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As an important tool to represent structured and uncertain data, graph models are widely used in computer networks and chemical molecular structure representation. If there are uncertainties in the vertices or the binary relationship between vertices in the graph, the membership function is introduced to the vertex set and the edge set, and the fuzzy graph is used to describe such uncertain structural features. The independent set is an important object to measure the topological structure of the graph, as the basis of many graph parameters. In this paper, a novel concept of independent set is proposed in bipolar fuzzy graph setting. The fuzzy topological parameters to measure its degree of independence are defined, and the characteristics of bipolar independence degree are obtained by using graph theory and fuzzy set theory, especially for two types of product bipolar fuzzy graphs. Finally, the algorithm for calculating the maximum (resp. minimum) positive (resp. negative) fuzzy subgraph with given positive (resp. negative) degree of independence is designed. [ABSTRACT FROM AUTHOR]

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]