Novel Face Index for Benzenoid Hydrocarbons

Abstract

A novel topological index, the face index ($FI$), is proposed in this paper. For a molecular graph G, face index is defined as $FI\left(G\right)={\sum }_{f\in F\left(G\right)}d\left(f\right)={\sum }_{v\sim f,f\in F\left(G\right)}d\left(v\right)$, where $d\left(v\right)$ is the degree of the vertex v. The index is very easy to calculate and improved the previously discussed correlation models for $\pi -electron$ energy and boiling point of benzenoid hydrocarbons. The study shows that the multiple linear regression involving the novel topological index can predict the $\pi$-electron energy and boiling points of the benzenoid hydrocarbons with correlation coefficient $r>0.99$. Moreover, the face indices of some planar molecular structures such as 2-dimensional graphene, triangular benzenoid, circumcoronene series of benzenoid are also investigated. The results suggest that the proposed index with good correlation ability and structural selectivity promised to be a useful parameter in QSPR/QSAR.

1. Introduction

In application of mathematical and statistical methods one of the most important purposes is to find a relationship between molecular structure and values of physical, chemical and biological properties. As a result, quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) have been studied.

In QSPR/QSAR studies topological indices(molecular descriptors) are key tools [1,2,3,4]. A topological index is a graph invariant number calculated from a graph associated to a molecule.

Properties estimation can help to minimize the time and cost in producing new chemical materials with desired properties. For estimation some of the statistical tests are discussed in [5]. The physiochemical properties of molecular compounds are important in many fields. A lot of work has done on prediction of physiochemical properties of different compounds [6,7,8,9,10,11].

Molecules can be represented by molecular graphs, where vertices represent the atoms and edges represent the bonds between them. For a molecular graph $G=(V,E)$, V and E represent the set of vertices and the set of edges, respectively. The graph theory-based structure descriptors can be determined by considering graph edges, vertices, or both. Rather simple arithmetic operations are carried out to get numerical indices. These indices are suppose to comprise information on properties/activities of the molecules.

The vertex-connectivity index $\chi \left(G\right)$ of a (molecular) graph G was introduced by Randić in 1975 [12]:

$$\chi \left(G\right)=\sum _{uv\in E\left(G\right)}{\left[d\left(u\right)d\left(v\right)\right]}^{-1/2}$$

where $d\left(u\right)$ is the degree of the vertex u in G.

The edge-connectivity index $ϵ\left(G\right)$ of a (molecular) graph G was introduced by Estrada [11]:

$$ϵ\left(G\right)=\sum _{e\sim f}{\left[d\left(e\right)d\left(f\right)\right]}^{-1/2}$$

where $e\sim f$ shows that the edges e and f are adjacent and $d\left(e\right)=d\left(u\right)+d\left(v\right)-2$ for $e=uv$.

In [13] Nikolić and Trinajstić made a comparison between the vertex and edge connectivity indices for benzenoid hydrocarbons. They showed that the $\pi$-electron energies (E) of benzenoids can be computed by means of either the vertex connectivity index or the edge connectivity index. Their best quadratic model based on the edge-connectivity index can predict the $\pi$-electron energies of benzenoids within the error range of 0.8%–2%. While their best structure-boiling point ($bp$) model was a quadratic model that can predict the boiling points of benzenoid hydrocarbons within the error range of 1.3%–4%. Their best models are as follows

$$E=2.257(\pm 0.020)ϵ+1.486(\pm 0.229)$$

(1)

$n=23;r=0.9993;r^2\left(adjusted\right)=0.9984;s=0.278;F=3276$

$$bp=-0.868(\pm 0.152){ϵ}^2+58.768(\pm 2.986)-81.692(\pm 13.998)$$

(2)

$n=23;r=0.9982;r^2\left(adjusted\right)=0.9962;s=8.0;F=2859$ where n is the number of benzenoids, r is the correlation coefficient, $r^2$(adjusted) is the adjusted correlation coefficient, s is the standard error of estimate and F is the Fisher ratio.

We introduced the novel topological index to improve the efficiency of the above equations.

Definition of the Face Index

When a connected graph can be drawn on the plane without crossing any edges, it is called a planar graph. When a planar graph is drawn in this way, it divides the plane into regions called faces. The unbounded region is called the infinite face.

