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

Authors: Haiying Wang; Chuantao Li
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

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 Rtα(G)=∑Pt=vi1vi2⋯vit+1⊆G[d(vi1)d(vi2)⋯d(vit+1)]α, 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 and medical 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 R3α(PAHn) for any real number α. These valuable results could produce strong guiding significance to these applied sciences.


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## Body

## 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 namedtopological descriptors ortopological 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 alltopological 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 letG=(V(G),E(G)) be a simple molecular graph with the vertex set V(G) and the edge set E(G). Denote the numbers of vertices and edges by |V(G)| and |E(G)| respectively. In physicochemical graph theory, the vertices and the edges correspond to the atoms and the bonds, respectively. Two vertices u and v areadjacent if there exists an edge e=uv between them in G. The number of its adjacent vertices is calleddegree of u, denoted by dG(u) or d(u). The set of all of neighbors of u is denoted by NG(u) or N(u). Specially, a vertex in G is calledpendant 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 consideredthe general(α,t)-path connectivity index of G as(1)RtαG=∑Pt=vi1vi2⋯vit+1⊆Gdvi1dvi2⋯dvit+1α,where the summation is taken over all possible paths of length t of G and we do not distinguish between two paths vi1vi2⋯vit+1 and vit+1⋯vi2vi1 ([4, 5]).According to the definition above, it is clear that the general(α,t)-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 (α,t)-path connectivity index.In this paper, we focus on the structures of Polycyclic Aromatic Hydrocarbons, for shortPAHn, which play a role in organic materials and medical science. Then we try to compute the general (α,3)-path connectivity indices of this family of hydrocarbon structures. Furthermore, we exactly derive the monotonicity and the extremal values of R3α(PAHn) for any real number α. The valuable results could produce strong guiding significance to these applied sciences.For convenience, it is necessary to simplify some basic concepts and notations inPAHn. An i-vertex denotes a vertex with degree i, and a (j,k)-edge stands for an edge connecting a j-vertex with a k-vertex. Let mjk denote the number of (j,k)-edge.Letvi0vi1⋯vit be a path Pt of length t in PAHn, that is, Pt=vi0vi1⋯vit. And (d(vi0),d(vi1), ⋯,d(vit)) is called its degree sequence. Then it is obvious that there are all three types (1,3,3,1), (1,3,3,3), and (3,3,3,3) of degree sequences of 3-paths in PAHn. Let m1331, m1333, and m3333 denote the numbers of the 3-paths of the degree sequence types (1,3,3,1) and (1,3,3,3) and (3,3,3,3) in PAHn, respectively.


## 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 namedtopological descriptors ortopological 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 alltopological 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 letG=(V(G),E(G)) be a simple molecular graph with the vertex set V(G) and the edge set E(G). Denote the numbers of vertices and edges by |V(G)| and |E(G)| respectively. In physicochemical graph theory, the vertices and the edges correspond to the atoms and the bonds, respectively. Two vertices u and v areadjacent if there exists an edge e=uv between them in G. The number of its adjacent vertices is calleddegree of u, denoted by dG(u) or d(u). The set of all of neighbors of u is denoted by NG(u) or N(u). Specially, a vertex in G is calledpendant 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 consideredthe general(α,t)-path connectivity index of G as(1)RtαG=∑Pt=vi1vi2⋯vit+1⊆Gdvi1dvi2⋯dvit+1α,where the summation is taken over all possible paths of length t of G and we do not distinguish between two paths vi1vi2⋯vit+1 and vit+1⋯vi2vi1 ([4, 5]).According to the definition above, it is clear that the general(α,t)-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 (α,t)-path connectivity index.In this paper, we focus on the structures of Polycyclic Aromatic Hydrocarbons, for shortPAHn, which play a role in organic materials and medical science. Then we try to compute the general (α,3)-path connectivity indices of this family of hydrocarbon structures. Furthermore, we exactly derive the monotonicity and the extremal values of R3α(PAHn) for any real number α. The valuable results could produce strong guiding significance to these applied sciences.For convenience, it is necessary to simplify some basic concepts and notations inPAHn. An i-vertex denotes a vertex with degree i, and a (j,k)-edge stands for an edge connecting a j-vertex with a k-vertex. Let mjk denote the number of (j,k)-edge.Letvi0vi1⋯vit be a path Pt of length t in PAHn, that is, Pt=vi0vi1⋯vit. And (d(vi0),d(vi1), ⋯,d(vit)) is called its degree sequence. Then it is obvious that there are all three types (1,3,3,1), (1,3,3,3), and (3,3,3,3) of degree sequences of 3-paths in PAHn. Let m1331, m1333, and m3333 denote the numbers of the 3-paths of the degree sequence types (1,3,3,1) and (1,3,3,3) and (3,3,3,3) in PAHn, respectively.


