As a generalisation of crisp set, Zadeh introduces the fuzzy set. The concept of partial truth between absolute true and absolute false was established with the introduction of the fuzzy set. His proposal inspired a large number of scholars who worked in a variety of scientific and technological fields.

The link between vertices and edges is described by a graph, which is a mathematical description of a network. Graph theory is used to model real-life occurrences, yet many phenomena are not adequately represented by graphs due to the inherent uncertainty of certain system properties. The fuzzy graphs were inspired by a variety of real-world occurrences. Using Zadeh's [1] fuzzy relation, Kauffman [4] introduced fuzzy graphs (FG). Networking, communication, data mining, clustering, picture capturing, image segmentation, planning, and scheduling are just a few of the many applications of FG theory. Rosenfeld [3] proposed a fuzzy analogy for various graph theoretical concepts. Bhattacharya [7] went on to explain a few points about FGs later on. The complement of a FG was established by Sunitha and Vijayakumar[8, 9], as well as certain fuzzy tree characterisation.

Mordeson and Nair [10] came up with a set of FG. Bhutani and Battou [11] proposed M-strong FGs with certain properties. Types of arcs in a FG were defined by Mathew and Sunitha [12]. Operations FGs were defined by Mordeson and Chang-Shyh [13]. Nagoor Gani and Radha[14 - 17] discussed the properties of FGs in conjunction, regular FGs, FG sequences, and the degree of vertex in some FGs. Soft graphs, bipolar FGs, and hypergraphs are all concepts defined by Akram et al.[18].

Intuitionistic Fuzzy Set (IFS) was first introduced by Atanassov [25]. Later on, he proposed the Intuitionistic FG theory. Karunambigai and Parvathi [26] introduced the intuitionistic FG as a particular example of Atanassov's IFG in their paper.

Arc analysis isn't very relevant in graph theory because all arcs are strong in the sense of [26]. However, in IFG, it is critical to understand the nature of arcs, and there is no such analysis in the literature.

Sampath Kumar [2] introduces the semigraph. The concept of semigraph is a broadening of the concept of graph. Many authors investigated semigraph after it was first introduced. Graphoidal Covers and Graphoidal Covering Number of a Graph were investigated by Acharya and E. Sampathkuma [29]. Graphoidal Covers of a Graph was examined by Armugam [30].

2. Preliminaires

Definition 2.1: [28]

Let X be a non-empty set. Then A is called an intuitionisticset (in short, IS) of X, if it is an object having the form    A =  such that  where  is called the set of members and  is
non-members of A.

Definition 2.2: [4]

A FG G = (s, μ) is a pair of function s : V → [0, 1],  μ : V × V → [0, 1] with 

μ(x, y) £ min{s(x), s (y)}, ∀ x, y ∈ V;  and σ , μ are fuzzy subset of V and symmetric fuzzy relation on σ.

Definition 2.3: [2]

Let G = < V, E >be an IFG, where

(i)          , where and  are the degree of membership and non-membership functions from V to [0, 1]of the element in V, and 0 ≤  ( +  ( ) ≤ 1 , 

(ii)          where  and such that  ( , )  (  ( ,  (  ( 

and , for every ( , , 

Definition 2.4: [2]

A semigraph  is an order pair  where  is a nonempty set whose elements are called vertices of , and  is a set of n-tuples, called edges of G, of distinct vertices, for various , satisfying the following conditions:

A semigraph is a type of graph G is an order pair  in which V is a nonempty set whose elements are termed vertices of G and X is a set of n-tuples, called edges of G, containing distinct vertices, for various , that satisfy the following conditions.

i.            Any two edges have at most one vertex in common.

ii.           Two edges  and  are considered to be equal iff      a. 

and     b. either  for , or for      .

Thus the edge  is same as the edge where and are called the end vertices which are identified as dark point and are called the middle vertices for , represented as hollow circles, while the middle-end vertex is denoted by hollow circle with a small tangent drawn to it marking the end of its adjacent edge. 

Definition 2.5: [2]

Adjacent vertices: [2]Two vertices in a semigraph are said to be adjacent if they belong to the same edge and are said to be consecutively adjacent if in addition they are consecutive in order as well. 

In a semigraph G, two vertices are said to be adjacent if they belong to the same edge, and sequentially adjacent if they are also adjacent in order.

