Abstract
The concepts of fuzzy labeling and fuzzy magic labeling graph are introduced. Fuzzy magic labeling for some graphs like path, cycle, and star graph is defined. It is proved that every fuzzy magic graph is a fuzzy labeling graph, but the converse is not true. We have shown that the removal of a fuzzy bridge from a fuzzy magic cycle with odd nodes reduces the strength of a fuzzy magic cycle. Some properties related to fuzzy bridge and fuzzy cut node have also been discussed.
1. Introduction
Fuzzy set is a newly emerging mathematical framework to exemplify the phenomenon of uncertainty in real life tribulations. It was introduced by Zadeh in 1965, and the concepts were pioneered by various independent researches, namely, Rosenfeld [1] and Bhutani and Battou [2] during 1970s. Bhattacharya has established the connectivity concepts between fuzzy cut nodes and fuzzy bridges entitled “Some remarks on fuzzy graphs [3].” Several fuzzy analogs of graph theoretic concepts such as paths, cycles, and connectedness were explored by them. There are many problems, which can be solved with the help of the fuzzy graphs.
Though it is very young, it has been growing fast and has numerous applications in various fields. Further, research on fuzzy graphs has been witnessing an exponential growth, both within mathematics and in its applications in science and Technology. A fuzzy graph is the generalization of the crisp graph. Therefore it is natural that many properties are similar to crisp graph and also it deviates at many places.
In crisp graph, a bijection that assigns to each vertex and/or edge if , a unique natural number is called a labeling. The concept of magic labeling in crisp graph was motivated by the notion of magic squares in number theory. The notion of magic graph was first introduced by Sunitha and Vijaya Kumar [4] in 1964. He defined a graph to be magic if it has an edge-labeling, within the range of real numbers, such that the sum of the labels around any vertex equals some constant, independent of the choice of vertex. This labeling has been studied by Stewart [5, 6] who called the labeling as super magic if the labels are consecutive integers, starting from 1. Several others have studied this labeling.
Kotzig and Rosa [7] defined a magic labeling to be a total labeling in which the labels are the integers from 1 to . The sum of labels on an edge and its two endpoints is constant. Recently Enomoto et al. [8] introduced the name super edge magic for magic labeling in the sense of Kotzig and Rosa, with the added property that the vertices receive the smaller labels. Many other researchers have investigated different forms of magic graphs; for example see Avadayappan et al. [9] Ngurah et al. [10], and Trenkler [11].
In this paper, Section 1 contains basic definitions and in Section 2 a new concept of fuzzy labeling and fuzzy magic labeling has been introduced and also fuzzy star graph is defined. In Section 2, fuzzy magic labeling for some graphs like path, cycle, and star is defined. In Section 3, some properties and results with fuzzy bridge and fuzzy cut nodes are discussed. The graphs which are considered in this paper are finite and connected.
We have used standard definitions and terminologies in this paper. For graphs considered in this paper, the readers are referred to [12–19].
1.1. Preliminaries
Let and be two sets. Then is said to be a fuzzy relation from into if is a fuzzy set of . A fuzzy graph is a pair of functions and , where for all , we have . A path in a fuzzy graph is a sequence of distinct nodes such that ; ; here is called the length of the path . The consecutive pairs are called the edge of the path. A path is called a cycle if and . The strength of a path is defined as . Let be a fuzzy graph. The degree of a vertex is defined as . Let be a fuzzy graph. The strong degree of a node is defined as the sum of membership values of all strong edges incident at . It is denoted by . Also if denote the set of all strong neighbours of , then . An edge is called a fuzzy bridge of if its removal reduces the strength of connectedness between some pair of nodes in . A node is a fuzzy cut node of if removal of it reduces the strength of connectedness between some other pairs of nodes.
Definition 1 (see, [20]). A graph is said to be a fuzzy labeling graph if and is bijective such that the membership value of edges and vertices are distinct and for all .
Example 2 (see, [20]). In Figure 1 and are bijective, such that no vertices and edges receive the same membership value.
Definition 3 (see, [20]). A fuzzy labeling graph is said to be a fuzzy magic graph if has a same magic value for all which is denoted as .
Example 4 (see, [20]). In Figure 2 , for all .
Definition 5. A star in a fuzzy graph consists of two node sets and with and , such that and , . It is denoted by .
Example 6. A fuzzy star graph is shown in Figure 3.
Definition 7 (see, [20]). The fuzzy labeling graph is called a fuzzy labeling subgraph of if for all and , for all .
2. Properties of Fuzzy Labeling Graphs
Proposition 8. For all , the path is a fuzzy magic graph.
