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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]

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]