Adjacent Edges:[2] Two edges  and in a semigraph  are said to be adjacent if they have a common vertex.

When two edges E and M of a semigraph G have a common vertex, then they are said to be adjacent.

3. Main Results

3.1. Definition

An Intuitionistic Semigraph (I-Semigraph)  corresponding to a Semigraph where is the set of vertices and  are the sets of n-tuples, called edges of of distinct vertices , satisfying the following conditions:

          IS(i):  Any two edges have at most one vertex in common.

          IS(ii): The edge and are the same edge of .

          IS(iii): ,  . By      we mean it is a complement of the edge . i.e. no two of s  are adjacent.

          The above definition is clarified with an example given below:

3.1. Example

Consider a Semigraph where and , figure (Figure 1) is given below.

                                     :

Figure 1

Then the I-Semigraph of G is describe as follows, the vertex set of is the same as of G and the edge set is 

=

The figure (Figure 2) is given below.

                                              :

Figure 2

Note that I-Semigraph corresponding to a Semigraph G is not unique.

3.2. Definition

Complement of I-Semigraphs: The complement of I-Semigraph  is an I-Semigraph , whose vertex set is the same set V and the edge set is the combination of the sets and . Thus . The following example (Figure 3) will illustrate this fact.

Figure 3

3.3. Definition

Union of two I-Semigraphs: The union of two I-Semigraphs  and  corresponding to the Semigraphs  and  respectively is the I-Semigraph , whose vertex set is  and the edge set is the combination of the sets and . An example is given below (Figure 4)

Figure 4

3.1. Proposition

Let and be any three Semigraphs. Then,  and  are I-Semigraphs corresponding to  and  respectively. We have,

(i)          For any I-Semigraph ,  

(ii)         For any two I-Semigraphs  and , 

(iii)       For any three I-Semigraphs  , and , 

3.4. Definition

     Intersection of two I-Semigraphs: The intersection of two I-Semigraphs  and  corresponding to the Semigraphs and  respectively is the I-Semigraph , whose vertex set is  and the edge set is the combination of the sets  and An example is given below.

Figure 5

3.2. Proposition

Let and be any three Semigraphs. Then,  and  are I-Semigraphs corresponding to  and  respectively. We have,

(i)          For any I-Semigraph ,  

(ii)         For any two I-Semigraphs  and , 

For any three I-Semigraphs  , and , 

3.5. Definition

Difference of two I-Semigraphs: Before define the difference of any two I-Semigraphs we define the difference of any two Semigraphs.

The difference of two I-Semigraphs  and  corresponding to the Semigraphs  and  respectively is the I-Semigraph , whose vertex set is  and the edge set is the combination of the sets and . An example is given below.

Figure 6

3.3. Proposition

Let and be any three Semigraphs. Then,  and  are I-Semigraphs corresponding to  and  respectively. We have,

(i)          For any two I-Semigraphs  and , if 

(ii)         For any two I-Semigraphs  and , 

(iii)      For any three I-Semigraphs  , and , 

3.6. Definition

Ring Sum of two I-Semigraphs: The ring sum of two I-Semigraphs  and  corresponding to the Semigraphs  and  respectively is the I-Semigraph , where    i.e. the symmetric difference of the edges sets and . And  is the complement of the set . An example is given below.

Figure 7

4. Conclusion:

In this paper basic concepts of intuitionistic semi graph is introduced. Union, intersection, complement of intuitionsticsemigraph is observed with examples. Also, some basic properties of intuitionistic semigraph is studied.

References

[1]  Zadeh, L.A. “Fuzzy sets”. Inf. Control 1965, 8, 338–353. 

[2] Sampathkumar, E. Deshpande C. M., Bam B. Y., Pushpalatha L. and Swaminathan V., “Semigraphs and their applications”, Academy of Discrete Mathematics and Applications India, (2019).

[3] Rosenfeld, A. “Fuzzy groups”. J. Math. Anal. Appl. 1971, 35, 512–517. 

[4] Kauffman, A. “Introduction a la Theorie des Sous-EmsemblesFlous”. Masson et Cie: Paris, French, 1973.

[5] Zadeh, L.A. “Similarity relations and fuzzy orderings”. Inf. Sci. 1971, 3, 177–200.

