Islam Sk Rabiul, Mohsin Bandar Bin, Pal Madhumangal
Department of Mathematics, Brainware University, 398, Ramkrishnapur Road, Jagadighata Market, Barasat, Kolkata-700125, India.
Department of Applied Mathematics, Vidyasagar University, Midnapore 721102, India.
Heliyon. 2024 Aug 15;10(16):e36110. doi: 10.1016/j.heliyon.2024.e36110. eCollection 2024 Aug 30.
The Zagreb indices (ZIs) are important graph invariants that are used extensively in many different fields in mathematics and chemistry, such as network theory, spectral graph theory, fuzzy graph theory (FGT) and molecular chemistry, etc. The hyper-ZI is introduced especially for fuzzy graphs (FGs) in this study. The study computes this index's bounds for a variety of FG types, including paths, cycles, stars, complete FGs and partial fuzzy subgraphs. It is shown that isomorphic FGs produce the same values for this index. Moreover, interesting connections are established between the hyper-ZI and the second ZI for FGs. Moreover, bounds on this index are found for the following operations: direct product, Cartesian product, composition, join, union, strong product and semi-strong product of two FGs. In the end, the effectiveness of this index is compared with three other topological indices: hyper-ZI for crisp graphs, first ZI for FGs and F-index for FGs, in an analysis of the crime "Murder" in India. While the hyper-ZI for FGs, first ZI for FGs and F-index for FGs yield similar outcomes, the hyper-ZI for FGs demonstrates superior realism in detecting crimes in India compared to its crisp graph counterpart.
萨格勒布指数（ZIs）是重要的图不变量，在数学和化学的许多不同领域中广泛应用，如网络理论、谱图理论、模糊图论（FGT）和分子化学等。本研究特别为模糊图（FGs）引入了超ZIs。该研究计算了各种FG类型（包括路径、圈、星、完全FG和部分模糊子图）的这一指数的界。结果表明，同构的FGs对于该指数产生相同的值。此外，还建立了FGs的超ZIs与第二ZIs之间的有趣联系。此外，还找到了该指数在以下运算中的界：两个FGs的直积、笛卡尔积、合成、联、并、强积和半强积。最后，在对印度“谋杀”犯罪的分析中，将该指数的有效性与其他三个拓扑指数进行了比较：清晰图的超ZIs、FGs的第一ZIs和FGs的F指数。虽然FGs的超ZIs、FGs的第一ZIs和FGs的F指数产生了相似的结果，但FGs的超ZIs在检测印度犯罪方面比其清晰图对应物表现出更高的现实性。
