Institute of Computing Science and Technology, Guangzhou University, Guangzhou, China.
Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore, Pakistan.
PLoS One. 2024 Jan 30;19(1):e0297197. doi: 10.1371/journal.pone.0297197. eCollection 2024.
Fuzzy graphs are very important when we are trying to understand and study complex systems with uncertain and not exact information. Among different types of fuzzy graphs, cubic fuzzy graphs are special due to their ability to represent the membership degree of both vertices and edges using intervals and fuzzy numbers, respectively. To figure out how things are connected in cubic fuzzy graphs, we need to know about cubic α-strong, cubic β-strong and cubic δ-weak edges. These concepts better help in making decisions, solving problems and analyzing things like transportation, social networks and communication systems. The applicability of connectivity and comprehension of cubic fuzzy graphs have urged us to discuss connectivity in the domain of cubic fuzzy graphs. In this paper, the terms partial cubic α-strong and partial cubic δ-weak edges are introduced for cubic fuzzy graphs. The bounds and exact expression of connectivity index for several cubic fuzzy graphs are estimated. The average connectivity index for cubic fuzzy graphs is also defined and some results pertaining to these concepts are proved in this paper. The results demonstrate that removing some vertices or edges may cause a change in the value of connectivity index or average connectivity index, but the change will not necessarily be related to both values. This paper also defines the concepts of partial cubic connectivity enhancing node and partial cubic connectivity reducing node and some related results are proved. Furthermore, the concepts of cubic α-strong, cubic β- strong, cubic δ-weak edge, partial cubic α-strong and partial cubic δ-weak edges are utilized to identify areas most affected by a tsunami resulting from an earthquake. Finally, the research findings are compared with the existing methods to demonstrate their suitability and creativity.
当我们试图理解和研究具有不确定和不精确信息的复杂系统时，模糊图非常重要。在不同类型的模糊图中，立方模糊图是特殊的，因为它们能够分别使用区间和模糊数来表示顶点和边的隶属度。为了弄清楚立方模糊图中事物是如何连接的，我们需要了解立方α-强、立方β-强和立方δ-弱边。这些概念有助于做出决策、解决问题，并分析交通、社交网络和通信系统等事物。连通性的适用性和立方模糊图的理解促使我们讨论立方模糊图中的连通性。在本文中，引入了部分立方α-强和部分立方δ-弱边的概念来描述立方模糊图。估计了几种立方模糊图的连通指数的上下界和精确表达式。还定义了立方模糊图的平均连通指数，并在本文中证明了这些概念的一些结果。结果表明，删除一些顶点或边可能会导致连通指数或平均连通指数的值发生变化，但这种变化不一定与这两个值都有关。本文还定义了部分立方增强连通节点和部分立方降低连通节点的概念，并证明了一些相关结果。此外，利用立方α-强、立方β-强、立方δ-弱边、部分立方α-强和部分立方δ-弱边的概念来识别地震引发海啸影响最严重的区域。最后，将研究结果与现有方法进行比较，以证明它们的适用性和创新性。