Let $G=\left(V\right(G),E(G),F(G\left)\right)$ be a finite simple connected planar graph, where $V\left(G\right)$, $E\left(G\right)$ and $F\left(G\right)$ represent the vertex, edge and face sets, respectively. A face $f\in F\left(G\right)$ is incident to an edge $e\in E\left(G\right)$ if e is one of those edges which surrounds the face. Similarly, a face $f\in F\left(G\right)$ is incident to a vertex v in G if v is at the end of one of those incident edges.

The molecular graph of benzene ring is a cycle consist of six vertices and edges each and two faces. The molecular graph of benzenoid hydrocarbons contain considerable number of faces.

In the present study, we introduced a new topological index, the face index ($FI$), using the degree of the vertices incident with the faces of a graph. For a planar graph G, $FI$ can be defined as

$$\begin{array}{c}\hfill FI\left(G\right)=\sum _{f\in F\left(G\right)}d\left(f\right)=\sum _{v\sim f,f\in F\left(G\right)}d\left(v\right)\end{array}$$

$v\sim f$ represents the incidency of the vertex v with the face f.

In benzenoid graphs ($BG$), vertices have either degree two or three; the vertices with degree two are incident with two faces and vertices with degree three are incident with three faces. Let $v_2$ and $v_3$ denote the number of vertices with degree two and three in $BG$, then the definition of $FI$ for $BG$ becomes

$$FI\left(BG\right)=2\times 2\times v_2+3\times 3\times v_3=4v_2+9v_3$$

Hence, $\mathit{FI}$ of any $BG$ can be investigated by just knowing the number of vertices of degree two and three. In case of perylene (P) as shown in Figure 1 $v_2=12$ and $v_3=8$, so $FI\left(P\right)=120$. Authors in [14], discussed theoretical study of singlet fission in oligorylenes.

2. Discussion

The novel topological index is used to construct the structure-energy and structure-boiling point models and the obtained results are comparable with the results obtained in [10,13,15]. We computed $FI$ for 21 common benzenoid hydrocarbons. The values of $FI$ for these hydrocarbons are given in

Table 1. The face index ($FI$), Randić index ($\chi$), edge-connectivity index ($\epsilon$), $\pi$-electron energies $\left(E\right)$ in $\beta$ and boiling points ($bp$) in ${}^{o}C$ of studied benzenoid hydrocarbons.

Benzenoid Hydrocarbons	$\mathit{FI}$	$\mathit{\chi }$	$\mathit{ϵ}$	E	$\mathit{bp}$
benzene	24	3	3	8	80.1
naphthalene	50	4.966	5.455	13.683	218.0
phenanthrene	76	6.95	7.926	19.448	338.0
anthracene	76	6.933	7.942	19.314	340.0
chrysene	102	8.933	10.247	25.192	431.0
benzanthracene	102	8.916	10.414	25.101	425.0
triphenylene	102	8.95	10.414	25.275	429.0
tetracene	102	8.899	10.43	25.188	440.0
benzo(a)pyrene	120	9.916	11.897	28.22	496.0
benzo(e)pyrene	120	9.933	11.897	28.336	493.0
perylene	120	9.933	11.897	28.245	497.0
anthanthrene	138	10.899	13.397	31.253	547.0
benzoperylene	146	10.916	13.379	31.425	542.0
dibenzo(a,c)anthracene	128	10.916	12.902	30.942	535.0
dibenzo(a,h)anthracene	128	10.899	12.885	30.881	535.0
dibenzo(a,i)anthracene	128	10.899	13.218	30.88	531.0
picene	128	10.915	12.686	30.943	519.0
coronene	156	11.899	14.863	34.572	590.0
dibenzo(a,h)pyrene	146	11.566	14.385	33.928	596.0
dibenzo(a,g)pyrene	146	11.491	14.385	33.954	594.0
pyrene	94	7.933	9.408	22.506	393.0

along with the $\pi$-electron energies ($\beta$) and boiling points (°C). The $\pi$-electron energies have been taken from the tabulation of Coulson and Streiwieser [16]. The experimental values of boiling points were taken from Basak et al. [17].

