Correlation of domination parameters with physicochemical properties of octane isomers

Let G = (V,E) be a graph. The number of vertices of G we denote by n and the number of edges we denote by m, thus |V(G)| = n and |E(G)| = m. For undefined terminologies, we refer the reader to [7].

Chemical graph theory is the branch of mathematical chemistry. It is concerned with handling chemical graphs that represent chemical system. Hence chemical graph theory deals with analysis of all consequences of connectivity in a chemical system. It has found to be a useful tool in QSAR (Quantitative Structure-Activity Relationships) and QSPR (Quantitative Structure-Property Relationship) [4,10,16]. Numerous studies have been made relating to the above mentioned fields by using what are called topological indices.In 1975, Randić [13] proposed a topological index that has become one of the most widely used in both QSAR and QSPR studies.

One of the fastest growing areas in graph theory is the study of domination and related subset problems such as independence, irredundance, covering and matching. An excellent treatment of fundamentals of domination in graphs is given by the book by Haynes et. al [5]. Surveys of several advances topics in domination are given in the book edited by Haynes et. al [6].

Let G = (V,E) be a graph. A subset S of V is called a dominating set of G if every vertex in V − S is adjacent to at least one vertex in S. The domination number γ(G) of a graph G is the minimum cardinality of a dominating set of G.

Sampathkumar and Walikar [15] have introduced an important domination invariant called connected domination number, which is defined as follows. Let G = (V,E) be a graph. A subset S of V is called a connected dominating set of G if every vertex in V − S is adjacent to at least one vertex in S and the subgraph induced by the set S is connected. The connected domination number γ_c(G) of a graph G is the minimum cardinality of a connected dominating set of G.

E. J. Cockayne et. al [3] have introduced the concept of total domination as follows. Let G = (V,E) be a graph. A subset S of V is called a total dominating set of G if every vertex G is adjacent some vertex in S. The total domination number γ_t(G) of a graph G is the minimum cardinality of a total dominating set of G.

The total edge domination is an analogues concept total domination, which is introduced and studied in [9]. For more details on domination see [1, 2].

Octane isomers have become an important set of organic molecules to test the applicability of various topological parameters in quantitative structure-property/activity relationships (QSPR/QSAR). These compounds are structurally diverse enough to yield considerable variation in shape, branching and non polarity [14]. In a comprehensive study of numerous properties of octane isomers, Randić et. al [10,12,14] have used single molecular descriptors and concluded that different physicochemical properties depend on different descriptors.

So far in the literature of chemical graph theory, domination parameters have not been used to predict the physical properties of chemical compounds. Therefore, in the present study an attempt has been made to study physical properties of octane isomers by using domination parameters.

The values of γ_c, γ_t and γt′$\begin{array}{} \displaystyle \gamma _t^\prime \end{array}$ of molecular graph of 2,2,3,3−tetramethyl butane are illustrated below.

From Fig. 1, it is clear that the minimal connected dominating set D_c = {2,5}, the total dominating set D_t = {2,5}, and the total edge dominating set Dt′={2,5,7}$\begin{array}{} \displaystyle D_t^\prime = \left\{ {2,5,7} \right\} \end{array}$. Hence, γ_c(G) = |D_c| = 2, γ_t(G) = |D_t| = 2, and γt′(G)=|Dt′|=3$\begin{array}{} \displaystyle \gamma _t^\prime \left( G \right) = \left| {D_t^\prime } \right| = 3 \end{array}$.

Fig. 1

Eight physicochemical properties of octane isomers have been selected on the availability [8] of a suitable body of data: boiling point (BP), critical temperature (CT), critical pressure (CP), entropy (S), density (D), mean radius (Rm2$\begin{array}{} \displaystyle R_m^2 \end{array}$), and heat of vaporization (H_ν), heat of formation (H_f). The values are compiled in Table 1.

Table 1

Domination parameters and physicochemical properties of octane isomers.

Next, we obtain a cross-correlation matrix of domination parameters, which is shown in Table 2.

Table 2

Cross correlation matrix of domination parameters.

Table 3 contains the correlation of domination parameters with physicochemical properties of octane isomers.

Table 3

Correlation of Domination parameters with physicochemical properties of octane isomers.

From Table 2, it is found that the cross correlation coefficient of domination parameters are found to be high (0.878 − 0.949). Also from Table 3, the correlation coefficient of domination parameters with physico-chemical properties of isomers are found to be good except critical pressure (CP) and density (D). For these two physicochemical properties of octane isomers the domination parameters are not well-correlated.

A generalized linear regression model has been proposed for the relationship of physicochemical properties of octane isomers with the domination parameters γ_c, γ_t, and γt′$\begin{array}{} \displaystyle \gamma _t^\prime \end{array}$, respectively.

P=a+∑i=13biγi,$$\begin{array}{} \displaystyle P = a + \sum\limits_{i = 1}^3 {{b_i}} {\gamma _i}, \end{array}$$(1)

where P refers to a physicochemical property, a is constant, and b_i is the sensitivity of γ_i towards P. For our convince, we assume that γ₁ = γ_c, γ₂ = γ_t, and γ3=γt′$\begin{array}{} \displaystyle {\gamma _3} = \gamma _t^\prime \end{array}$.

