 , y2 ) = −0.3,

(B) (ν1 ∪ ν2 )(xy) = ν1 (xy) if xy ∈ E1 ∩ E2 , (ν1 ∪ ν2 )(xy) = ν2 (xy) if xy ∈ E2 ∩ E1 , (ν1 ∪ν2 )(xy) = max(ν1 (xy), ν2 (xy)) if xy ∈ E1 ∩ E2 .

(ν1 ◦ ν2 )((x1 , x2 )(x1 , y2 )) = −0.2, (ν1 ◦ ν2 )((x1 , x2 )(y1 , x2 )) = −0.1, (ν1 ◦ ν2 )((y1 , x2 )(y1 , y2 )) = −0.2,

Definition 11. Let G1 and G2 be two N -graphs. The degree of a vertex in G1 ∪ G2 can be defined as follows:

(ν1 ◦ ν2 )((x1 , y2 )(y1 , y2 )) = −0.1. (ν1 ◦ ν2 )((x1 , x2 )(y1 , y2 )) = −0.2, (ν1 ◦ ν2 )((y1 , x2 )(x1 , y2 )) = −0.2. x1

x2

−0.2

−0.35

−0.1

−0.2

Case 1: When x ∈ V1 or x ∈ V2 but not Pin both. If x ∈ V1 , then dG1 ∪G2 (x) = xy∈E1 ν1 (xy), P If x ∈ V2 , then dG1 ∪G2 (x) = xy∈E2 ν2 (xy). Case 2: When x ∈ V1 ∩ V2 but no edge incident at x lies in E1 ∩ E2 . dG1 ∪G2 (x) = dG1 (x) + dG2 (x). Case 3: When x ∈ V1 ∪ V2 and some edges incident at x lies in E1 ∩ E2 . P dG1 ∪G2 (x) = dG1 (x) + dG2 (x)xy∈E1 ∩E2 max(ν1 (xy), ν2 (xy).

−0.3

−0.4

y1

y2

G1

G2

(x1 , x2 )

(x1 , y2 )

Example 12. Let G∗1 = (V1 , E1 ) and G∗2 = (V2 , E2 ) be two graphs, where V1 = {a, b, c, d, e} and V2 = {a, b, c, d, f } be underlying sets. Let µ1 and µ2 be N functions of V1 and V2 and let ν1 and ν2 be N -functions of E1 and E2 , respectively. We define µ1 : V1 → [−1, 0], µ2 : V2 → [−1, 0], ν1 : E1 → [−1, 0] and ν2 : E2 → [−1, 0] by

−0.2 −0.2

−0.2

−0.1

−0.1 −

2 0.

0. 2

−

µ1 (a) = −0.2, µ1 (b) = −0.1, µ1 (c) = −0.3, µ1 (d) = −0.3, µ1(e) = −0.4,

−0.3

−0.3

µ2 (a) = −0.1, µ2 (b) = −0.2, µ2 (c) = −0.3,

−0.2

(y1 , x2 )

(y1 , y2 )

µ2 (d) = −0.5, µ2 (f ) = −0.4,

G1 [G2 ]

ν1 (ab) = −0.1, ν1 (bc) = −0.1, ν1 (ce) = −0.2, 4

World Appl. Sci. J., 22 (Special Issue of Applied Math): 1-9, 2013

ν1 (be) = −0.1, ν1 (ad) = −0.1, ν1 (de) = −0.2,

As a consequence of above propositions, we obtain

ν2 (ab) = −0.1, ν2 (bc) = −0.1,

Proposition 16. Let G1 = (µ1 , ν1 ) and G2 = (µ2 , ν2 ) be N -graphs of the graphs G∗1 and G∗2 and let V1 ∩ V2 = ∅ . Then join G1 +G2 = (µ1 +µ2 , ν1 +ν2 ) is an N -graph of G∗ if and only if G1 = (µ1 , ν1 ) and G2 = (µ2 , ν2 ) are N -graphs of the graphs G∗1 and G∗2 , respectively.

ν2 (cf ) = −0.2, ν2 (bd) = −0.2, ν2 (bf ) = −0.1. It is easy to see that G1 = (µ1 , ν1 ) and G2 = (µ2 , ν2 ) are N -graphs of G∗1 and G∗2 , respectively. Routine computations give

We now discuss isomorphism of N -graphs.

