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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 fractional factor theory of graphs originated from the feasible flow problem in communication networks. With the development of computer networks, this problem has been highly valued by scholars from the field of computer science and mathematics. This paper studies the sufficient conditions for the existence of fractional factors in the different setting of network from a theoretical perspective. These theoretical results provide the basis for the initial network designing. We first study the relationship between vulnerable parameter in networks and the existence of fractional factor, and an isolated toughness condition for a graph to be fractional (g; f; n')- critical is determined. Then, we illustrate some neighborhood union conditions for independent-set-deletable deleted graphs, and point out that the conditions are best in some senses. Moreover, we present an independent set condition for a graph to be fractional ID-(g; f;m)-deleted. Also, the result is tight on independent set degree condition. Finally, we introduce the concept of all fractional ID-(g; f;m)-deleted graph, and several conclusions are obtained from the known results. [ABSTRACT FROM AUTHOR]

In computer network, the structure of network is modelled by a graph, and toughness, as a graph parameter, is employed to detect the vulnerability of the network. A fractional k-factor exists to satisfy h(e) = 1, if for arbitrary e ∈ E(G), graph G is fractional k-covered. This work studies the mutual influence of toughness and fractional fractional k-covered graph, and obtains the result that a graph G is fractional k-covered if t(G) ≥ k - 1/k. This conclusion implies that the best toughness bound for the existence of the fractional k-factor is also the tight t(G) condition of the fractional k-coverage graph. Finally, some extended conclusions are presented. [ABSTRACT FROM AUTHOR]