2.1. Relationships between the Face Index and Vertex and Edge Connectivity Indices for Benzenoid Hydrocarbons

We first considered the relationships between the face index and the vertex-connectivity index and between the face index and the edge connectivity index for benzenoid hydrocarbons. Plots of $FI$ versus $\chi$ and $FI$ versus $ϵ$ for benzenoids is given in Figure 2 and Figure 3 and the obtained mathematical models are:

$$FI=0.68(\pm 0.002)\chi +1.741(\pm 0.262)$$

(3)

$n=21;r=0.9896;r^2\left(adjusted\right)=0.9783;s=4.8962;F=906.241$

$$FI=0.091(\pm 0.002)ϵ+1.023(\pm 0.222)$$

(4)

$n=21;r=0.996;r^2\left(adjusted\right)=0.991;s=3.1325;F=2241.371$

From above equations we can see that the face index is more closely related with edge connectivity index rather the vertex connectivity index.

2.2. Linear Mathematical Models of the Face Index ($FI$) for the $\pi$-Electron Energy (E)

In this section, we constructed the linear and multiple linear relationships between the $\pi$-electron energies of considered benzenoid hydrocarbons and the face index. The linear and multiple linear correlation plots are shown in Figure 4 and Figure 5.

The regression analysis gives the following relationships for $\pi$-electron energy (E):

Linear Correlation:

$$E=0.074(\pm 0.003)FI+5.491(\pm 0.766)$$

(5)

$n=21;r=0.9933;r^2\left(adjusted\right)=0.9859;s=0.8129;F=1368.287$

Multivariate Correlation:

$$E=0.002(\pm 0.016)FI+1.123(\pm 0.308)ϵ+1.480(\pm 0.261)\chi +0.095(\pm 0.294)$$

(6)

$n=21;r=0.9996;r^2\left(adjusted\right)=0.9991;s=0.2015;F=7709.838$

The Equation (6) gives the best correlation with energy.

2.3. Linear Mathematical Models of the Face Index ($FI$) for the Boiling Points ($bp$).

We considered the linear and multiple linear relationships between the boiling points ($bp$) of benzenoid hydrocarbons and the face index. The corresponding plots of linear and multiple linear correlation is shown in Figure 6 and Figure 7.

The regression analysis gives the following relationships for the boiling points (°C)of benzenoid hydrocarbons:

Linear Correlation:

$$bp=3.822(\pm 0.136)FI+30.881(\pm 15.878)$$

(7)

$n=21;r=0.9886;r^2\left(adjusted\right)=0.9782;s=19.8306;F=784.290$

Multivariate Correlation:

$$bp=-0.008(\pm 0.001)F{I}^2+1.867(\pm 0.752)FI+38.195(\pm 6.687)ϵ-68.627(\pm 11.542)$$

(8)

$n=21;r=0.9985;r^2\left(adjusted\right)=0.9965;s=7.5990;F=1904.216$

Clearly, Equation (8) gives the best result.

3. Computational Techniques for the Face Index

In this section, we computed the face index of some planar molecular graphs. Method of partitioning of face set based on the degrees of the faces in the graph is used to find the face index. $f^{\infty }$ will denote the unbounded face of the graph. The face index of graphene, triangular benzenoid and circumcoronene series of benzenoid is computed in the following subsections. There are several papers in which authors used different techniques to calculate the certain topological indices of some special molecular graphs [18,19,20,21,22,23,24,25].

3.1. Face Index of Graphene $G(n,s)$

Graphene is a 2-dimensional planar sheet of carbon atoms which is densely packed in a honeycomb crystal lattice, and it is the major element of certain carbon allotropes including charcoal, fullerenes and graphite, see Figure 8. It is represented by $G(s,n)$ where n is number of rows and s is number of benzene rings in each row. In the review [26], authors gave a brief introduction on the recent advances in the study of graphene edges, including edge formation energy, edge reconstruction, method of graphene edge synthesis and the recent progress on metal-passivated graphene edges.

Theorem 1.

Let $G(n,s)$ be a graphene structure, where ‘n’ is the number of rows and ‘s’ is the benzene rings in each row and $n\ge 1$ and $s\ge 1$. The face index of $G(n,s)$ is given as

$$FI\left(G(n,s)\right)=\left\{\begin{array}{cc}18ns+8(n+s)-10,\hfill & ifn\ne 1,\hfill \\ 26s-2,\hfill & ifn=1.\hfill \end{array}\right.$$

Proof. 