(µ1 ∪ µ2 )(a) = −0.1, (µ1 ∪ µ2 )(b) = −0.1,

Definition 17. Let G1 = (µ1 , ν1 ) and G2 = (µ2 , ν2 ) be N -graphs. A homomorphism f : G1 → G2 is a mapping f : V1 → V2 such that

(µ1 ∪ µ2 )(c) = −0.3, (µ1 ∪ µ2 )(d) = −0.3, (µ1 ∪ µ2 )(e) = −0.4, (µ1 ∪ µ2 )(f ) = −0.4, (ν1 ∪ ν2 )(ab) = −0.1, (ν1 ∪ ν2 )(bc) = −0.1,

(i) µ1 (x1 ) ≥ µ2 (f (x1 )),

(ν1 ∪ ν2 )(ce) = −0.2, (ν1 ∪ ν2 )(be) = −0.1,

(ii) ν1 (x1 y1 ) ≥ ν2 (f (x1 )f (y1 ))

(ν1 ∪ ν2 )(ad) = −0.1, (ν1 ∪ ν2 )(de) = −0.2,

for all x1 ∈ V1 , x1 y1 ∈ E1 . A bijective homomorphism with the property

(ν1 ∪ ν2 )(bd) = −0.2, (ν1 ∪ ν2 )(de) = −0.1. Clearly, (µ1 ∪µ2 , ν1 ∪ν2 ) is an N - graph of G∗1 ∪G∗2 .

(iii) µ1 (x1 ) = µ2 (f (x1 ))

Definition 13. Let µ1 and µ2 be N -functions of V1 and V2 and let ν1 and ν2 be N -functions of E1 and E2 , respectively. Then join of two N -graphs G1 and G2 of the graphs G∗1 and G∗2 is denoted by G1 + G2 = (µ1 + µ2 , ν1 + ν2 ) and is defined as follows:

is called a strong isomorphism. A strong isomorphism preserves the weights of the nodes but not necessarily the weights of the arcs. A bijective homomorphism preserving the weights of the arcs but not necessarily the weights of nodes, i.e., a bijective homomorphism f : G1 → G2 such that

• (µ1 + µ2 )(x) = (µ1 ∪ µ2 )(x) if x ∈ V1 ∪ V2 ,

(iv) ν1 (x1 y1 ) = ν2 (f (x1 )f (y1 ))

• (ν1 + ν2 )(xy) = (ν1 ∪ ν2 )(xy) = ν1 (xy) if xy ∈ E1 ∪ E2 ,

for all x1 y1 ∈ V1 is called a strong co-isomorphism. A bijective mapping f : G1 → G2 satisfying (iii) and (iv) is called an isomorphism.

′

• (ν1 + ν2 )(xy) = max(µ1 (x), µ2 (y)) if xy ∈ E . Proposition 14. If G1 and G2 are the N -graphs, then G1 × G2 , G1 [G2 ], G1 ∪ G2 and G1 + G2 are N -graphs.

Proposition 18. An isomorphism between N -graphs is an equivalence relation.

We formulate the following characterizations. Proposition 15. Let G1 = (µ1 , ν1 ) and G2 = (µ2 , ν2 ) be N -graphs of the graphs G∗1 and G∗2 and let V1 ∩ V2 = ∅. Then union G1 ∪ G2 = (µ1 ∪ µ2 , ν1 ∪ ν2 ) is an N -graph of G∗ if and only if G1 = (µ1 , ν1 ) and G2 = (µ2 , ν2 ) are N -graphs of the graphs G∗1 and G∗2 , respectively.

Proof. The reflexivity and symmetry are obvious. To prove the transitivity, we let f : V1 → V2 and g : V2 → V3 be the isomorphisms of G1 onto G2 and G2 onto G3 , respectively. Then g ◦ f : V1 → V3 is a bijective map from V1 to V3 , where (g ◦ f )(x1 ) = g(f (x1 )) for all x1 ∈ V1 . Since a map f : V1 → V2 defined by f (x1 ) = x2 for x1 ∈ V1 is an isomorphism, so we have

Proof. Suppose that G1 ∪ G2 is an N -graph. Let xy ∈ E1 . Then xy ∈ / E2 and x, y ∈ V1 − V2 . Thus

µ1 (x1 ) = µ2 (f (x1 )) = µ2 (x2 ) for all x1 ∈ V1 · · · (A),

ν1 (xy)

= (ν1 ∩ ν2 )(xy)

ν1 (x1 y1 ) =

≥ max((µ1 ∩ µ2 )(x), (µ1 ∩ µ2 )(y))

=

ν2 (f (x1 )f (y1 )) ν2 (x2 y2 ) for all x1 y1 ∈ E1 · · · (B).

