The General ( α , 3 )-Path Connectivity Indices of Polycyclic Aromatic Hydrocarbons

The general (α, t)-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 tRα(G) = ∑Pt=Vi1 Vi2 ⋅⋅⋅Vit+1⊆G[d(Vi1 )d(Vi2 ) ⋅ ⋅ ⋅ d(Vit+1 )] , where the summation is taken over all possible paths of length t of G and we do not distinguish between the paths Vi1Vi2 ⋅ ⋅ ⋅ Vit+1 and Vit+1 ⋅ ⋅ ⋅ Vi2Vi1 . In this paper, we focus on the structures of Polycyclic Aromatic Hydrocarbons (PAHn), which play a role in organic materials andmedical sciences.We try to compute the exact general (α, 3)-path connectivity indices of this family of hydrocarbon structures. Furthermore, we exactly derive the monotonicity and the extremal values of 3Rα(PAHn) for any real number α. These valuable results could produce strong guiding significance to these applied sciences.


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][2][3].
In the last decades, as a powerful approach, these twodimensional topological indices have been used to discover many new drugs such as Anticonvulsants, Anineoplastics, Antimalarials or Antiallergics, and Silico generation ( [4][5][6][7][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][10][11][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 V are adjacent if there exists an edge  = V 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  as    () = ∑ where the summation is taken over all possible paths of length  of  and we do not distinguish between two paths 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 (, 3)-path connectivity indices of this family of hydrocarbon structures.Furthermore, we exactly derive the monotonicity and the extremal values of 3   (  ) for any real number .The valuable results could produce strong guiding significance to these applied sciences.

Polycyclic Aromatic Hydrocarbons (𝑃𝐴𝐻 𝑛 )
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][10][11][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  1 is called Benzene with 6 carbon atoms () and 6 hydrogen () atoms,  2 the Coronene with 24 carbon atoms and 12 hydrogen atoms,  3 Circumcoronene with 54 carbon atoms, and 18 hydrogen atoms.
From Figure 1, there are 6 2 carbon atoms and 6 hydrogen atoms in   , denoted () = 6 and () = 6 2 , respectively.Thus, this molecular graph has 6 2 + 6 atoms (or vertices); satisfying the degrees of each hydrogen atom is 1 and each carbon atom is 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, (1, 3)-edge set and (3, 3)-edge set.Thus and If one goes along the perimeter of Polycyclic Aromatic Hydrocarbon System, then there are only two types of structures, which are named  and V, respectively.A  is a structural feature formed by a 1-vertex (hydrogen atom), followed by two consecutive 3-vertices (carbon atoms) and then followed by a 1-vertex (hydrogen atom).A V is a structural feature formed by a 1-vertex (hydrogen atom), followed by three consecutive 3-vertices (carbon atoms) and then followed by a 1-vertex (hydrogen atom).See Figure 3.
A 1-vertex (hydrogen atom) is  if it just belong to coves, not on any bay.Denote the numbers of its bays, coves and proper 1-vertices by (bays), (coves), and (proper 1-vertices), respectively.According to the structures of   with  ≥ 2, we have and

Main Results on the Value 3 𝑅 𝛼 (𝑃𝐴𝐻 𝑛 )
Let   be the general representation of Polycyclic Aromatic Hydrocarbons molecules for any positive integer  (see Figure 1).Let  3 = V  0 V  1 V  2 V  3 be any 3-path in which  1 ,  2 , and  3 are the edges of this 3-path and , and V  2 V  3 =  3 .And we call  2 the midedge of this 3path.
In this section, we compute the general (, 3)-path connectivity indices of a family of Polycyclic Aromatic Hydrocarbons.The indices should reflect directly the material natural properties.

Figure 3 :
Figure 3: Structure of a bay and a cove.