Comparison Between Two Kinds of Connectivity Indices for Measuring the π-Electronic Energies of Benzenoid Hydrocarbons

1 Introduction

Suppose G is a simple graph with edge set E(G). Let d_u and d_v denote the degrees of the vertices u and v in G, respectively.

The connectivity index of G, proposed by Randić [1] in 1975, is one of the most famous molecular descriptors whose chemical and mathematical properties have been extensively studied [2], [3], [4], [5], [6]. It is defined as

With the intention of extending the applicability of the connectivity index, Bollobás and Erdös [7] in 1998 generalised the connectivity index to the following general connectivity index:

where α is a real number not equal to 0.

There are also many contributions on the general connectivity index χα(G) in mathematical and/or chemical literature [3], [8], [9], [10], [11].

Recently, two kinds of new connectivity indices were introduced [12], [13]. For a graph G, the sum-connectivity index and its generalisation the general sum-connectivity index of G are defined as follows:

and

respectively, where α is a real number not equal to 0.

More applications of the sum-connectivity index and the general sum-connectivity index can be found in the lieterature [12], [13], [14], [15].

Obviously, χ−0.5(G)=χ(G) and χ−0.5s(G)=χs(G). Moreover, corresponding to the sum-connectivity index χs and the general sum-connectivity index χαs, χ and χ_α are often called the product-connectivity index and the general product-connectivity index, respectively.

In the Hückel theory, the π-electronic energy Eπ(G) of a bipartite graph G is defined as the sum Eπ(G)=∑i=1n|λi| of the absolute values of the eigenvalues λ1≤λ2≤⋯≤λn of the adjacency matrix of G. This energy is in good linear correlation with the observed heats of formations of the corresponding conjugated hydrocarbons and is also related to other relevant chemical invariants [16], [17].

2 Materials and Methods

Benzenoid systems are natural graph representations of benzenoid hydrocarbons. A benzenoid system is a finite, connected plane graph without cut vertices, in which every interior face is bounded by a regular hexagon of side of length 1.

The following definitions were introduced in [18]. Suppose H is a benzenoid system with h hexagons and n vertices. We can associate with each path u1−u2−⋯−uk+1 of length k(k≥1,k∈ℕ) in H the vertex degree sequence (du1,du2,…,duk+1). If one goes along the perimeter of H, then a fissure, bay, cove, and fjord, are, respectively, paths of degree sequences (2, 3, 2), (2, 3, 3, 2), (2, 3, 3, 3, 2), and (2, 3, 3, 3, 3, 2). See Figure 1.

Figure 1:

Bay, cove, fissure, and fjord in a hexagonal system.

Fissures, bays, coves, and fjords are called various types of inlets. The number of inlets is defined as the sum of the numbers of fissures, bays, coves, and fjords.

Suppose H is a benzenoid system with n vertices, h hexagons, and r inlets. Let m_ij denote the number of edges in H which have two ends u, v satisfying d_u = i and d_v = j, where d_u and d_v are the degrees of u and v, respectively. By Lemma 1 in [18]

By (2) and (5), the general product-connectivity index of the benzenoid system H is equal to

Similarly, H has the general sum-connectivity index

In (6) and (7), n, h, and r are the numbers of vertices, hexagons, and inlets in H, and α is a real number not equal to 0.

In Table 1, the π-electronic energy E_π, the general product-connectivity index χ_α, and the general sum-connectivity index χαs for 30 lower benzenoid hydrocarbons are listed.

Table 1:

Molecular structure, π-electronic energy E_π, the general product-connectivity index χ_α, and the general sum-connectivity index χαs of 30 lower benzenoids.