Consider the graphene structure, $G(n,s)$, with ‘n’ rows and each row has ‘s’ benzene rings. Let $f_j$ denotes a face having degree j, i.e., $\sum _{v\sim f}d\left(v\right)=j$ and $|f_j|$ denotes the number of faces with degree j. 2-dimensional structure of graphene (shown in Figure 8) contains five types of internal faces $f_{14}$, $f_{15}$, $f_{16}$, $f_{17}$, $f_{18}$ and an external face. For $n=1$ sum of degrees of vertices of external face is $10s+2$, For $n=2$ sum of degrees of vertices of external face is $10s+12$, for $n=3$ sum of degrees of vertices of external face is $10s+22$ and similarly when graphene structure has n rows then sum of degrees of vertices of external face is $10(n+s)-8$.

Table 2. Numbers of $f_{14}$, $f_{15}$, $f_{16}$, $f_{17}$, $f_{18}$ in each row.

Rows	$|{\mathit{f}}_{14}|$	$|{\mathit{f}}_{15}|$	$|{\mathit{f}}_{16}|$	$|{\mathit{f}}_{17}|$	$|{\mathit{f}}_{18}|$
1	2	-	s − 2	-	-
2	-	2	2	2(s − 2)	-
3	-	2	3	2(s − 2)	(s − 1)
4	-	2	4	2(s − 2)	2(s − 1)
.	.	.	.	.	.
.	.	.	.	.	.
.	.	.	.	.	.
n	-	2	n	2(s − 2)	(n − 2)(s − 1)

shows the details of the number of faces with a degree in each row.

Case 1. From the definition of face index and Table 2 for $n\ne 1$, we have

$$\begin{array}{cc}\hfill FI\left(G\right(n,s\left)\right)& =\sum _{v\sim f\in F\left(G\right)}d\left(v\right)\hfill \\ \hfill & =\sum _{v\sim f_{15}\in F\left(G\right)}d\left(v\right)+\sum _{v\sim f_{16}\in F\left(G\right)}d\left(v\right)+\sum _{v\sim f_{17}\in F\left(G\right)}d\left(v\right)+\sum _{v\sim f_{18}\in F\left(G\right)}d\left(v\right)\hfill \\ & +\sum _{v\sim f^{\infty }\in F\left(G\right)}d\left(v\right)\hfill \\ \hfill & =|f_{15}|\left(15\right)+|f_{16}|\left(16\right)+|f_{17}|\left(17\right)+|f_{18}|\left(18\right)+10n+10s-8\hfill \\ \hfill & =2\left(15\right)+n\left(16\right)+2(s-2)\left(17\right)+(n-2)(s-1)\left(18\right)+10n+10s-8\hfill \\ \hfill & =18ns+8(n+s)-10\hfill \end{array}$$

Case 2. When 2-dimensional structure of graphene has just 1 row then it has two types of internal faces $|f_{14}|$, $|f_{16}|$ and one external face whose sum of degrees is $10s+2$.

For $n=1$, $|f_{14}|=2$ and $|f_{16}|=s-2$ as shown in Figure 9, and the face index is

$$\begin{array}{cc}\hfill FI\left(G\right(n,s\left)\right)& =\sum _{v\sim f\in F\left(G\right)}d\left(v\right)\hfill \\ \hfill & =\sum _{v\sim f_{14}\in F\left(G\right)}d\left(v\right)+\sum _{v\sim f_{16}\in F\left(G\right)}d\left(v\right)\hfill \\ \hfill & =|f_{14}|\left(14\right)+|f_{16}|\left(16\right)+10s+2\hfill \\ \hfill & =2\left(14\right)+(s-2)\left(16\right)+10s+2\hfill \\ \hfill & =26s-2\hfill \end{array}$$

This completes the proof. □

3.2. Face index of Triangular Benzenoid $T\left(n\right)$

Now, we calculate the face index of a triangular benzenoid graph $T\left(n\right)$ where n represents the number of rows depicted in Figure 10. For synthesis and characterization of $\pi$-extended triangulene we refer [27].

