A fuzzy graph is one of the versatile application tools in the field of mathematics, which allows the user to easily describe the fuzzy relation between any objects. The nature of fuzziness is favorable for any environment, which supports to predict the problem and solving it. Fuzzy graphs are beneficial to give more precision and flexibility to the system as compared to the classical model (i.e.,) crisp theory. A topological index is a numerical quantity for the structural graph of the molecule and it can be represented through Graph theory. Moreover, its application not only in the field of chemistry can also be applied in areas including computer science, networking, etc. A lot of topological indices are available in chemical-graph theory and H. Wiener proposed the first index to estimate the boiling point of alkanes called ‘Wiener index’. Many topological indices exist only in the crisp but it’s new to the fuzzy graph environment. The main aim of this paper is to define the topological indices in fuzzy graphs. Here, indices defined in fuzzy graphs are Modified Wiener index, Hyper Wiener index, Schultz index, Gutman index, Zagreb indices, Harmonic index, and Randić index with illustrations. Bounds for some of the indices are proved. The algorithms for distance matrix and MWI are shown. Finally, the application of these indices is discussed.

中文翻译：

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模糊图中的一些拓扑指标

模糊图是数学领域中通用的应用工具之一，它使用户可以轻松地描述任何对象之间的模糊关系。模糊性对于任何支持预测问题和解决问题的环境都是有利的。与经典模型（即清脆理论）相比，模糊图有利于提高系统的精度和灵活性。拓扑指数是分子结构图的数值，可以通过图论表示。此外，其不仅在化学领域中的应用，还可以在计算机科学，网络等领域中应用。化学图论中有许多拓扑指标可用，H。Wiener提出了第一个估算沸点的指标称为“维纳指数”的烷烃 许多拓扑指数仅存在于清晰的区域中，但对于模糊图环境来说是新的。本文的主要目的是在模糊图中定义拓扑指数。在此，模糊图中定义的索引是带插图的修正Wiener索引，Hyper Wiener索引，Schultz索引，Gutman索引，Zagreb索引，Harmonic索引和Randić索引。证明了某些指标的界限。显示了距离矩阵和MWI的算法。最后，讨论了这些指标的应用。以及带有插图的Randić索引。证明了某些指标的界限。显示了距离矩阵和MWI的算法。最后，讨论了这些指标的应用。以及带有插图的Randić索引。证明了某些指标的界限。显示了距离矩阵和MWI的算法。最后，讨论了这些指标的应用。