## 2. Polycyclic Aromatic Hydrocarbons(PAHn)

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 integern, let PAHn be the general representation of this Polycyclic Aromatic Hydrocarbon shown in Figure 1. To understand more structures of PAHn, the first three members of this hydrocarbon family are given in Figure 2, where PAH1 is called Benzene with 6 carbon atoms (C) and 6 hydrogen (H) atoms, PAH2 the Coronene with 24 carbon atoms and 12 hydrogen atoms, PAH3 Circumcoronene with 54 carbon atoms, and 18 hydrogen atoms.Figure 1
The general representation of Polycyclic Aromatic Hydrocarbon.Figure 2
The first three graphs of Polycyclic Aromatic Hydrocarbon.From Figure1, there are 6n2 carbon atoms and 6n hydrogen atoms in PAHn, denoted N(H)=6n and N(C)=6n2, respectively. Thus, this molecular graph has 6n2+6n atoms (or vertices); satisfying the degrees of each hydrogen atom is 1 and each carbon atom is 3. That is, VPAHn=6n2+6n and EPAHn=3×6n2+1×6n/2=9n2+3n.In Figure1, 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(2)m13=NH=6n,and(3)m33=EPAHn-NH=9n2-3n.If one goes along the perimeter of Polycyclic Aromatic Hydrocarbon System, then there are only two types of structures, which are namedbays and coves, respectively. A bay 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 cove 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.Figure 3
Structure of a bay and a cove.A1-vertex (hydrogen atom) is proper if it just belong to coves, not on any bay. Denote the numbers of its bays, coves and proper 1-vertices by N(bays), N(coves), and N(proper 1-vertices), respectively. According to the structures of PAHn with n≥2, we have(4)Nbays=6Ncoves=6n-1and(5)Nproper1-vertices=NH-2Nbays=6n-2·6=6n-2.