Theorem 2.

Let $T\left(n\right)$ be a triangular benzenoid gragh where n represents the number of rows and $n\ge 1$. The face index of $T\left(n\right)$ is equal to

$$FI\left(T\left(n\right)\right)=9n^2+21n-6.$$

Proof. 

Let $f_j$ denotes the face having degree j, i.e., $\sum _{v\sim f_j}d\left(v\right)=j$. Triangular benzenoid graph contains four types of internal faces $f_{14}$, $f_{15}$, $f_{17}$ and $f_{18}$ and an external face. When triangular benzenoid has just 1 row then sum of degrees of vertices of external face is 12, when triangular benzenoid has 2 rows then sum of degrees of vertices of external face is 27, when triangular benzenoid has 3 rows then sum of degrees of vertices of external face is 42, and similarly when triangular benzenoid has n rows then sum of degrees of vertices of external face is $15n-3$. The number of internal faces with the degree in each row is mentioned in

Table 3. Number of $f_{14}$, $f_{15}$, $f_{17}$, $f_{18}$ in each row.

Rows	$|{\mathit{f}}_{12}|$	$|{\mathit{f}}_{15}|$	$|{\mathit{f}}_{17}|$	$|{\mathit{f}}_{18}|$
1	1	-	-	-
2	-	3	-	-
3	-	3	3	-
4	-	3	6	1
5	-	3	9	3
6	-	3	12	6
.	.	.	.	.
.	.	.	.	.
.	.	.	.	.
n	-	3	3(n − 2)	$\frac{1}{2}(n-3)(n-2)$

.

From Table 3 and according to the definition of face index, we have

$$\begin{array}{cc}\hfill FI\left(T\right(n\left)\right)& =\sum _{v\sim f\in F\left(G\right)}d\left(v\right)\hfill \\ \hfill & =\sum _{v\sim f_{15}\in F\left(G\right)}d\left(v\right)+\sum _{v\sim f_{17}\in F\left(G\right)}d\left(v\right)+\sum _{v\sim f_{18}\in F\left(G\right)}d\left(v\right)+\sum _{v\sim f^{\infty }\in F\left(G\right)}d\left(v\right)\hfill \\ \hfill & =|f_{15}|\left(15\right)+|f_{17}|\left(17\right)+|f_{18}|\left(18\right)+15n-3\hfill \\ \hfill & =3\left(15\right)+3(n-2)\left(17\right)+{\displaystyle \frac{n^2-5n+6}{2}}\left(18\right)+15n-3\hfill \\ \hfill & =9n^2+21n-6\hfill \end{array}$$

Hence, this is our required result. □

3.3. Face Index of Circumcoronene Series of Benzeniod $H_k$

A circumcoronene homologous series of benzenoid $H_k$ (k is the number of generations) consist of many copies of benzene C₆ on circumference. First three molecule of this series is shown in Figure 11 and the generalized circumcoronene series is shown in Figure 12.

Theorem 3.

Let $H_k$ be a molecular graph of circumcoronene series of benzenoid, where $k\ge 1$. The face index of this graph is given by

$$FI\left(H_k\right)=54k^2-30k.$$

Proof. 

First we consider a circumcoronene series of benzeniod $H_k$ for $k=1,2,3$ presented in Figure 11 where k denotes the number of generations. Let $f_j$ denotes a face having ${\displaystyle \sum _{v\sim f_j}}d\left(v\right)=j$ and $|f_j|$ denotes the number of faces having degree j. In $H_1$ both internal and external faces have degree 12. In $H_2$ there is two types of internal faces $f_{18}$, $f_{16}$ and 1 external face $f_{42}$. In $H_3$ there are 3 types of internal faces $f_{18}$, $f_{17}$, $f_{16}$ and 1 external face $f_{72}$. Figure 12 shown the circumcoronene series of benzenoid $H_k$ for $k\ge 1$.