= max(µ1 (x), µ1 (y)). Since a map g : V2 → V3 defined by g(x2 ) = x3 for x2 ∈ V2 is an isomorphism, so

This shows that G1 = (µ1 , ν1 ) is an N -graph. Similarly, we can show that G2 = (µ2 , ν2 ) is an N -graph. The converse part is obvious.

µ2 (x2 ) = µ3 (g(x2 )) = µ3 (x3 ) for all x2 ∈ V2 · · · (C), 5

World Appl. Sci. J., 22 (Special Issue of Applied Math): 1-9, 2013

ν2 (x2 y2 )

= ν3 (g(x2 )g(y2 ))

(iii)

= ν3 (x3 y3 ) for all x2 y2 ∈ E2 · · · (D). ν(xy) =

From (A), (C) and f (x1 ) = x2 , x1 ∈ V1 , we have µ1 (x1 )

(

0 max(ν(x), ν(y))

if ν(xy) > 0, if if ν(xy) = 0.

= µ2 (f (x1 )) = µ2 (x2 )

Definition 21. An N -graph G is called self complemen= µ3 (g(x2 )) = µ3 (g(f (x1 ))), for all x1 ∈ V1 , tary if G ≈ G.

From (B) and (D), we have ν1 (x1 y1 ) =

The following propositions are obvious. ν2 (f (x1 )f (y1 )) = ν2 (x2 y2 )

=

ν3 (g(x2 )g(y2 ))

=

ν3 (g(f (x1 ))g(f (y1 )))

Proposition 22. Let G be a self complementary N graph. Then X

for all x1 y1 ∈ E1 .

x6=y

Therefore, g ◦ f is an isomorphism between G1 and G3 . This completes the proof.

1X max(µ(x), µ(y)). 2 x6=y

Proposition 23. Let G be an N -graph. If ν(xy) = max(µ(x), µ(y)) for all x, y ∈ V , then G is self complementary.

Proposition 19. A weak isomorphism (co-isomorphism) between N -graphs is a partial ordering relation.

Proposition 24. Let G1 and G2 be N -graphs. Then G1 ∼ = G2 if and only if G1 ∼ = G2 .

Proof. The reflexivity and transitivity are obvious. To prove the anti symmetry, we let f : V1 → V2 be a strong isomorphism of G1 onto G2 . Then f is a bijective map defined by f (x1 ) = x2 for all x1 ∈ V1 satisfying

Proof. Assume that G1 and G2 are isomorphic, there exists a bijective map f : V1 → V2 satisfying ν1 (x) = µ2 (f (x)) for all x ∈ V1 ,

µ1 (x1 ) = µ2 (f (x1 )) for all x1 ∈ V1 ,

ν1 (xy) = µ2 (f (x)f (y)) for all xy ∈ E1 .

ν1 (x1 y1 ) ≥ ν2 (f (x1 )f (y1 )) for all x1 y1 ∈ E1 · · · (E).

By definition of complement, we have

Let g : V2 → V1 be a strong isomorphism of G2 onto G1 . Then g is a bijective map defined by g(x2 ) = x1 for all x2 ∈ V2 satisfying

ν 1 (xy) = max(µ1 (x), µ1 (y) = max(µ2 (f (x)), µ2 (f (y))) = µ2 (f (x)f (y)) for all xy ∈ E1 . Hence G1 ∼ = G2 . The proof of converse part is straightforward. This completes the proof.

µ2 (x2 ) = µ1 (g(x2 )) for all x2 ∈ V2 , ν2 (x2 y2 ) ≥ ν1 (g(x2 )g(y2 )) for all x2 y2 ∈ E2 · · · (F ).

Proposition 25. Let G1 and G2 be N -graphs. If there is a strong isomorphism between G1 and G2 , then there is a strong isomorphism between G1 and G2 .

