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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]

By modeling the network structure with a graph, the corresponding data transmission problem can be transformed into a fractional flow problem on the graph. Under certain specific conditions, the transmission path of data is required to pass through certain special channels, thus the corresponding mathematical framework is transformed into the existence of fractional covered graphs under the special network graph structure. In this paper, by means of the necessary and sufficient conditions for fractional (g; f; n';m)- critical covered graphs, we obtain some sufficient conditions on graph parameters of fractional covered graphs. Furthermore, we point out that these results are tight in terms of some counterexamples, and the results have potential guiding value for network designing. [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]

There are a lot of nanomaterials and related chemical substances synthesized in the laboratory every year, which makes the test of their performance has become a hard work. The theory of nanoscience from the perspective of graph theory provides an excellent idea, characteristics of the nanomaterials can be obtained by calculating topological indices in their corresponding molecular graphs, and have attracted the attention of scholars in the field of nanoscience. In this paper, we learn the characteristics of nanostructures from mathematical point of view. Some important nanomaterials are selected and their multiplicative atom-bond connectivity indices are determined by edge set divided trick. These theoretical results can be considered as a guideline in nanoengineering. [ABSTRACT FROM AUTHOR]

There are a lot of nanomaterials and related chemical substances synthesized in the laboratory every year, which makes the test of their performance has become a hard work. The theory of nanoscience from the perspective of graph theory provides an excellent idea, characteristics of the nanomaterials can be obtained by calculating topological indices in their corresponding molecular graphs, and have attracted the attention of scholars in the field of nanoscience. In this paper, we learn the characteristics of nanostructures from mathematical point of view. Some important nanomaterials are selected and their multiplicative atom-bond connectivity indices are determined by edge set divided trick. These theoretical results can be considered as a guideline in nanoengineering. [ABSTRACT FROM AUTHOR]

In theoretical chemistry, chemical compounds and drugs are modeled as graphs in which each vertex represents an atom of molecule and covalent bounds between atoms are represented by edges between the corresponding vertices. Such a graph constructed according to a chemical compounds is often called its molecular graph and the indices defined over this molecular graph have been shown to be strongly correlated to various chemical properties of the compounds. In this paper, in terms of analyzing the structure of chemical molecular graph and the techniques of edge dividing, we report several distance-based indices for four families of dendrimer Nanostars, including second ABC index, second GA related indices, PI related indices and polynomials and Szeged related indices and polynomials. Furthermore, we discuss the distance-based indices of three kinds of widely appeared molecular structures: TC4C8 (R) Nanotorus, H-Naphtalenic nanotubes and 1,3-adamantane array. The formulations of second ABC index, second GA related indices, vertex PI and Szeged polynomials are manifested as additional conclusions. [ABSTRACT FROM AUTHOR]

In theoretical chemistry, chemical compounds and drugs are modeled as graphs in which each vertex represents an atom of molecule and covalent bounds between atoms are represented by edges between the corresponding vertices. Such a graph constructed according to a chemical compounds is often called its molecular graph and the indices defined over this molecular graph have been shown to be strongly correlated to various chemical properties of the compounds. In this paper, in terms of analyzing the structure of chemical molecular graph and the techniques of edge dividing, we report several distance-based indices for four families of dendrimer Nanostars, including second ABC index, second GA related indices, PI related indices and polynomials and Szeged related indices and polynomials. Furthermore, we discuss the distance-based indices of three kinds of widely appeared molecular structures: TC4C8 (R) Nanotorus, H-Naphtalenic nanotubes and 1,3-adamantane array. The formulations of second ABC index, second GA related indices, vertex PI and Szeged polynomials are manifested as additional conclusions. [ABSTRACT FROM AUTHOR]

In theoretical chemistry, the Szeged index and Szeged related indices were introduced to measure the stability of alkanes and the strain energy of cycloalkanes. In this paper, we determine the edge-vertex Szeged index and vertex-edge Szeged index of unilateral polyomino chain and unilateral hexagonal chain. Furthermore, the total Szeged indices, third atom bond connectivity indices, PI polynomials, vertex PI polynomials, Szeged polynomials and edge Szeged polynomials of these chemical structures are presented. [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]

The learning and optimizing of the scoring function are widely used in neural network, information retrieval and protein analysis. The multi-dividing ontology algorithms have drawn plenty of attention recent years. In this paper, we consider the Bayes-optimal scoring function in multi-dividing setting. By virtue of conditional risk, proper loss theory and derivative computing, we determine the scoring function in multi-dividing setting for certain special case. The results achieved in our paper illustrate the promising application prospects for multi-dividing ontology algorithm. [ABSTRACT FROM AUTHOR]