# A Study of Regular and Irregular Neutrosophic Graphs with Real Life Applications
## 5 Complete Bipartite Neutrosophic Graphs
\right.\]  
Here, in both cases, we see that for every two adjacent nodes \(v_{i}\) and \(v_{i+1}\) in \(G\), we get \(d_{m}(v_{i})\
eq d_{m}(v_{i+1})\), and so, \(G\) is a \(\left[\frac{k}{2}\right]\)-HING. Similarly, for the indeterministic and false membership function, for each arc, it is proven. Hence the proof. Consider the the neutrosophic graph \(G=(N,M)\) on graph \(G^{*}=(V,E)\) shown in Figure 6.  
Figure 5: \(m\)-highly irregular neutrosophic graph. _We see that \(T_{M}(ab)<T_{M}(bc)<T_{M}(cd)\), and we get \(td_{2}(a)=(0.1+0.4,0.4+0.4)=(0.5,0.8)\) \(td_{2}(b)=(0.2+0.3,0.5+0.3)=(0.5,0.8)\) \(td_{2}(c)=(0.1+0.4,0.4+0.2)=(0.5,0.6)\) \(td_{2}(d)=(0.2+0.5,0.5+0.4)=(0.7,0.9)\)_  
_Since for two adjacent nodes a and b, \(td_{2}(a)=td_{2}(b)\). Then, we see that \(G\) is not a two-HTING._  
**Definition 29**. : _Let \(G=(N,M)\) be a neutrosophic graph. The graph \(G\) is said to be a star neutrosophic graph if it is a complete bipartite neutrosophic graph such that one partition of \(V\) contains only one node._  
**Theorem 15**. : _Let \(G=(N,M)\) be a star neutrosophic graph with \(k\geq 4\) nodes such that the node \(v_{1}\) is incident to the other nodes. If for all \(i=1,2,...,k\), \(T_{M}(v_{i}v_{i+1})<T_{M}(v_{i+1}v_{i+2})\), or \(I_{M}(v_{i}v_{i+1})>I_{M}(v_{i+1}v_{i+2})\), or \(F_{M}(v_{i}v_{i+1})>F_{M}(v_{i+1}v_{i+2})\). Then, \(G\) is a \([1,2]\)-HING._  
Proof. : Suppose that \(v_{1},v_{2},...,v_{k},v_{1}\) is the arcs of \(G\), where \(T_{M}(v_{1}v_{2})<T_{M}(v_{2}v_{3})<...<T_{M}(v_{i}v_{i+1})<T_{M}(v_{i+1}v_{i+2 })<...<T_{M}(v_{k}v_{1})\). Then, \(\sum_{v_{i}\
eq v_{j}\in V}T_{M}^{1}(v_{1}v_{j})=\sum_{j=2}^{k}T_{M}(v_{1}v_{ j})\) and \(\sum_{v_{i}\
eq v_{j}\in V}T_{M}^{1}(v_{1}v_{j})=T_{M}(v_{1}v_{j})\); \(i=2,3,...,k\). Hence, \(d_{1}(v_{1})\
eq d_{1}(v_{2})\
eq d_{1}(v_{3})\
eq....\
eq d_{1}(v_{k})\). There, \(G\) is a one-HING. In addition, we see that \(d_{1}(v_{1})=(0,1)\) and \(\sum_{v_{i}\
eq v_{j}\in V}T_{M}^{2}(v_{1}v_{j})=(k-i)t_{M}(v_{1}v_{i})+\sum_{ j=2}^{i-1}T_{M}(v_{1}v_{j})\). Hence, for every \(i=2,...,k\), \(d_{1}(v_{i})\
eq d_{1}(v_{1})\), and so, \(G\) is a two-HING. Similarly, the proof is established for the indeterminacy and false membership function of each arc. Hence the proof. **7. Some Applications of Regular and Bipartite Neutrosophic Graphs**  
There are many networks that are used in real-life scenarios. Graph theory plays a vital role in the application parts of these networks. Some networks are designed for distributing commodities or services. The power grid, Internet-related issues, and road transportation networks are some examples of technological networks. There has been a phenomenal change in network research in recent years from a small graphical structure to a wide statistical analysis of large-scale networks. Most of the real-life networks/structures are ambiguous and vague in nature and not well defined. Indeterministic information exists in almost every real-life scenario. The neutrosophic graph is more flexible and compatible to model the indeterministic information in a network when compared to the fuzzy graph or vague graph. A graph is a classical way of modeling real-life networks, which consist of relationships between entities. In a graph, the entities are expressed by nodes and relationships by arcs. Figure 6: Neutrosophic graph. If there exist uncertainties due to indeterministic or vague information about the entities, or relations, or both, then the neutrosophic graph model is very efficient to design such networks. Neutrosophic graphs are widespread and important in the area of mathematical modeling, pattern recognition, relational mapping, biological, social networks, and information systems. It can be applied to artificial intelligence with neutrosophic rules in the inference system. These graphs can be represented in different data structures in different applications internally. In those real-life applications, the basic mathematical model is that of a neutrosophic graph.