[7] Bhattacharya, P. “Some remarks on fuzzy graphs”. Pattern Recognit. Lett. 1987, 6, 297–302

[8] Sunitha, M.S., Vijayakumar, A. “Complement of a fuzzy graph”. Indian J. Pure Appl. Math. 2002, 33, 1451–1464. 

[9] Sunitha, M.S., Vijayakumar, A. A. “Characterization of fuzzy trees”. Inf. Sci. 1999, 113, 293–300. 

[10] Mordeson, J.N., Nair, P.S. “Fuzzy Graphs and Fuzzy Hypergraphs”. Springer: Heidelberg, Germany, 2000; ISBN 978-3-7908-1854-3. 

[11] Bhutani, K.R., Battou, A. “On M-strong fuzzy graphs”. Inf. Sci. 2003, 155, 103–109.

[12] Mathew, S., Sunitha, M.S. “Types of arcs in a fuzzy graph”. Inf. Sci. 2009, 179, 1760–1768. 

[13] Mordeson, J.N., Chang-Shyh, P. “Operations on fuzzy graphs”. Inf. Sci. 1994, 79, 159–170. 

[14] Nagoor Gani, A. Radha, K. “On regular fuzzy graphs”. J. Phys. Sci. 2008, 12, 33–44.

[15] Nagoor Gani, A., Radha, K. “Conjunction of two fuzzy graphs”. Int. Rev. Fuzzy Math. 2008, 3, 61–71. 

[16] Nagoor Gani, A., Radha, K. “Some sequences in fuzzy graphs”. Far East J. Appl. Math. 2008, 31, 321–335. 

[17] Nagoor Gani, A., Radha, K. “The degree of a vertex in some fuzzy graphs”. Int. J. Algorithms Comput. Math. 2009, 2, 107–116. 

[18] Akram, M., Dudek, W.A. “Regular bipolar fuzzy graphs”. Neural Comput. Appl. 2012, 21, 197–205. 

[19] Akram, M., Nawaz, S. “Operations on soft graphs”. Fuzzy Inf. Eng. 2015, 7, 423–449. 

[20] Akram, M., Luqman, A. “Certain concepts of bipolar fuzzy directed hypergraphs”. Mathematics 2017, 5, 17. 

[21] Sarwar, M., Akram, M.; Alshehri, N.O. “A new method to decision-making with fuzzy competition hypergraphs”. Symmetry 2018, 10, 404.

[22] Sampathkumar, E. “Generalized graph structures”. Bull. Kerala Math. Assoc. 2006, 3, 65–123.

[23] Dinesh, T. A Study on Graph Structures. “Incidence Algebras and Their Fuzzy Analogues”. Ph.D. Thesis, Kannur University, Kannur, India, 2011.

[24] Ramakrishnan, R.V.; Dinesh, T. “On generalised fuzzy graph structures”. Appl. Math. Sci. 2011, 5, 173–180. 

[25] Atanassov, K. “Intuitionistic Fuzzy Sets: Theory and Applications”. Springer Physica-Verlag, Berlin, 1999. 

[26] Karunambigai, M. G., Parvathi R., “Intuitionistic Fuzzy Graphs, Proceedings of 9th Fuzzy Days International Conference on Computational Intelligence, Advances in soft computing: Computational Intelligence,Theory and Applications, Springer-Verlag, Vol. 20, 2006, 139– 150.

[27] Bhutani, K. R., Rosenfeld A. “Strong arcs in fuzzy graphs”. Information Sciences, Vol.152, 2003, 319–322. 57.

[28] Coker D. “A note on intuitionistic sets and intuitionistic points”. Tr. J. of Mathematics 20 (1996) 343–351.

[29] Acharya B. D. and Sampathkumar E. “Graphoidal Covers and Graphoidal Covering Number of a Graph”. J. Pure ppl. Math., 18(10) (1987), 882–890. 

[30] Armugam S., Acharya B. D.  and Sampathkumar E. “Graphoidal Covers of a Graph: A Creative Review, Proc. National Workshop on Graph Theory and its Applications”. ManonmanianSundarnar University, Tirunelveli, 1–26 (1996), Tata-McGraw Hill, New Delhi, 1997.