# A Study of Regular and Irregular Neutrosophic Graphs with Real Life Applications
## 1 Introduction
Recently, many researchers have more actively worked on neutrosophic graph theory; for instance, [PERSON] [29], [PERSON] et al. [30], [PERSON] et al. [31], [PERSON] [15], and [PERSON] [32, 33, 34, 35]. The vertex degree is a significant way to represent the total number of relations of a vertex in a graph, and the vertex degree can be used to analyze the graph. [PERSON] and [PERSON] [36] proposed the concept of irregularity, total irregularity, and total degree in a fuzzy graph. [PERSON] and [PERSON] [37] proposed the notation of the \(d_{2}\)-vertex in a fuzzy graph and also described several properties on the \(d_{2}\)-vertex degree of a fuzzy graph. [PERSON] et al. [38] presented the idea of the \(d_{m}\)-regular vague graph, \(td_{m}\)-regular vague graph, \(m\)-highly irregular vague graph, and \(m\)-highly total irregular vague graph, and they discussed some properties of those graphs. They described some real-life applications (e.g., fullerene molecules, wireless networks, and road networks) of regular vague graphs. Neutrosophic graphs are more effective, precise, flexible, and compatible when modeling uncertain real-life problemscompared to fuzzy graphs or vague graphs. Thus, the use of the neutrosophic graph is inevitable for modeling optimization problems [29; 34; 39; 40; 41] in real-life scenarios, and it is essential to present some new properties and theories for neutrosophic graphs. This idea motivates us to introduce different types of neutrosophic graphs (regular, bipartite, isomorphic, and \(\mu\) complement neutrosophic graphs) and their related theorems. The concept of the regularity and degree of nodes has a significant role in both theories and applications (e.g., social network analysis, road transpotation network, wireless multihop network, and the assignment problem) in neutrosophic graph theory. The main contributions of this manuscript are as follows. 1. As far as we know, there exists no research work on the regularity of the neutrosophic graph until now. Therefore, in this manuscript, we present the definition of the regular neutrosophic graph, star neutrosophic graph, regular strong neutrosophic graph, and complete bipartite neutrosophic graph. 2. We introduce the two types of degree, \(d_{m}\) and total \(d_{m}\)-degrees, of a node in a neutrosophic graph. The definition of busy and free nodes in a regular neutrosophic graph are presented here. 3. Depending on the degree of the node, we introduce the three types of regularity of a neutrosophic graph, \(d_{m}\)-regular, \(td_{m}\)-regular, and \(m\)-highly-irregular neutrosophic graphs (HING), and some properties are also described. We also investigate the concept of \(m\)-HING on the cycle graph and path graph in this manuscript. 4. We have presented the definition of the \(\mu\)-complement and \(h\)-morphism of a neutrosophic graph. Some properties of the complement and isomorphic regular neutrosophic graph are also presented here. 5. Some real-life applications of the regular neutrosophic graph and complete bipartite neutrosophic graph such as the road transport network, social networks, and assignment problem are described in this paper.