The inequalities (E) and (F) hold on the finite sets V1 and V2 only when G1 and G2 have the same number of edges and the corresponding edges have same weight. Hence G1 and G2 are identical. Therefore, g ◦ f is a strong isomorphism between G1 and G3 . This completes the proof.

Proof. Let f be a strong isomorphism between G1 and G2 , then f : V1 → V2 is a bijective map that satisfies f (x1 ) = x2 for all x1 ∈ V1 ,

Definition 20. The complement of a weak negativevalued fuzzy graph G = (µ, ν) of G∗ = (V, E) is a weak N -graph G = (µ, ν) on G∗ , is defined by

µ1 (x1 ) = µ2 (f (x1 )) for all x1 ∈ V1 , µ1 (x1 y1 ) ≥ µ2 (f (x1 )f (y1 )) for all x1 y1 ∈ E1 . Since f : V1 → V2 is a bijective map, f −1 : V2 → V1 is also bijective map such that f −1 (x2 ) = x1 for all x2 ∈ V2 . Thus

(i) V = V, (ii) µ(x) = µ(x)

ν(xy) =

for all x ∈ V,

µ1 (f −1 (x2 )) = µ2 (x2 ) for all x2 ∈ V2 . 6

World Appl. Sci. J., 22 (Special Issue of Applied Math): 1-9, 2013

Example 30. Consider a graph G∗ = (V, E) such that V = {v1 , v2 , v3 , v4 } and E = {x1 = v1 v2 , x2 = v2 v3 , x3 = v3 v4 , x4 = v4 v1 }. Let µ1 be an N -function of V and let ν1 be an N -functions of E defined by

By definition of complement, we have ν 1 (x1 y1 ) =

max(µ1(x1 ), µ1 (y1 )

≥

max(µ2 (f (x2 )), µ2 (f (y2 )))

=

max(µ2 (x2 ), µ2 (y2 ))

=

ν 2 (x2 y2 ). µ1

Thus, f −1 : V2 → V1 is a bijective map which is a strong isomorphism between G1 and G2 . This ends the proof.

v1 -0.5

v2 -0.4

v3 -0.5

v4 -0.3

v1

ν1

x1 -0.2

x2 -0.3

v2 −0.2

−0.5

−0.4

The following Proposition is obvious. Proposition 26. Let G1 and G2 be N -graphs. If there is a co-strong isomorphism between G1 and G2 , then there is a homomorphism between G1 and G2 .

G

−0.2

−0.3

We now discuss N -line graphs. −0.3

Definition 27. Let P (S) = (S, T ) be an intersection graph of a simple graph G∗ = (V, E). Let G = (µ1 , ν1 ) be an N -graph of G∗ . We define an N -intersection graph P (G) = (µ2 , ν2 ) of P (S) as follows:

−0.5 −0.2

v4

v3

By routine computations, it is easy to see that G is an N -graph. Consider a line graph L(G∗ ) = (Z, W ) such that

(1) µ2 and ν2 are N -functions of S and T , respectively, (2) µ2 (Si ) = µ1 (vi ),

Z = {Sx1 , Sx2 , Sx3 , Sx4 }

(3) ν2 (Si Sj ) = ν2 (vi vj ) and

for all Si , Sj ∈ S, Si Sj ∈ T . That is, any N -graph of P (S) is called an N -intersection graph.

W = {Sx1 Sx2 , Sx2 Sx3 , Sx3 Sx4 , Sx4 Sx1 }.

The following Proposition is obvious.

Let µ2 and ν2 be N -functions on Z and W , respectively. Then, by routine computations, we have

∗

Proposition 28. Let G = (µ1 , ν1 ) be an N -graph of G . Then

µ2 (Sx1 ) = −0.2, µ2 (Sx2 ) = −0.3,

• P (G) = (µ2 , ν2 ) is an N -graph of P (S),

µ2 (Sx3 ) = −0.2, µ2 (Sx4 ) = −0.2.

• G ≃ P (G).

ν2 (Sx1 Sx2 ) = −0.2, ν2 (Sx2 Sx3 ) = −0.2,

This Proposition shows that any N -graph is isomorphic to an N -intersection graph.

