Abstract
The general -path connectivity index of a molecular graph originates from many practical problems such as three-dimensional quantitative structure-activity (3D QSAR) and molecular chirality. It is defined as , where the summation is taken over all possible paths of length of and we do not distinguish between the paths and . In this paper, we focus on the structures of Polycyclic Aromatic Hydrocarbons (), which play a role in organic materials and medical sciences. We try to compute the exact general -path connectivity indices of this family of hydrocarbon structures. Furthermore, we exactly derive the monotonicity and the extremal values of for any real number . These valuable results could produce strong guiding significance to these applied sciences.
1. Introduction
1.1. Application Background
In many fields like physics, chemistry, and electric network, the boiling point, the melting point, the chemical bonds, and the bond energy are all important quantifiable parameters in their fields.
To understand physicochemical properties of chemical compounds or network structures, we abstractly define different concepts, collectively named topological descriptors or topological indices after mathematical modelings. We called them different names such as Randić index and Zagreb index. Different index represents its corresponding chemical structures in graph-theoretical terms via arbitrary molecular graph. A large number of articles about related all topological indices are proposed and based on edges or vertices in molecular graph [1–3].
In the last decades, as a powerful approach, these two-dimensional topological indices have been used to discover many new drugs such as Anticonvulsants, Anineoplastics, Antimalarials or Antiallergics, and Silico generation ([4–8]). Therefore, the practice has proven that the topological indices and quantitative structure-activity relationships (QSAR) have moved from an attractive possibility to representing a foundation stone in the process of drug discovery and other research areas ([9–12]).
Most importantly, with the further study of chemical indices and drug design and discovery, three-dimensional molecular features (topographic indices) and molecular chirality are also presented. It is more and more urgent to study the three-dimensional quantitative structure-activity (3D QSAR) such as molecular chirality. Actually, so far there have been few results expect that one related definition which is generally mentioned in [7].
1.2. Notations
Throughout this paper, we always let be a simple molecular graph with the vertex set and the edge set . Denote the numbers of vertices and edges by and respectively. In physicochemical graph theory, the vertices and the edges correspond to the atoms and the bonds, respectively. Two vertices and are adjacent if there exists an edge between them in . The number of its adjacent vertices is called degree of , denoted by or . The set of all of neighbors of is denoted by or . Specially, a vertex in is called pendant if its degree is one. All other notations and terminologies are referred to [13].
With the intention of extending the applicability of the general Randić index, L.B.Kier, L.H.Hall, E. Estrada, and coworkers considered the general -path connectivity index of aswhere the summation is taken over all possible paths of length of and we do not distinguish between two paths and ([4, 5]).
According to the definition above, it is clear that the general -path connectivity index of a graph is a real number and an important invariant under graph automorphism. It is closely related to the structures of a molecular graph. For any molecular material, only by mastering its structure can we calculate its exact value of the general -path connectivity index.
In this paper, we focus on the structures of Polycyclic Aromatic Hydrocarbons, for short , which play a role in organic materials and medical science. Then we try to compute the general -path connectivity indices of this family of hydrocarbon structures. Furthermore, we exactly derive the monotonicity and the extremal values of for any real number . The valuable results could produce strong guiding significance to these applied sciences.
For convenience, it is necessary to simplify some basic concepts and notations in . An -vertex denotes a vertex with degree , and a -edge stands for an edge connecting a -vertex with a -vertex. Let denote the number of -edge.
Let be a path of length in , that is, . And , is called its degree sequence. Then it is obvious that there are all three types , , and of degree sequences of -paths in . Let , , and denote the numbers of the -paths of the degree sequence types and and in , respectively.
2. Polycyclic Aromatic Hydrocarbons
Large Polycyclic Aromatic Hydrocarbons are ubiquitous combustion products and belong to more important hydrocarbon molecules. They have been implicated as carcinogens and play a role in organic materials and medical science [14].
As we known, Polycyclic Aromatic Hydrocarbons have great significance as molecular analogues of graphite as candidates for interstellar species and as building blocks of functional materials for device applications. In addition, synthetic routes to Polycyclic Aromatic Hydrocarbons are available. Therefore, much detailed knowledge of all molecular features would be necessary for the tuning of molecular properties towards specific applications.
Polycyclic Aromatic Hydrocarbons can be regarded as graphene sheets composed of free radicals of saturated suspended bonds and vice versa; graphene sheets can be interpreted as an infinite number of PAH molecules. The successful application of Polycyclic Aromatic Hydrocarbons in modeling of graphite surface has been reported and references have been provided. The family of Polycyclic Aromatic Hydrocarbons have similar properties with Benzenoid system (Circumcoronene Homologous Series of Benzenoid) ([9–12]). Thus, molecular structures of Polycyclic Aromatic Hydrocarbons play a key role in particular.
