Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, 721102, India.
Department of Mathematics, Ramkrishna Mahato Govt. Engg. College, Purulia, 723103, India.
Sci Rep. 2023 Apr 20;13(1):6452. doi: 10.1038/s41598-023-33071-9.
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.
在清晰图论中，等距是一个众所周知的事实。但是，在模糊环境下的等距最近才被提出并研究了许多事实。在 m-极性模糊图中，我们必须为每个节点和边考虑 m 个分量。由于在我们的考虑中，我们为每个节点和边考虑 m 个分量，因此我们不能使用模糊模型来处理这种情况，因为这个概念只有一个分量。此外，我们不能应用双极或直觉模糊图模型，因为每条边或节点只有两个分量。因此，这些 mPFG 模型比其他模糊模型提供更有效的模糊结果。此外，用示例和相关定理来开发和分析这种类型的 mPFG 非常有趣。考虑到所有这些因素，我们在 m-极性模糊环境下提出了等距。在本文中，我们讨论了等距的 m-极性模糊图以及与之相关的许多令人兴奋的事实。度量空间性质也已经应用于 m-极性模糊等距图上。我们还引入了一种广义模糊图，即对极 m-极性模糊图，并讨论了几个问题。它的度以及边正则性性质也被提出。我们还给出了 m-极性模糊对极图与其基础清晰图之间的关系。它的性质也在 m-极性模糊奇环和偶环、完全图等上进行了讨论。最后，还提出了一个在 m-极性模糊环境下使用 [公式：见文本]-距离概念的道路交通网络系统的实际应用。
