OPERATIONS ON M-STRONG FUZZY GRAPHS

M. Rostamy Malkhalifeh 1 , * H. Saleh 2 and F. Falahati Nezhad 3 . 1. Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran. 2. Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran. 3. Young Researchers and Elite Club, Safadasht Branch, Islamic Azad University, Tehran, Iran. ...................................................................................................................... Manuscript Info Abstract ......................... ........................................................................ Manuscript History

In current paper, deleted lexicographical product, disjunction and symmetric difference on fuzzy graphs are defined and some of their properties are discussed. Moreover, the concept of M-strong fuzzy graphs are investigated for mentioned operations. These results also are illustrated with some examples.
A graph is anordered pair , where is the set of vertices of and is the set of edges of where .Graph theory has been used to study modern science such as operations research, transportationand cluster analysis.
In 1975, Rosenfeld introduced fuzzy graphs [5] based on fuzzy set.Fuzzy graph theory plays essential roles in various disciplinesincluding information theory, neural networks, clustering problems and control theory, etc.Fuzzy models is more compatible to the systemin compare with classical models [9,10].
Bhutani and Rosenfeld introduced the notion of M-strong fuzzy graphs and studied some of their properties. [1,4] Many interesting graphs are obtained fromcomposing simpler graphs via several operations. For more information on graph operations see [3].
In this paper,we definedeleted lexicographical product,disjunction and symmetric difference oftwo fuzzy graphsand prove that new graphs constructed from mentioned operations are fuzzy graph. Also we show that deleted lexicographical product,disjunction and symmetric difference of two M-strong fuzzy graphs are also M-strong fuzzy graph.Finally we prove that if ̅ , and are M-strong fuzzy graphs, then at least one factor must be M-strong fuzzy graph.All properties are illustrated with examples.
123 Preliminaries:-In this section, we list some necessary definitions as follows: Definition 2.1 [6]. A fuzzy set is a set of ordered pairs { } where is universal set. is a map from to which is called a membership function or degree of membership of in . Definition 2.2 [5].A fuzzy graph is a pair of functions ( , ) where is a fuzzy subset of and is symmetric fuzzy relation on such that { }. Throughout the paper, we use instead of for an element of . Definition 2.3 [2].The deleted lexicographical product of two graphs and is defined as a graph ̅ with the vertex set andvertex isadjacent with vertex whenever is adjacent with in or and is adjacent with in . Definition 2.4 [3].The disjunction of two fuzzy graphs and is defined as a fuzzy graph withthevertexset andvertex isadjacentwithvertex whenever is adjacent with or is adjacent with in or both of them. Definition 2.5 [3].The symmetric difference of two fuzzy graphs and is defined as a fuzzy graph withthevertexset andvertex isadjacent with vertex whenever is adjacent with in or is adjacent with in but not both.

M-strong fuzzy graphs:-
In this section, we prove some theorems to showthat if and are M-strong fuzzy graphs, then new fuzzy graph constructed from them areM-strong fuzzy graph too. Also,if ̅ , and are M-strong fuzzy graphs, then at least one factor must be M-strong fuzzy graph. All computations are illustrated with examples. Theorem 4.1. Let be a deleted lexicographical product of two M-strong fuzzy graphs and , then is a Mstrong fuzzy graph. Proof.The first part is taken over all edges such that and . Using the fact that is a M-strong fuzzy graph, we have The second part is taken over alledges such that and .Using the fact that is a M-strong fuzzy graph, we have   It is easy to see that ̅ and are strong fuzzy graphs but is not.   It is easy to see that the disjunction of and is a strong fuzzy graph. Proof.It is straightforward.
Theorem 4.6.Let be a symmetric difference of two M-strong fuzzy graphs and , then is a M-strong fuzzy graph.
Proof.It is straightforward. In next example, we will illustrate that the opposite is not necessarily true.

Conclusion:-
Inpresent paper, specific operations on fuzzy graphs have been introduced and some theorems are discussed. Some properties of M-strongfuzzy graphs are investigated. Fuzzy graph theory is highly utilized in various areas. In future work, we can focus on Intuitionistic, bipolar and hyperfuzzy graphs and attempt to investigate many properties on them.