For any positive integer , let be the general representation of this Polycyclic Aromatic Hydrocarbon shown in Figure 1. To understand more structures of , the first three members of this hydrocarbon family are given in Figure 2, where is called Benzene with 6 carbon atoms () and 6 hydrogen () atoms, the Coronene with carbon atoms and hydrogen atoms, Circumcoronene with carbon atoms, and hydrogen atoms.
From Figure 1, there are carbon atoms and hydrogen atoms in , denoted and , respectively. Thus, this molecular graph has atoms (or vertices); satisfying the degrees of each hydrogen atom is and each carbon atom is . That is, and .
In Figure 1, each hydrogen atom has just one edge/bond between only one carbon atom and any carbon atom just has three bonds with carbon atoms or hydrogen atom. The edge/bond set of Polycyclic Aromatic Hydrocarbon can be divided into two partitions, -edge set and -edge set. Thusand
If one goes along the perimeter of Polycyclic Aromatic Hydrocarbon System, then there are only two types of structures, which are named and , respectively. A is a structural feature formed by a -vertex (hydrogen atom), followed by two consecutive -vertices (carbon atoms) and then followed by a -vertex (hydrogen atom). A is a structural feature formed by a -vertex (hydrogen atom), followed by three consecutive -vertices (carbon atoms) and then followed by a -vertex (hydrogen atom). See Figure 3.
A -vertex (hydrogen atom) is if it just belong to coves, not on any bay. Denote the numbers of its bays, coves and proper -vertices by bays, coves, and proper -vertices), respectively. According to the structures of with , we haveand
3. Main Results on the Value
Let be the general representation of Polycyclic Aromatic Hydrocarbons molecules for any positive integer (see Figure 1). Let be any -path in which , and are the edges of this -path and , , and . And we call the midedge of this -path.
In this section, we compute the general -path connectivity indices of a family of Polycyclic Aromatic Hydrocarbons. The indices should reflect directly the material natural properties.
Theorem 1. For any real number and positive integer , the general -path connectivity indices of are equal to
Proof. In Polycyclic Aromatic Hydrocarbon with , there are all three types of -paths in which degree sequences are and , respectively. Assume that is an arbitrary -path in which degree sequence is , and . Denote , and , where is its midedge. It is clear that is not -edge.
(1) . By the structures of Benzene (see Figure 2), it is easy to obtainThus(2) . First, we compute the total number of distinct -paths in . For any , which is -edge and not -edge, there exist four distinct -paths in which midedges are the same but other parts are different -edges. Only when traverses all -edge one time in , will it produce all distinct -paths in since any -paths with distinct midedges will be different. ThenSecond, we consider . According to the structure of Polycyclic Aromatic Hydrocarbon , only bays can produce -paths of the type . And one produces one -path of this type. Thus,Consider . It is clear that each -vertex (hydrogen atom) can produce different -paths of the type .
(i) For any improper -vertex which belongs to some bay, it can produce three -paths of the type . Since each bay has two different -vertices (hydrogen atoms), each bay goes with all six -paths of the type . Thus, all improper vertices (hydrogen atoms) produce a total of thirty-six -paths of the type .
(ii) For any proper -vertex (hydrogen atom), it may produce four -paths of the type . Then all proper -vertices (hydrogen atoms) go with all-paths of the type .
Combining (i) and (ii), we haveConsider . Since there are all three types of -paths in which degree sequences are , and in , we havewhich inducesBy usage of the definition, we can compute the general -path connectivity index of Polycyclic Aromatic Hydrocarbon as follows:
4. The Monotonicity and the Extremal Values of
Let be the general representation of the Polycyclic Aromatic Hydrocarbons molecules for any positive integer . In this section, we approach the monotonicity and the extremal values of for any real number and .
Let
Denote
For any real number , we have . Then has the same monotonicity with the function .
Consider the quadratic function on a variable with and . Since for any , the quadratic function is increasing in the interval , where . For any , So the general -path connectivity indices of Polycyclic Aromatic Hydrocarbons are increasing in the interval .
Theorem 2. For any , the general -path connectivity indices of Polycyclic Aromatic Hydrocarbons are increasing in the interval . Therefore, the minimal value happens when .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research is supported by the Natural Science Foundation of China (no. 11701530) and the Fundamental Research Funds for the Central Universities (nos. 2652015193 and 2652017146).