We first consider the regression model containing single descriptors connected domination number γ_c, total domination number γ_t and total edge domination γt′$\begin{array}{} \displaystyle \gamma _t^\prime \end{array}$ separately.

P=a+bγc.$$\begin{array}{} \displaystyle P{\rm{ }} = a + b{\gamma _c}. \end{array}$$(2)

P=a+bγt.$$\begin{array}{} \displaystyle P{\rm{ }} = a + b{\gamma _t}. \end{array}$$(3)

P=a+bγt′.$$\begin{array}{} \displaystyle P{\rm{\;}} = a + b\gamma _t^\prime . \end{array}$$(4)

In Tables 4, 5, and 6, statistical parameters for the linear QSPR models in Eqs. (2)-(4) are given.

Table 4

Statistical parameters for the linear QSPR model (2).

Table 5

Statistical parameters for the linear QSPR model (3).

Table 6

Statistical parameters for the linear QSPR model (4).

Next, we consider the multiple regression model containing two descriptors of the combination of connected domination number γ_c, total domination number γ_t, and total edge domination γt′$\begin{array}{} \displaystyle \gamma _t^\prime \end{array}$, separately.

P=a+∑i=12biγi.$$\begin{array}{} \displaystyle P = a + \sum\limits_{i = 1}^2 {{b_i}} {\gamma _i}. \end{array}$$(5)

In Table 7, statistical parameters for the regression model containing two descriptors γ_c and γ_t are provided. Table 8 collects the statistical parameters for the regression model containing two descriptors γ_c and γt′$\begin{array}{} \displaystyle \gamma _t^\prime \end{array}$. In Table 9, statistical parameters for the regression model containing two descriptors γ_t and γt′$\begin{array}{} \displaystyle \gamma _t^\prime \end{array}$ appear.

Table 7

Statistical parameters for the QSPR model (5).

Table 8

Statistical parameters for the QSPR model (5).

Table 9

Statistical parameters for the QSPR model (5).

Finally, we consider the multiple regression model containing three descriptors, connected domination number γ_c, total domination number γ_t, and total edge domination γt′$\begin{array}{} \displaystyle \gamma _t^\prime \end{array}$.

P=a+∑i=13biγi.$$\begin{array}{} \displaystyle P = a + \sum\limits_{i = 1}^3 {{b_i}} {\gamma _i}. \end{array}$$(6)

In Table 10, statistical parameters for the regression model containing three descriptors γ_c, γ_t, and γt′$\begin{array}{} \displaystyle \gamma _t^\prime \end{array}$ are shown.

Table 10

Statistical parameters for the QSPR model (6).

The correlation coefficients of physicochemical properties with individual γ_i values show some significant results. The entropy and the mean radius values, which generally do not show good relationships with any single descriptor, are found to have a good correlation coefficient (0.825 − 0.902) and (0.766 − 0.903), respectively. The critical temperature,pressure, density, and enthalpy of vaporization values are found to have poor correlation coefficient (−0.599 − 0.446).

The regression analysis of models (2)-(4) with single descriptors, reveals some interesting results for mean radius Rm2$\begin{array}{} \displaystyle R_m^2 \end{array}$ and entropy S.

From Table 4, we can see that for entropy S, R = 0.902 (standard error of 2.008°C) and F = 70.017, and for mean radius Rm2$\begin{array}{} \displaystyle R_m^2 \end{array}$, R = 0.903 (standard error of 0.0800°C) and F = 70.944.

From Table 5, we can see that for entropy S, R = 0.840 (standard error of 2.5277°C) and F = 38.299, and for mean radius Rm2$\begin{array}{} \displaystyle R_m^2 \end{array}$, R = 0.766 (standard error of 0.1200°C) and F = 22.696.

From Table 6, we can see that for entropy S, R = 0.825 (standard error of 2.6297°C) and F = 34.168, and for mean radius Rm2$\begin{array}{} \displaystyle R_m^2 \end{array}$, R = 0.895 (standard error of 0.0831°C) and F = 64.657.

The use of two descriptors, the combination of γ_c, γ_t, and γt′$\begin{array}{} \displaystyle \gamma _t^\prime \end{array}$ gave a good correlation coefficients for entropy ranging from R = (0.860 − 0.907) having standard error ranging from (2.0211 − 2.4573) and F = (21.228 − 34.964). For mean radius Rm2$\begin{array}{} \displaystyle R_m^2 \end{array}$, R = (0.896 − 0.922) having standard error ranging from (0.0744 − 0.0854) and F = (30.663 − 42.743).

However, the addition of third descriptor does not produce any significant improvement in the regression model. Hence, for other physical properties of the octane isomers, the relationship with domination parameters can be seen in Tables 4-10.

The results of QSPR studies reveals that the regression model (2) is the most significant model to predict the physicochemical properties like entropy and mean radius of isomers. Hence, domination parameters be used as candidate to represent the molecular structure for predicting physicochemical properties.