Solution of road network problem with the help of m-polar fuzzy graph using isometric and antipodal concept

The isometry in crisp graph theory is a well-known fact. But, isometry under a fuzzy environment was developed recently and studied many facts. In a m-polar fuzzy graph, we have to think m components for each node and edge. Since, in our consideration, we consider m components for each nodes as well as edges, therefore we can not handle this type of situation using fuzzy model as their is a single components for this concept. Again, we can not apply bipolar or intuitionistic fuzzy graph model as each edges or nodes have just two components. Thus, these mPFG models give more efficient fuzziness results than other fuzzy model. Also, it is very interesting to develop and analyze such types of mPFGs with examples and related theorems. Considering all those things together, we have presented isometry under a m-polar fuzzy environment. In this paper, we have discussed the isometric m-polar fuzzy graph along with many exciting facts about it. Metric space properties have also been implemented on m-polar fuzzy isometric graph. We also have initiated a generalized fuzzy graph, namely antipodal m-polar fuzzy graphs, along with several issues. The degree of it is also presented along with edge regularity properties. We also give a relation between m-polar fuzzy antipodal graphs and their underlying crisp graphs. Its properties have also been discussed on m-polar fuzzy odd as well as even cycles, complete graphs, etc. Finally, a real-life application on a road network system in a m-polar fuzzy environment using the [Formula: see text]-distance concept is also presented.