Let
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
Sampathkumar and Walikar [15] have introduced an important domination invariant called connected domination number, which is defined as follows. Let
E. J. Cockayne et. al [3] have introduced the concept of total domination as follows. Let
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
From Fig. 1, it is clear that the minimal connected dominating set
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 (
Domination parameters and physicochemical properties of octane isomers.Alkane BP TC PC S D −Δ −Δ 6 5 4 125.70 296.20 24.64 111.67 0.7025 2.0449 208.6 41.49 2M 5 4 4 117.6 288.0 24.80 109.84 0.6980 1.8913 215.4 39.67 3M 5 4 4 118.9 292.0 25.60 111.26 0.7058 1.7984 212.5 39.83 4M 5 5 4 117.7 290.0 25.60 109.32 0.7046 1.7673 210.7 39.64 3E 5 5 4 118.5 292.0 25.74 109.43 0.7136 1.7673 210.7 39.64 22MM 4 4 3 106.8 279.0 25.60 103.42 0.6953 1.6744 224.6 37.28 23MM 4 4 3 115.6 293.0 26.60 108.02 0.7121 1.6464 213.8 38.78 24MM 4 4 3 109.4 282.0 25.80 106.98 0.7004 1.6142 219.2 37.76 25MM 4 4 3 109.1 279.0 25.00 105.72 0.6935 1.6449 222.5 37.85 33MM 4 4 3 112.0 290.8 27.20 104.74 0.7100 1.7377 220.0 37.53 34MM 4 4 3 117.7 298.0 27.40 106.59 0.7200 1.5230 212.8 38.97 2M3E 4 4 3 115.6 295.0 27.40 106.06 0.7193 1.5525 211.0 38.52 3M3E 4 4 3 118.3 305.0 28.90 101.48 0.7274 1.5212 214.8 37.99 223MMM 3 3 2 109.8 294.0 28.20 101.31 0.7161 1.4306 220.0 36.91 224MMM 3 3 2 99.24 271.1 25.50 104.09 0.6919 1.4010 224.0 35.14 233MM 3 3 2 114.8 303.0 29.00 102.06 0.7262 1.4931 216.3 37.27 234MMM 3 3 2 113.5 295.0 27.60 102.39 0.7191 1.3698 217.3 37.75 2233MMMM 2 2 2 106.5 270.8 24.50 93.06 0.8242 1.4612 225.6 42.90
Next, we obtain a cross-correlation matrix of domination parameters, which is shown in Table 2.
Cross correlation matrix of domination parameters. 1.000 0.926 1.000 0.949 0.878 1.000
Table 3 contains the correlation of domination parameters with physicochemical properties of octane isomers.
Correlation of Domination parameters with physicochemical properties of octane isomers. BP TC PC S D Δ Δ 0.730 0.316 -0.349 0.902 -0.579 0.903 0.716 0.305 0.644 0.334 -0.216 0.840 -0.599 0.766 0.682 0.159 0.683 0.196 -0.449 0.825 -0.415 0.895 0.653 0.446
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
where
We first consider the regression model containing single descriptors connected domination number
In Tables 4, 5, and 6, statistical parameters for the linear QSPR models in Eqs. (2)-(4) are given.
Statistical parameters for the linear QSPR model (2).Physical property R S F BP 0.730 4.3097 18.307 TC 0.316 9.6182 1.778 PC 0.349 1.3882 2.223 S 0.902 2.008 70.017 0.579 0.0247 8.063 0.903 0.0800 70.944 −Δ 0.716 3.7360 16.826 Δ 0.305 1.7463 1.638
Statistical parameters for the linear QSPR model (3).Physical property R S F BP 0.644 4.8274 11.343 TC 0.334 9.5547 2.015 PC 0.216 1.4467 0.780 S 0.840 2.5277 38.299 0.599 0.0242 8.973 0.766 0.1200 22.696 −Δ 0.682 3.9151 13.891 Δ 0.159 1.1802 0.415
Statistical parameters for the linear QSPR model (4).Physical property R S F BP 0.683 4.6065 14.028 TC 0.196 9.9424 0.637 PC 0.449 1.3236 4.046 S 0.825 2.6297 34.168 0.415 0.0275 3.320 0.895 0.0831 64.657 −Δ 0.653 4.0529 11.893 Δ 0.446 1.6408 3.979
Next, we consider the multiple regression model containing two descriptors of the combination of connected domination number
In Table 7, statistical parameters for the regression model containing two descriptors
Statistical parameters for the QSPR model (5).Physical property R S F BP 0.735 4.4162 8.836 TC 0.335 9.8663 0.947 PC 0.451 1.3657 1.914 S 0.902 2.0734 32.852 0.603 0.0249 4.280 0.922 0.0744 42.743 −Δ 0.718 3.8486 7.966 Δ 0.446 1.6947 1.864
Statistical parameters for the QSPR model (5).Physical property R S F BP 0.731 4.4467 8.613 TC 0.457 9.3140 1.979 PC 0.511 1.3153 2.650 S 0.907 2.0211 34.964 0.719 0.0217 8.007 0.911 0.0793 36.801 −Δ 0.721 3.8312 8.107 Δ 0.583 1.5385 3.861
Statistical parameters for the QSPR model (5).Physical property R S F BP 0.690 4.7201 6.800 TC 0.392 9.6316 1.364 PC 0.585 1.2411 3.900 S 0.860 2.4573 21.228 0.644 0.0239 5.302 0.896 0.0854 30.663 −Δ 0.691 3.9946 6.856 Δ 0.661 1.4213 5.812
Finally, we consider the multiple regression model containing three descriptors, connected domination number
In Table 10, statistical parameters for the regression model containing three descriptors
Statistical parameters for the QSPR model (6).Physical property R S F BP 0.736 4.5666 5.518 TC 0.470 9.5666 1.323 PC 0.585 1.2844 2.430 S 0.907 2.0912 21.777 0.738 0.0218 5.580 0.930 0.0731 30.018 −Δ 0.722 3.9554 5.095 Δ 0.668 1.4589 3.757
The correlation coefficients of physicochemical properties with individual
The regression analysis of models (2)-(4) with single descriptors, reveals some interesting results for mean radius
From Table 4, we can see that for entropy
From Table 5, we can see that for entropy
From Table 6, we can see that for entropy
The use of two descriptors, the combination of
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.