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Bipolar objects are widespread in nature, and their two attributes describe the opposition and unity of the things. Motivated by characterizing fuzzy topological structures by means of fuzzy graphs, we propose a bipolar fuzzy topological graph to measure the features of bipolar fuzzy systems. The topological characteristics in bipolar neutrosophic and bipolar interval-valued fuzzy settings are discussed as well. Several instances are manifested to clarify new definitions and conclusions. Furthermore, the properties of edge calculating and graph isomorphic are determined. [ABSTRACT FROM AUTHOR]

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]

As a critical field of theoretical chemistry, chemical graph theory works as a key part in the model topology calculation and prediction based on chemical molecular structures. The paper can contribute to the research in the following aspects: (1) We propose the concept of bipolar fuzzy influence graph, using positive and negative membership functions to describe positive and negative uncertainties respectively; (2) Based on the new concept, we give a hypothesis to explain the memory of water. The hypothesis is based on two salient facts of the bipolar fuzzy influence graph. Initially, the non-collapsibility of the positive and negative membership functions ensures that the information after using fuzzy coding will not change with the material layer. Then, the influence pairs connect the vertices and edges between different components in the graph through bipolar uncertain influence. At the end of this paper, the conditions for the establishment of the hypothesis and the facts to be verified in the future are analyzed. [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]

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]

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]

Ontology is an effective tool for processing concept semantics, and in the ontology learning algorithm, all the semantic information of each vertex is expressed by a multidimensional vector. The essence of ontology learning algorithm is to obtain ontology function in terms of ontology data samples, so as to map each concept in ontology to a real number. Stability is the foundation of the ontology learning algorithm and the guarantee of its generalization ability. This article relaxes the original uniformly stable hypothesis and proposes the concept of locally ontology relaxed stability. And under the setting of reproducing kernel Hilbert space, the upper bound of stability is verified. Under the framework of random ontology algorithm, the original concept is redefined. The error bounds in general, the reproducing kernel Hilbert space and the stochastic ontology learning algorithm frameworks are obtained in terms of their respective stability definitions. [ABSTRACT FROM AUTHOR]