Multiplicative Version of First Zagreb Index in Fuzzy Graph and its Application in Crime Analysis

IF 0.8 4区综合性期刊 Q3 MULTIDISCIPLINARY SCIENCES Proceedings of the National Academy of Sciences, India Section A: Physical Sciences Pub Date : 2023-12-23 DOI:10.1007/s40010-023-00868-2

Sk Rabiul Islam, Madhumangal Pal

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Abstract

The multiplicative first Zagreb index is discussed in this paper and studied for fuzzy graphs. Bounds of this index are calculated for path, cycle, star, complete fuzzy graph, partial fuzzy subgraph, etc. For an isomorphic fuzzy graph, it is shown that the value of this index is the same. Some interesting relations are established among the first Zagreb index, F-index and multiplicative first Zagreb index for a fuzzy graph. The bounds of this index are studied for some fuzzy graph operations. Also, the crime of kidnapping and abduction in metropolitan cities in India is analyzed by this index. Significant Statement: Topological indices for a crisp graph have applications in real life. But sometimes, it is seen that most real-life problems cannot be modeled using crisp graphs. Also, laboratory testing of chemicals to understand their different properties are costly. To overcome this, many topological indices have been presented in theoretical chemistry. Those topological indices are needed to define for a fuzzy graph for these circumstances.

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模糊图中第一萨格勒布指数的乘法版本及其在犯罪分析中的应用

本文讨论了乘法第一萨格勒布指数，并对模糊图进行了研究。计算了路径、循环、星形、完整模糊图、部分模糊子图等的该指数的界限。对于同构模糊图，该指数的值是相同的。在模糊图的第一萨格勒布指数、F 指数和乘法第一萨格勒布指数之间建立了一些有趣的关系。研究了该指数在某些模糊图操作中的界限。此外，该指数还分析了印度大都市中的绑架和诱拐犯罪。重要声明：清晰图的拓扑指数在现实生活中也有应用。但有时，我们会发现现实生活中的大多数问题都无法用清晰图来建模。此外，为了解化学品的不同特性而进行的实验室测试成本高昂。为了解决这个问题，理论化学中提出了许多拓扑指数。在这种情况下，需要用这些拓扑指数来定义模糊图。

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