# A Study of Regular and Irregular Neutrosophic Graphs with Real Life Applications
## 1 Introduction  
Graph theory has many real-life applications for problems in computer applications, systems analysis, computer networks, transportation, operations research, and economics. A graph is basically a model of relations, and it is a used to represent the real-life problem consisting of relationshipsbetween objects. The vertices and edges of the graph are used to represent the objects and the relations between objects, respectively. In many optimization problems, the available information is inexact or imprecise for various reasons such as the loss of information, a lack of evidence, imperfect statistical data and insufficient information. Generally, the uncertainty in real-life problems may be present in the information that defines the problem. Classical graph theory uses the basic concept of classical set theory, which was presented by [PERSON]. In a classical graph, for any vertex or edge, there are two possibilities: it is either in the graph or it is not in the graph. Therefore, classical graphs cannot model uncertain optimization problems. Real-life problems are often uncertain, which are hard to model using classical graphs. The fuzzy set [1] is an extended version of the classical set, where the objects have varying membership degrees. A fuzzy set gives its objects different membership degrees between zero and one. The membership degree is not the same as probability; rather, it describes membership in vaguely-defined sets. The concept of fuzziness in graph theory [2] was described by [PERSON] [3] using the fuzzy relation. [PERSON] [4] introduced some concepts such as bridges, cycles, paths, trees, and the connectedness of the fuzzy graph and described some of the properties of the fuzzy graph. [PERSON] and [PERSON] [5] presented the notation of the regular fuzzy graph. Many other researchers, such as [PERSON] and [PERSON] [6], [PERSON] and [PERSON] [7], [PERSON] et al. [8, 9], [PERSON] [10], [PERSON] [11], [PERSON] and [PERSON] [12], [PERSON] et. al [13], and [PERSON] [14] presented much work in the domain of the fuzzy graph and its several applications in real life. [PERSON] and [PERSON] [6] and [PERSON] and [PERSON] [15] presented the concept of the irregular and regular fuzzy graph. They also described some applications of those graphs. [PERSON] [16, 17, 18] presented the idea of the intuitionistic fuzzy set (IFS) as a modified version of the classical fuzzy set. The classical fuzzy set uses only the membership grade (degree), but IFS uses independent membership grade and non-membership grade for any entity, and the only requirement is that the sum of non-membership and membership degree values be no greater than one. [PERSON] and [PERSON] [19] presented the idea of the IFS relation and the intuitionistic fuzzy graph (IFG) and discussed many theorems, proofs, and proprieties [19]. [PERSON] et al. [20, 21, 22] presented many different operations such as the join, union, and product of two IFGs. [PERSON] et al. [23] presented some products such as strong, direct, and lexicographic products for two IFGs. For further detailed descriptions of IFGs, please refer to [24, 25, 26, 27]. Although the fuzzy graph and IFG have been used to model many real-word problems, uncertainties due to inconsistent and indeterminate information about a problem cannot be represented properly by the fuzzy graph or IFG. Therefore, a new idea is needed to deal with such scenarios. [PERSON] [28] introduced the idea of the neutrosophic set, by modifying the concept of the fuzzy set. The neutrosophic set can work with uncertain, indeterminate, vague, and inconsistent information of any uncertain real-life problem. It is basically a modified version of the crisp set, Type 1 fuzzy set, and IFS. It is described by the truth, indeterminate, and false membership degrees of any object. These three membership degrees are independent and always lie within \(]^{-}0,1^{+}[\), i.e., a nonstandard unit interval. The neutrosophic graph [12] can efficiently model the inconsistent information about any real-life problem.