Theoretical concepts of graphs are highly utilized by computer science applications. Especially in research areas of computer science such as data mining, image segmentation, clustering, image capturing and networking. In this paper, we discussed some properties of the μ-complement of bipolar fuzzy graphs. Self μ-complement bipolar fuzzy graphs and self weak μ-complement bipolar fuzzy graphs are defined and a necessary condition for a bipolar fuzzy graph to be self μ-complement is given. We defined busy vertices and free vertices in bipolar fuzzy graphs and studied their image under an isomorphism. Categorical properties of bipolar fuzzy graphs are discussed. Also, we investigated some properties of isomorphism on bipolar fuzzy graphs.

In recent years, the concept of domination has been the backbone of research activities in graph theory. The application of graphic domination has become widespread in different areas to solve human-life issues, including social media theories, radio channels, commuter train transportation, earth measurement, internet transportation systems, and pharmacy. The purpose of this paper was to generalize the idea of bondage set (BS) and non-bondage set (NBS), bondage number $\alpha \left(G\right)$, and non-bondage number ${\alpha }_k\left(G\right)$, respectively, in the intuitionistic fuzzy graph (IFG). The BS is based on a strong arc (SA) in the fuzzy graph (FG). In this research, a new definition of SA in connection with the strength of connectivity in IFGs was applied. Additionally, the BS, $\alpha \left(G\right)$, NBS, and ${\alpha }_k\left(G\right)$ concepts were presented in IFGs. Three different examples were described to show the informative development procedure by applying the idea to IFGs. Considering the examples, some results were developed. Also, the applications were utilized in water supply systems. The present study was conducted to make daily life more useful and productive.

The concepts of graph theory are applied in many areas of computer science including data mining, image segmentation, clustering, image capturing and networking. It is also known that lots of uncertainties occur in these areas. To handle the uncertainty that occurs in graph theory, fuzzy graph theory is successfully used in many problems. A bipolar fuzzy set is a generalization of the fuzzy set. In this paper, two new operations on bipolar fuzzy graphs, viz. normal product and tensor product, are defined. Also, the degrees of the vertices of the resultant graphs which are obtained from two given bipolar fuzzy graphs $G_1$ and $G_2$ using the operations Cartesian product, composition, tensor and normal product are determined.

The concept of interval-valued intuitionistic fuzzy set was introduced by Atanassov (See [6 ]). Interval-valued intuitionistic fuzzy sets provide a more adequate description of uncertainty than the traditional fuzzy sets. It has many applications in

A vague graph is a generalized structure of a fuzzy graph that gives more precision, flexibility and compatibility to a system when compared with systems that are designed using fuzzy graphs. In this paper, we defined the notions of product and complete product vague graphs. We investigated some properties of union, join and complement on product vague graphs and introduced the multiplication of the product vague graphs. Finally we give an application of product vague graphs. Product and complete product vague graphs are highly utilized by computer science, geometry, algebra, number theory and operation research. Likewise, these properties will also be helpful to study large vague graph as a combination of small, vague graphs and to derive its properties from those of the smaller ones.