ν2 (Sx3 Sx4 ) = −0.2, ν2 (Sx4 Sx1 ) = −0.2. Sx 1

Definition 29. Let L(G∗ ) = (Z, W ) be a line graph of a simple graph G∗ = (V, E). Let G = (µ1 , ν1 ) be an N - graph of G∗ . We define an N -line graph L(G) = (µ2 , ν2 ) of G as follows:

Sx 2 −0.2

−0.2

(4) µ2 and ν2 are N -functions of Z and W , respectively,

−0.3

L(G)

−0.2

−0.2

(5) µ2 (Sx ) = ν1 (x) = ν1 (ux vx ), −0.2

(6) ν2 (Sx Sy ) = max(ν1 (x), ν1 (y))

−0.2 −0.2

Sx 4

for all Sx , Sy ∈ Z, Sx Sy ∈ W . 7

Sx 3

x3 -0.2

x4 -0.2

World Appl. Sci. J., 22 (Special Issue of Applied Math): 1-9, 2013

By routine computations, it is clear that L(G) is an N line graph.

(a) Now

The following propositions are obvious.



r´i ≥ max(´ si , s´i+1 ), i = 1, 2, · · · , n.

Z = {Sx1 , Sx1 , Sx2 , · · · , Sxn }

Proposition 31. L(G) is an N -line graph corresponding to N -graph G.

W = {Sx1 Sx2 , Sx2 Sx3 , · · · , Sxn Sx1 }. Also for rn+1 = r1 ,

Proposition 32. If L(G) is an N -line graph of N -graph G. Then L(G∗ ) is the line graph of G∗ .

µ2 (Sxi ) = µ1 (xi ) = µ1 (vi vi+1 ) = r´i ,

Proposition 33. L(G) is an N -line graph of some N graph G if and only if

ν2 (Sxi Sxi+1 ) =

ν2 (Sx Sy ) = max(µ2 (Sx ), µ2 (Sy )) for all Sx Sy ∈ W. Proof. Assume that ν2 (Sx Sy ) = max(µ2 (Sx ), µ2 (Sy )) for all Sx Sy ∈ W . We define µ1 (x) = µ2 (Sx ) for all x ∈ E. Then

max(ν1 (xi ), ν1 (xi+1 ))

=

max(µ1 (vi vi+1 ), ν1 (vi+1 vi+2 ))

=

max(´ ri , r´i+1 )

for i = 1, 2, · · · , n, vn+1 = v1 , vn+2 = v2 . Since f ν2 (Sx Sy ) = max(µ2 (Sx ), µ2 (Sy )) = max(µ1 (x), µ1 (y)). is an isomorphism of G∗ onto L(G∗ ), f maps V one-toone and onto Z. Also f preserves adjacency. Hence f An N -function (µ1 , ν1 ) that yields that the property induces a permutation π of {1, 2, · · · , n} such that ν1 (xy) ≥ max(µ1 (x), µ1 (y)) f (vi ) = Sxπ(i) = Sxπ(i) Sxπ(i+1) will suffice. The converse part is obvious. and Proposition 34. L(G) is an N -line graph if and only if L(G∗ ) is a line graph and xi = vi vi+1 → f (vi )f (vi+1 ) = Svπ(i) Svπ(i+1) Svπ(i+2) ,

ν2 (uv) = max(µ2 (u), µ2 (v)) for all uv ∈ W.

i = 1, 2, · · · , n − 1.

Now

Proposition 35. Let G1 and G2 be N -graphs. If f is a strong isomorphism of G1 onto G2 , then f is an isomorphism of G∗1 onto G∗2 .

s´i = µ1 (vi ) ≥ µ2 (f (vi )) = µ2 (Svπ(i) vπ(i+1) ) = r´π(i) ,

Theorem 36. Let L(G) = (µ2 , ν2 ) be the N -line graph corresponding to N -graph G = (µ1 , ν1 ). Suppose that G∗ = (V, E) is connected. Then

r´i = ν1 (vi vi+1 ) ≥

(1) there exists a strong isomorphism of G onto L(G) if and only if G∗ is a cyclic and for all v ∈ V , x ∈ E, µ1 (v) = ν1 (x), i.e., µ1 and ν1 are constant functions on V and E, respectively, taking on the same value.

