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The fuzzy membership function has non-collapsability and can be used to characterize uncertainties that cannot be explored by sudden events. As the membership function does not satisfy the complementarity law, a single membership function cannot simultaneously characterize the positive and negative aspects of things. This article considers structured fuzzy data with negative uncertainty information, which is modeled using bipolar fuzzy graphs. On this basis, we consider the connectivity remainder of the model and obtain the corresponding theoretical results. In addition, we have discussed some related fractional factor issues. [ABSTRACT FROM AUTHOR]

In graph-based modelled networks, connectivity and toughness are two types of parameters to measure the robustness of graphs (corresponding networks), and recent studies have shown that these variables are explicitly associated with the existence of H-factors in specific settings, where the latter is inner connected to the feasibility of data transmission in networks. However, the previous contributions presented in the form of several surfaces of connectivity and toughness (isolated toughness or sun toughness) lead to a "choice dilemma" for decision-makers in the network designing stage. In this work, to overcome this problem, we propose a new approach in terms of knee point identification to determine the best combination of connectivity and toughness related parameters. The family of knee points on curves are implemented which vary with respect to m or k. [ABSTRACT FROM AUTHOR]