## 3. Main Results on the ValueR3α(PAHn)

LetPAHn be the general representation of Polycyclic Aromatic Hydrocarbons molecules for any positive integer n (see Figure 1). Let P3=vi0vi1vi2vi3 be any 3-path in which e1,e2, and e3 are the edges of this 3-path and vi0vi1=e1, vi1vi2=e2, and vi2vi3=e3. And we call e2 the midedge of this 3-path.In this section, we compute the general(α,3)-path connectivity indices of a family of Polycyclic Aromatic Hydrocarbons. The indices should reflect directly the material natural properties.Theorem 1.
For any real numberα and positive integer n≥1, the general (α,3)-path connectivity indices of PAHn are equal to(6)R3αPAHn=2·32α+1·2·32α+1·n2+2·3α·2-3α+1·n+1-2·3α+32α.Proof.
In Polycyclic Aromatic HydrocarbonPAHn with n≥1, there are all three types of 3-paths in which degree sequences are (1,3,3,1)(1,3,3,3) and (3,3,3,3), respectively. Assume that P3=vi0vi1vi2vi3 is an arbitrary 3-path in which degree sequence is (d(vi0),d(vi1),d(vi2), and d(vi3)). Denote vi0vi1=e1, vi1vi2=e2 and vi2vi3=e3, where e2 is its midedge. It is clear that e2 is not (1,3)-edge.
(1)n=1. By the structures of Benzene (see Figure 2), it is easy to obtain(7)m1331=6;m1333=2·NH=12;m3333=6.Thus(8)R3αPAH1=∑P3=vi1vi2vi3vi4⊆PAH1dvi1dvi2dvi3dvi4α=m1331·1×3×3×1α+m1333·1×3×3×3α+m3333·3×3×3×3α=6·32α+12·33α+6·34α=2·32α+1·1+2·3α+32α.
(2)n≥2. First, we compute the total number of distinct 3-paths in PAHn. For any e2, which is (3,3)-edge and not (1,3)-edge, there exist four distinct 3-paths in which midedges are the same e2 but other parts are different (3,3)-edges. Only when e2 traverses all (3,3)-edge one time in PAHn, will it produce all distinct 3-paths in PAHn since any 3-paths with distinct midedges will be different. Then(9)NP3=4m33=4·EPAHn-NH=4·9n2+3n-6·n=12n3n-1.
Second, we considerm1331. According to the structure of Polycyclic Aromatic Hydrocarbon PAHn, only bays can produce 3-paths of the type (1,3,3,1). And one produces one 3-path of this type. Thus,(10)m1331=Nbays=6.
Considerm1333. It is clear that each 1-vertex (hydrogen atom) can produce different 3-paths of the type (1,3,3,3).
(i) For any improper1-vertex H which belongs to some bay, it can produce three 3-paths of the type (1,3,3,3). Since each bay has two different 1-vertices (hydrogen atoms), each bay goes with all six 3-paths of the type (1,3,3,3). Thus, all improper vertices (hydrogen atoms) produce a total of thirty-six 3-paths of the type (1,3,3,3).
(ii) For any proper1-vertex (hydrogen atom), it may produce four 3-paths of the type (1,3,3,3). Then all proper 1-vertices (hydrogen atoms) go with all(11)Nproper  1-vertices·43-paths of the type (1,3,3,3).
Combining (i) and (ii), we have(12)m1333=36+Nproper  1-vertices·4=36+6n-2·4=24n-12.
Considerm3333. Since there are all three types of 3-paths in which degree sequences are (1,3,3,1), (1,3,3,3) and (3,3,3,3) in PAHn, we have(13)m1331+m1333+m3333=NP3,which induces(14)m3333=NP3-m1331-m1333=12n3n-1-6-24n-12=36n2-36n+6.
By usage of the definition, we can compute the general(α,3)-path connectivity index of Polycyclic Aromatic Hydrocarbon PAHn as follows:(15)R3αPAHn=∑P3=vi1vi2vi3vi4⊆PAHndvi1dvi2dvi3dvi4α=m1331·1×3×3×1α+m1333·1×3×3×3α+m3333·3×3×3×3α=6·32α+24n-12·33α+36n2-36n+6·34α=32α·4·32α+2·n2+4·3α+1·2-3α+1·n+6-4·3α+1+2·32α+1=2·32α+1·2·32α+1·n2+2·3α·2-3α+1·n+1-2·3α+32α.


## 4. The Monotonicity and the Extremal Values ofR3α(PAHn)

LetPAHn be the general representation of the Polycyclic Aromatic Hydrocarbons molecules for any positive integer n. In this section, we approach the monotonicity and the extremal values of R3α(PAHn) for any real number α and n≥3.Let(16)R3αPAHn=2·32α+1·2·32α+1·n2+2·3α·2-3α+1·n+1-2·3α+32α.Denote(17)fn=2·32α+1·n2+2·3α·2-3α+1·n+1-2·3α+32α.For any real numberα, we have 2·32α+1>0. Then R3α(PAHn) has the same monotonicity with the function f(n).Consider the quadratic functionf(n) on a variable n with a=2·32α+1 and b=2·3α·(2-3α+1). Since a=2·32α+1>0 for any α∈(-∞,+∞), the quadratic function f(n) is increasing in the interval [n0,+∞), where n0=max⁡{1,-b/2a}. For any α∈(-∞,+∞), -b/2a=1/2-1/3α+1<1/2<1. So the general (α,3)-path connectivity indices of Polycyclic Aromatic Hydrocarbons PAHn are increasing in the interval [1,+∞).Theorem 2.
For anyα∈(-∞,+∞), the general (α,3)-path connectivity indices of Polycyclic Aromatic Hydrocarbons PAHn are increasing in the interval [1,+∞). Therefore, the minimal value happens when n=1.



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