Proof. Let be any path with length and and are the nodes and edges of . Let such that one can choose if and if . Such fuzzy labeling is defined as follows.
When length is odd:
Case (i). is even.
Then for any positive integer and for each edge ,
Case (ii). is odd.
Then for any positive integer and for each edge
When length is even:
Case (i). is even.
Then for any positive integer and for each edge
Case (ii). is odd.
Then for any positive integer and for each edge Therefore in both the cases the magic value is same and unique. Thus is fuzzy magic graph for all .
Proposition 9. If is odd, then the cycle is a fuzzy magic graph.
Proof. Let be any cycle with odd number of nodes and and , be the nodes and edges of . Let such that one can choose if and if . The fuzzy labeling for cycle is defined as follows:
Case (i). is even.
Then for any positive integer and for each edge
Case (ii). is odd.
Then for any positive integer and for each edge
Therefore from above cases is a fuzzy magic graph if is odd.
Proposition 10. For any , star is a fuzzy magic graph.
Proof. Let be a star graph with as nodes and as edges.
Let such that one can choose if and if . Such a fuzzy labeling is defined as follows:
Case (i). is even.
Then for any positive integer and for each edge
Case (ii). is odd.
Then for any positive integer and for each edge
From the above cases one can easily verify that all star graphs are fuzzy magic graphs.
Remark 11. One can observe the same labeling holds well if we choose the value of as 0.03, 0.05, and so forth, for the Propositions 8, 9, and 10.
Remark 12. (1) If is a fuzzy magic graph, then for any pair of nodes and .
(2) For any fuzzy magic graph, .
(3) Sum of the degree of all nodes in a fuzzy magic graph is equal to twice the sum of membership values of all edges (i.e., ).
(4) Sum of strong degree of all nodes in a fuzzy magic graph is equal to twice the sum of the membership values of all strong arcs in (i.e., ).
3. Properties of Fuzzy Magic Graphs
Proposition 13. Every fuzzy magic graph is a fuzzy labeling graph, but the converse is not true.
Proof. This is immediate from Definition 3.
Proposition 14. For every fuzzy magic graph , there exists at least one fuzzy bridge.
Proof. Let be a fuzzy magic graph, such that there exists only one edge with maximum value, since is bijective. Now we claim that is a fuzzy bridge. If we remove the edge from , then in its subgraph we have , which implies is a fuzzy bridge.
Proposition 15. Removal of a fuzzy cut node from a fuzzy magic path is also a fuzzy magic graph.
Proof. Let be any fuzzy magic path with length . Then there must be a fuzzy cut node; if we remove that cut node from then it either becomes a smaller path or disconnected path, anyway it remains to be a path with odd or even length; by Proposition 8, it is concluded that removal of a fuzzy cut node from a fuzzy magic path is also a fuzzy magic graph.
Proposition 16. When is odd, removal of a fuzzy bridge from a fuzzy magic cycle is a fuzzy magic graph.
Proof. Let be any fuzzy magic cycle with odd nodes. If we choose any path then there must be at least one fuzzy bridge, whose removal from will result as a path of odd or even length. By Proposition 8, the removal of a fuzzy bridge from a fuzzy magic cycle is also a fuzzy magic graph.
Remark 17. (1) Removal of a fuzzy cut node from the cycle is also a fuzzy magic graph.
(2) For all fuzzy magic cycles with odd nodes, there exists at least one pair of nodes and such that .
Proposition 18. Removal of a fuzzy bridge from a fuzzy magic cycle will reduce the strength of the fuzzy magic cycle .
Proof. Let be a fuzzy magic cycle with odd number of nodes. Now choose any path from , and then it is obvious that there exists at least one fuzzy bridge . Removal of this fuzzy bridge will reduce the strength of connectedness between and . This implies that the removal of fuzzy bridge from the fuzzy magic cycle will reduce its strength.
4. Concluding Remarks
Fuzzy graph theory is finding an increasing number of applications in modeling real time systems where the level of information inherent in the system varies with different levels of precision. Fuzzy models are becoming useful because of their aim in reducing the differences between the traditional numerical models used in engineering and sciences and the symbolic models used in expert systems. In this paper, the concept of fuzzy labeling and fuzzy magic labeling graphs has been introduced. We plan to extend our research work to (1) bipolar fuzzy labeling and bipolar fuzzy magic labeling graphs and (2) fuzzy labeling and fuzzy magic labeling hypergraphs.
Conflict of Interests
The authors declare that they do not have any conflict of interests regarding the publication of this paper.