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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 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]

Graph theory is used for modeling, designing, analysis and understanding chemical structures or chemical networks and their properties. The molecular graph is a graph consisting of atoms called vertices and the chemical bond between atoms called edges. In this article, we study the chemical graphs of carbon graphite and crystal structure of cubic carbon. Moreover, we compute and give closed formulas of degree based additive topological indices, namely hyper-Zagreb index, first multiple and second multiple Zagreb indices, and first and second Zagreb polynomials. [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]

A graph G is called a fractional (k; n ' ;m)-critical deleted graph if any n ' vertices are removed from G the resulting graph is a fractional (k;m)-deleted graph. In this paper, we prove that for integers for each pair of non-adjacent vertices x, y of G, then G is a fractional (k; n ' ;m)-critical deleted graph. The bounds for neighborhood union condition, the order n and the minimum degree δ(G) of G are all sharp. [ABSTRACT FROM AUTHOR]

In the setting of software definition network (SDN), whether there exists an available transmission plan between any two sites is equal to whether there exists a fractional factor after deleting certain vertices and edges in the corresponding graph. A graphGis called a fractional ID-(a, b,m)-deleted graph if after deleting any independent-set I, the remaining graph G - I is a fractional (a, b,m)-deleted graph. A fractional (g, f)-factor F of a graph G is called a Hamiltonian fractional (g, f)-factor if F includes a Hamiltonian cycle. Furthermore, we say that G has a ID-Hamiltonian fractional (g, f)-factor if after deleting any independent set ofGthe remaining graph ofGincludes a Hamiltonian fractional (g, f)-factor. In this paper, we first give a binding number condition for a graph to be a fractional ID-(a, b,m)-deleted graph, and then two sufficient conditions for graphs to have ID-Hamiltonian fractional (g, f)-factors are obtained. [ABSTRACT FROM AUTHOR]

Strong connections between the chemical characteristics of chemical compounds, materials and drugs and their topological structures have been proved by lots of previous research. For instance, they get the connections in melting point and boiling point. The chemical indices from these molecular topological structures turn out to be favorable for chemists, material and medical scientists, when they try to get the relevant chemical reactivity, biological activity and physical features. As a result, the shortage of the experiments can be covered and made up, if the conduct the study of the topological indices on the molecular structures. Meanwhile, the study can also make contributions by providing the theoretical evidence in chemical, pharmaceutical and material engineering. On the basis of graph analysis and computation derivation, some degree and distance mixing indices of some classes of Harary graphs and coronene polycyclic aromatic hydrocarbons are determined in the paper. Therefore, the study lay a theoretical foundation for the material properties. [ABSTRACT FROM AUTHOR]