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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]

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]

Ontology, a common tool in various fields of natural sciences, aims to get the optimal ontology similarity calculation function. Favored by researchers from semantic query and other disciplines, it is used to calculate the similarity between ontology concepts. The semantic information of each concept is expressed by a d-dimensional vector, and the similarity calculation is transformed into the geometric distance calculation of the two corresponding vectors. Using the Mahalanobis distance calculation formula, the ontology algorithm can contribute to getting the optimal distance matrix. In this paper, in terms of the coordinate descent trick and iterative method, we get the ontology distance matrix learning algorithm, and then apply it to ontology similarity computation and ontology mapping. Moreover, the ontology distance matrix learning approach in the manifold setting is discussed, and its kernel solution is studied as well. The main ontology learning algorithm is illustrated by a comparison of other ontology algorithmic data in a specific application context. [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]

Ontology, as a useful tool, has been widely applied in various fields, and ontology concept similarity calculation is an essential problem in these application algorithms. A recent method to get similarity between vertices on ontology is not by pairwise computation but based on a function which maps ontology graph into a line and maps every vertex in graph into a real-value, the similarity is measured by the difference of their corresponding scores. The multi-dividing method is suitable for ontology problem and plays a key role in achieving this. Such ontology function is given by learning a training sample which contains a subset of vertices with k classes from ontology graph. In this paper, we propose a new multi-dividing ontology algorithm framework, which is designed to avoid the choice of loss function. Meanwhile, there is such a vertex selection policy in new multi-dividing ontology algorithm that it guarantees that the new algorithm can be employed for an ontology graph with its structure rather than a tree. We provide some theoretical characteristics of the new multi-dividing ontology algorithm, and show that the new algorithm is convergent. [ABSTRACT FROM AUTHOR]

In recent years, many learning technologies have been applied in ontology similarity measuring and ontology mapping via learning an ontology function f : V → R which maps an ontology graph to the real line. In these settings, all the information for an ontology vertex (corresponding to concept) is expressed as a vector. However, in a special application, the value of ontology function for each ontology vertex is determined by a few components of the vector. The aim of feature extraction for ontology vector is to obtain these components to fix the index set of the vector, and such a procedure is equivalent to learning an ontology sparse vector in which most components are zero. In this paper, we raise an ontology sparse vector learning model for ontology similarity measuring and ontology mapping in terms of SOCP. The balance term consists of Ω norm, and the directed acyclic graph is employed in ontology setting for backward and forward procedure. Then, the active index set algorithm is designed to moderate the value of p, thus applications will be extended. Finally, five experiments are presented on various fields to verify the efficiency of the new ontology algorithm for ontology similarity measuring and ontology mapping in multidisciplinary research. [ABSTRACT FROM AUTHOR]

Along with the arrival of the era of large amount of data, many machine learning methods have been applied to the ontology similarity calculation and ontology mapping. In this paper, we raise an infinite push model for ontology similarity measuring and ontology mapping in multi-dividing setting. The iterative algorithm and generalization bound are given by virtue of the dual solution for optimization and the trick of covering number approach. Furthermore, the fast ontology algorithm for standard ontology SVM by virtue of infinite push multi-dividing ontology algorithm is obtained. Finally, four experiments presented on various fields verify the efficiency of the new computation model for ontology similarity measuring and ontology mapping applications in multi-dividing setting. [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]

As a semantic representation model, ontology has penetrated into all areas of natural science and social science. The core issue of ontology applications is similarity computing between ontology concepts. In this article, we report a sparse vector learning algorithm for ontology similarity measure and ontology mapping in terms of sub-gradient calculating and iterative computation. The main procedure of our iterative computation is based on the tricks of fuse and divide. The simulation experimental results show that the new proposed algorithm has high efficiency and accuracy in ontology similarity measure and ontology mapping in biology and physical education science. [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]

Ontology learning problem has raised much attention in semantic structure expression and information retrieval. As a powerful tool, ontology is evenly employed in various subjects, such as neuroscience, medical science, pharmacopedia, chemistry, education and other social science. Ontology similarity measuring plays a vital role in practical implementations since essential issues of ontology mapping are also similarity calculating. In ontology function learning, one learns a real valued score function that assigns scores to each ontology vertex which corresponds to a concept. Thus, the similarity between vertices is determined by means of the absolute value of difference between their corresponding scores. In this paper, we report the new optimization algorithms for obtaining ontology function in view of ontology sparse vector learning. The implementation of ontology algorithms is mainly based on iterative calculation in which we consider the whole matrix version of framework and ontology sparse vector are updated in each iterative. The data results obtained from four simulation experiments reveal that our newly proposed ontology approach has high efficiency and accuracy in biology and plant science with regard to ontology similarity measure, and humanoid robotics and education science with regard to ontology mapping. [ABSTRACT FROM AUTHOR]

Ontology, as an effective structured information storage, classification, and statistical tool, has been attached importance from researchers of various disciplines. At present, ontology has become the core study field of semantic network, artificial intelligence, and information retrieval. The core topic of ontology algorithm is to calculate the similarity between concepts, so as to obtain the potential information of each other. In this paper, we present the boosting based iterative learning algorithm to compute the ontology sparse vector, and thus get the ontology function which maps each ontology concept to a real number. Since the advantage of ontology sparse vector is dimensionality reduction, the algorithm can extract the key information from the high dimensional ontology representation data. At last, we select four classical ontologies to test the efficiency of new ontology algorithm, and the rest give the positive answer. [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]