Molecule	Molecular structure	Eπ	χα	χαs
Benzene		8.0000	6⋅4α	6⋅4α
Naphthalene		13.6832	6⋅4α+4⋅6α+9α	6⋅4α+4⋅5α+6α
Anthracene		19.3137	6⋅4α+8⋅6α+2⋅9α	6⋅4α+8⋅5α+2⋅6α
Phenanthrene		19.4483	7⋅4α+6⋅6α+3⋅9α	7⋅4α+6⋅5α+3⋅6α
Tetracene		24.9308	6⋅4α+12⋅6α+3⋅9α	6⋅4α+12⋅5α+3⋅6α
Benzo[c]phenanthrene		25.1875	8⋅4α+8⋅6α+5⋅9α	8⋅4α+8⋅5α+5⋅6α
Benzo[a]anthracene		25.1012	7⋅4α+10⋅6α+4⋅9α	7⋅4α+10⋅5α+4⋅6α
Chrysene		25.1922	8⋅4α+8⋅6α+5⋅9α	8⋅4α+8⋅5α+5⋅6α
Triphenylene		25.2745	9⋅4α+6⋅6α+6⋅9α	9⋅4α+6⋅5α+6⋅6α
Pyrene		22.5055	6⋅4α+8⋅6α+5⋅9α	6⋅4α+8⋅5α+5⋅6α
Pentacene		30.5440	6⋅4α+16⋅6α+4⋅9α	6⋅4α+16⋅5α+4⋅6α
Benzo[a]tetracene		30.7255	7⋅4α+14⋅6α+5⋅9α	7⋅4α+14⋅5α+5⋅6α
Dibenzo[a,h]anthracene		30.8805	8⋅4α+12⋅6α+6⋅9α	8⋅4α+12⋅5α+6⋅6α
Dibenzo[a,j]anthracene		30.8795	8⋅4α+12⋅6α+6⋅9α	8⋅4α+12⋅5α+6⋅6α
Pentaphene		30.7627	7⋅4α+14⋅6α+5⋅9α	7⋅4α+14⋅5α+5⋅6α
Benzo[g]chrysene		30.9990	10⋅4α+8⋅6α+8⋅9α	10⋅4α+8⋅5α+8⋅6α
Pentahelicene		30.9362	9⋅4α+10⋅6α+7⋅9α	9⋅4α+10⋅5α+7⋅6α
Benzo[c]chrysene		30.9386	9⋅4α+10⋅6α+7⋅9α	9⋅4α+10⋅5α+7⋅6α
Picene		30.9432	9⋅4α+10⋅6α+7⋅9α	9⋅4α+10⋅5α+7⋅6α
Benzo[b]chrysene		30.8390	8⋅4α+12⋅6α+6⋅9α	8⋅4α+12⋅5α+6⋅6α
Dibenzo[a,c]anthracene		30.9418	9⋅4α+10⋅6α+7⋅9α	9⋅4α+10⋅5α+7⋅6α
Dibenzo[b,g]phenanthrene		30.8336	8⋅4α+12⋅6α+6⋅9α	8⋅4α+12⋅5α+6⋅6α
Perylene		28.2453	8⋅4α+8⋅6α+8⋅9α	8⋅4α+8⋅5α+8⋅6α
Benzo[e]pyrene		28.3361	8⋅4α+8⋅6α+8⋅9α	8⋅4α+8⋅5α+8⋅6α
Benzo[a]pyrene		28.2220	7⋅4α+10⋅6α+7⋅9α	7⋅4α+10⋅5α+7⋅6α
Hexahelicene		36.6814	10⋅4α+12⋅6α+9⋅9α	10⋅4α+12⋅5α+9⋅6α
Benzo[ghi]perylene		31.4251	7⋅4α+10⋅6α+10⋅9α	7⋅4α+10⋅5α+10⋅6α
Hexacene		36.1557	6⋅4α+20⋅6α+5⋅9α	6⋅4α+20⋅5α+5⋅6α
Coronene		34.5718	6⋅4α+12⋅6α+12⋅9α	6⋅4α+12⋅5α+12⋅6α
Ovalene		46.4974	6⋅4α+16⋅6α+19⋅9α	6⋅4α+16⋅5α+19⋅6α

3 Results and Discussion

Results presented in this section show that for α in some interval, the general product-connectivity index χ_α and the general sum-connectivity index χαs can reproduce rather accurately the π-electronic energies of lower benzenoid hydrocarbons.

Based on the data in Table 1, we can obtain two curves, as shown in Figure 2. For 30 lower benzenoid hydrocarbons, the curve of the correlation coefficient for their E_π and χ_α is shown by the solid line. The curve of the correlation coefficient for their E_π and χαs is shown by the dashed line.

For α∈(−0.3461,0), the general product-connectivity index χ_α is a better measure than the sum-connectivity index χαs of the π-electronic energies for benzenoid hydrocarbons. For other α, the conclusion is completely opposite.

In statistics, for a series of n measurements of X and Y, written as x_i and y_i (i=1,2,…,n), the correlation coefficient of X and Y is defined by

where x¯=1n∑i=1nxi and y¯=1n∑i=1nyi.

Using the data x_i and y_i(i=1,2,…,n), we can calculate a regression liney′=ax+b, where a and b are real numbers. This is also called a line of best fit or the least square line. The standard error of fit is defined by

where y′i=axi+b (the predicted value from the regression line).

Both the correlation coefficient and standard error of fit are key goodness-of-fit measures for regression analysis. They can be easily obtained using many mathematical or statistical software.

For α in some interval, there exists good correlation between E_π and χ_α. For example, when α∈[−0.5,0], the correlation coefficient of E_π and χ_α is more than 0.999127. Similarly, there is also good correlation between E_π and χαs for some α. Especially, for α=−0.5, the correlation coefficients and standard errors of fit between E_π and χ−0.5 and χ−0.5s are, respectively

where ρ and s are the correlation coefficient and standard error of fit, respectively. These results were also obtained in [19].

By Figure 2, we have that, for 30 lower benzenoids, χ−0.2661 has the best linear correlation with E_π among all product-connectivity indices and χ−0.5601s has the best linear correlation with E_π among all sum-connectivity indices. The linear correlations between E_π and χ−0.2661 and χ−0.5601s, respectively, are given below:

where ρ and s are, respectively, the correlation coefficient and standard error of fit.

The scatter plots between E_π and χ−0.2661 and E_π and χ−0.5601s are shown in Figure 3 for 30 lower benzenoids.

Figure 2:

Curves of correlation coefficients for lower benzenoids.

Figure 3:

Scatter plots between Eπ and χ−0.2661 and E_π and χ−0.5601s for lower benzenoids.

By (12) and (13), it is easy to see that the sum-connectivity index χ−0.5601s is the best one for measuring the π-electronic energies among all connectivity indices.

4 Conclusion

In this paper, we showed that, for α in some interval, the general product-connectivity index χ_α and the general sum-connectivity index χαs are both closely related molecular descriptors for measuring the π-electronic energies of benzenoid hydrocarbons. Moreover, for α∈(−0.3461,0), χα(G) is a better measure than χαs(G) of the π-electronic energy of G. For other α, the conclusion is completely opposite. A special case (that is, α=−0.5) of our discussion implies the previous results obtained by Lučić et al. [19].