$$\begin{array}{cc}\hfill FI\left(H_1\right)& =\sum _{w\sim f_{12}\in F\left(G\right)}{d}_w+\sum _{w\sim f^{\infty }\in F\left(G\right)}{d}_w\hfill \\ \hfill & =|f_{12}|\left(12\right)+12=24\hfill \\ \hfill FI\left(H_2\right)& =\sum _{w\sim f_{18}\in F\left(G\right)}{d}_w+\sum _{w\sim f_{16}\in F\left(G\right)}{d}_w+\sum _{w\sim f^{\infty }\in F\left(G\right)}{d}_w\hfill \\ \hfill & =|f_{18}|\left(18\right)+|f_{16}|\left(16\right)+|f_{42}|\left(42\right)\hfill \\ \hfill & =1\left(18\right)+6\left(16\right)+42\hfill \\ \hfill & =156\hfill \end{array}$$

Now we calculate the face index $FI$ for $H_1$, $H_2$ ... $H_5$.

$$\begin{array}{cc}\hfill FI\left(H_3\right)& =\sum _{w\sim f_{18}\in F\left(G\right)}{d}_w+\sum _{w\sim f_{16}\in F\left(G\right)}{d}_w+\sum _{w\sim f_{17}\in F\left(G\right)}{d}_w+\sum _{w\sim f^{\infty }\in F\left(G\right)}{d}_w\hfill \\ \hfill & =|f_{18}|\left(18\right)+|f_{16}|\left(16\right)+|f_{17}|\left(17\right)+|f_{72}|\left(72\right)\hfill \\ \hfill & =7\left(18\right)+6\left(16\right)+6\left(17\right)+72\hfill \\ \hfill & =396\hfill \\ \hfill FI\left(H_4\right)& =\sum _{w\sim f_{18}\in F\left(G\right)}{d}_w+\sum _{w\sim f_{16}\in F\left(G\right)}{d}_w+\sum _{w\sim f_{17}\in F\left(G\right)}{d}_w+\sum _{v\sim f^{\infty }\in F\left(G\right)}{d}_w\hfill \\ \hfill & =|f_{18}|\left(18\right)+|f_{16}|\left(16\right)+|f_{17}|\left(17\right)+|f_{102}|\left(102\right)\hfill \\ \hfill & =19\left(18\right)+6\left(16\right)+12\left(17\right)+102\hfill \\ \hfill & =744\hfill \\ \hfill FI\left(H_5\right)& =\sum _{w\sim f_{18}\in F\left(G\right)}{d}_w+\sum _{w\sim f_{16}\in F\left(G\right)}{d}_w+\sum _{w\sim f_{17}\in F\left(G\right)}{d}_w+\sum _{w\sim f^{\infty }\in F\left(G\right)}{d}_w\hfill \\ \hfill & =|f_{18}|\left(18\right)+|f_{16}|\left(16\right)+|f_{17}|\left(17\right)+|f_{132}|\left(132\right)\hfill \\ \hfill & =37\left(18\right)+6\left(16\right)+18\left(17\right)+132\hfill \\ \hfill & =1200\hfill \end{array}$$

From above information we get a recurrence relation;

$$\begin{array}{cc}\hfill FI\left(H_k\right)& =FI\left(H_{k-1}\right)+108k-84\hfill \end{array}$$

By solving this recurrence relation we get the following;

$$\begin{array}{c}\hfill FI\left(H_k\right)=54k^2-30k\end{array}$$

This is our required result. □

From above result we can obtain the face index of circumcoronene with a window which is $FI\left(C{W}_k\right)=54k^2-30k-232$. Where $C{W}_k$ is the molecular structure of circumcoronene with a window as shown in Figure 13.

4. Conclusions

In this paper a new topological index, the face index ($FI$), is proposed which is based on the degrees of the vertices incident to the face of a graph. We showed that novel face index can be computed by the vertex and edge connectivity indices within the error of 6%–9% and 2%–6%, respectively. Which shows that the novel index is more closely correlate with edge connectivity index. The model Equation (6) involve the face index that improved the Nikolić Equation (1) and can predict the energy of the benzenoid within the error range of 0.6%–1.5%. Equation (8) improved the Nikolić Equation (2) and can compute the boiling point within the range of 1.2%–3.5%. The results warrant further studies on the properties and uses of the face index. In the later section, we considered 2-dimensional graphene, triangular benzenoid and circumcoronene series of benzenoid to study the topological property using their faces. The analytical closed formulas of $FI$ for these molecular graphs are determined.