ν2 (f (vi )f (vi+1 ))

=

ν2 (Svπ(i) Svπ(i)+1 Svπ(i+1)+1 )

=

max(ν1 (vπ(i) vπ(i)+1 ), ν1 (vπ(i)+1 vπ(i+1)+1 ))

=

max(´ rπ(i) , r´π(i+1) )

for i = 1, 2, · · · , n. That is, s´i ≥ r´π (i)

(2) If f is a strong isomorphism of G onto L(G), then f is an isomorphism.

and (b)

Proof. Assume that f is a strong isomorphism of G onto L(G). From Proposition 3.31, it follows that G∗ = (V, E) is a cycle [12, Theorem 8.2, p.72]. Let V = {v1 , v2 , · · · , vn } and E = {x1 = v1 v2 , x2 = v2 v3 , · · · , xn = vn v1 }, where v1 v2 v3 · · · vn v1 is a cyclic. Define N -functions



r´i ≥ max(´ rπ(i) , r´π(i+1) ).

By (b), we have r´i ≥ r´π(i) for i = 1, 2, · · · , n and so r´π (i) ≤ r´π(π(i)) for i = 1, 2, · · · , n. Continuing, we have r´i ≥ r´π(i) ≥ · · · ≥ r´πj (i) ≥ r´i

and so ri = rπ(i) , r´i = r´π(i) , i = 1, 2, · · · , n, where j+1 is the identity map. Again, by (b), we have µ1 (vi ) = s´i , ν1 (vi vi+1 ) = r´i , i = 1, 2, · · · , n, vn+1 = v1 . π r´i ≥ r´π(i+1) = r´i+1 , i = 1, 2, · · · , r´n+1 = r´1 .

Then for s´n+1 = s´1 , 8

World Appl. Sci. J., 22 (Special Issue of Applied Math): 1-9, 2013

12. Sunitha, M. S. and K. Sameena, 2008. Characterization of g-self centered fuzzy graphs, The Journal of Fuzzy Mathematics, 16: 787-791.

Hence by (a) and (b), r´1 = · · · = r´n = s´1 = · · · = s´n . Thus we have not only proved the conclusion about µ1 and ν1 being constant function, but we have also shown that (2) holds. The converse part is obvious.

13. Zadeh, L.A. 1965. Fuzzy sets, Information and Control, 8:338-353. 14. Zadeh, L.A. 1971. Similarity relations and fuzzy orderings, Information Sciences, 3:177-200

We state the following Theorem without proof. Theorem 37. Let G and H be N -graphs of G∗ and H ∗ , respectively, such that G∗ and H ∗ are connected. Let L(G) and L(H) be the N -line graphs corresponding to G and H, respectively. Suppose that it is not the case that one of G∗ and H∗ is complete graph K3 and other is bipartite complete graph K1,3 . If L(G) and L(H) are isomorphic, then G and H are line-isomorphic. REFERENCES 1. Jun, Y.B., K.J. Lee and S.Z. Song. 2009. Nideals of BCK/BCI-algerbas, Journal of the ChungcheongMathematical Society, 22: 417-437. 2. Rosenfeld, A. 1975. Fuzzy graphs, Fuzzy Sets and their Applications (L.A. Zadeh, K.S. Fu, M. Shimura, Eds.), Academic Press, New York, 77-95. 3. Kauffman, A. 1973. Introduction a la Theorie des Sous-emsembles Flous, Masson et Cie, Vol.1. 4. Bhattacharya, P. 1987. Some remarks on fuzzy graphs, Pattern Recognition Letter, 6: 297-302. 5. Akram, M. 2011. Bipolar fuzzy graphs, Information Sciences, 181: 5548-5564. 6. Akram,M. 2012. Interval-valued fuzzy line graphs, Neural Computing and Applications, 21:145-150. 7. Akram, M. and W.A. Dudek, 2011. Interval-valued fuzzy graphs, Computers Math. Applications, 61: 289-299. 8. Akram, M. and M.G. Karunambigai, 2012. Metric in bipolar fuzzy graphs,World Applied Sciences Journal, 14:1920-1927. 9. Akram, M. 2013. Bipolar fuzzy graphs with applications, Knowledge-Based Systems, 39: 1-8. 10. Buckley, F. 1989. Self-centered graphs, Graph Theory and Its Applications: East and West. Ann. New York Acad. Sci., 576:71-78. 11. Mordeson, J.N. and P.S. Nair, 2001. Fuzzy graphs and fuzzy hypergraphs, Physica Verlag, Heidelberg. 9