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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]

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]

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 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]