In mathematical chemistry, a large number of topological indices are used to predict the physicochemical properties of compounds, especially in the study of quantitative structure-proerty relationship (QSPR).
However, many topological indices have almost the same predictive ability. In this paper, we focus on how to use fewer topological indices to predict the physicochemical properties of compounds through the QSPR analysis of connectivity indices of benzene hydrocarbons.
In mathematical chemistry, topological indices (or molecular structure descriptors) have been extensively studied in various areas of mathematics [1,2], physics [3], informatics [4], biology [5], especially in chemistry [6-9], such as chemical documentation, isomer discrimination, molecular complexity, chirality, similarity/dissimilarity, drug design, database selection, lead optimization, quantitative structure-activity/proerty relationship (QSAR/QSPR), and etc. A degree-based topological indices is a common type of topological index which is calculated using the degrees of a chemical graph, such as Zagreb indices [10], Narumi-Katayama index [11], Albertson index [12], reduced second Zagreb index [13], the second Hyper-Zagreb index [14], the Hyper-Zagreb index [15], the forgotten topological index [16], Sombor index [17], the face index [18]. Some of the degree-based topological indices are called the connectivity indices or branching indices originating from Randić’s seminal paper [19], which includes Randić index [19], harmonic index [20], sum-connectivity indices [21], atom-bond-connectivity index [22] and atom-bond sum-connectivity index [23]. The connectivity indices are the best known and widely applied in mathematical chemistry.
Let \(G\) be a simple undirected connected graph with vertex set \(V(G)\) and edge set \(E(G)\). The degree of a vertex \(u\in V(G)\) is the number of edges that are incident to \(u\), denoted by \(d_u(G)\), \(d_u\) for short. The maximum degree of \(G\) are denoted by \(\Delta(G)\), or simply \(\Delta\). A chemical graph (or molecular graph) is a graph with \(2\leq \Delta\leq 4\). The first Zagreb index (\(M_1\)), Randić index (\(R\)) and atom-bond-connectivity index (\(ABC\)) are classical degree-based topological indices, and defined as \[\begin{aligned} M_1(G) & = & \sum_{uv\in E(G)}(d_u+d_v),\\ R(G) & = & \sum_{uv\in E(G)}\frac{1}{\sqrt{d_ud_v}},\\ ABC(G) & = & \sum_{uv\in E(G)}\sqrt{\frac{d_u+d_v-2}{d_ud_v}}, \end{aligned}\] which are frequently used to predict the physicochemical properties and biological activity of chemical compounds in QSAR/QSPR.
The Sombor index (\(SO\)), harmonic index (\(H\)), sum-connectivity index (\(SCI\)) and atom-bond sum-connectivity index (\(ABS\)) are introduced and defined as \[\begin{aligned} SO(G) & = & \sum_{uv\in E(G)}\sqrt{d^2_u+d^2_v},\\ H(G) & = & \sum_{uv\in E(G)}\frac{2}{d_u+d_v},\\ SCI(G) & = & \sum_{uv\in E(G)}\frac{1}{\sqrt{d_u+d_v}},\\ ABS(G) & = & \sum_{uv\in E(G)}\sqrt{1-\frac{2}{d_u+d_v}}. \end{aligned}\] In particular, research shows that the Sombor index, sum-connectivity index and atom-bond sum-connectivity index are useful in predicting physicochemical properties for octane isomers (see [21,24,25]). However, professor Gutman [26] believes that Sombor index is not of great applicability in QSPR and QSAR studies. Indeed, Sombor index and the first Zagreb index are almost the same correlation with physicochemical properties for octane isomers (see [25]) and lower benzenoid hydrocarbons (see Table 2.3 in Second 2).
From a mathematical perspective, it is not difficult to find such a fact that \[\frac{1}{\sqrt{2}}(a+b)\leq \sqrt{a^2+b^2}\leq \frac{5}{6}(a+b)\] for any integer number \(1\leq a\leq 4\) and \(1\leq b\leq 4\). Thus, for the chemical graph, we have \[\frac{1}{\sqrt{2}}M_1\leq SO\leq \frac{5}{6}M_1.\] Similarly, we consider connectivity indices. Note that the degree of benzenoid hydrocarbons is only 2 and 3. We obtain the following relationship of connectivity indices for benzenoid hydrocarbons: \[\frac{40}{41}R<H\leq R,\] \[R\leq SCI \leq \frac{\sqrt{30}}{5}R,\] \[ABC\leq ABS \leq \frac{\sqrt{30}}{5}ABC.\] On the other hand, we find that the difference between these corresponding indices is very small for lower benzenoid hydrocarbons. Let \(e_{i,j}\) be the number of edges in \(G\) joining vertices of degree \(i\) and \(j\). For benzenoid hydrocarbons, we have \[\begin{aligned} M_1 & = & 4e_{2,2}+5e_{2,3}+6e_{3,3},\\ \sqrt{2}SO & = & 4e_{2,2}+\sqrt{26}e_{2,3}+6e_{3,3},\\ R & = & \frac{1}{2}e_{2,2}+\frac{1}{\sqrt{6}}e_{2,3}+\frac{1}{3}e_{3,3},\\ H & = & \frac{1}{2}e_{2,2}+\frac{2}{5}e_{2,3}+\frac{1}{3}e_{3,3},\\ SCI & = & \frac{1}{2}e_{2,2}+\frac{1}{\sqrt{5}}e_{2,3}+\frac{1}{\sqrt{6}}e_{3,3},\\ ABC & = & \frac{1}{\sqrt{2}}e_{2,2}+\frac{1}{\sqrt{2}}e_{2,3}+\frac{2}{3}e_{3,3},\\ ABS & = & \frac{1}{\sqrt{2}}e_{2,2}+\frac{\sqrt{15}}{5}e_{2,3}+\frac{\sqrt{6}}{3}e_{3,3}. \end{aligned}\] If a benzenoid graph contains \(n\) vertices, \(h\) hexagons and \(r\) inlets, then \(e_{2,2}=n-2h-r+2\), \(e_{2,3}=2r\) and \(e_{3,3}=3h-r-3\) (see [27]). Thus we obtain \[\begin{aligned} M_1 & = & 4n+10h-10,\\ \sqrt{2}SO & = & 4n+10h-10+ 0.198039r,\\ R & = & 0.5n-0.016837r,\\ H & = & 0.5n-0.033333r,\\ SCI & = & 0.5n+0.224745(h-1)-0.013821r,\\ ABC & = & 0.5n+0.585786(h-1)+0.040440r,\\ ABS & = & 0.5n+1.035276(h-1)+0.025590r. \end{aligned}\]
The above discussion reveals to us why these corresponding indices have almost the same correlation and predictive ability for lower benzenoid hydrocarbons (see Table 2.3 and Table 2.4 in Section 2). This means that using connectivity indices to study the QSPR of benzene hydrocarbons, only calculating \(R\) and \(ABC\) is sufficient. Therefore, we should first consider the degree composition of the chemical graph in order to select the degree-based topological indices with significant differences for predicting physicochemical properties of compounds, which has guiding significance for the study of quantitative structure-proerty relationship of chemical compounds. Meanwhile, we should make reasonable use of existing topological indices (especially classical topological indices) to reduce the generation of similar topological indices from chemical point of view.
In this section, we study the correlation between some degree-based topological indices and physicochemical properties of twenty-two lower benzenoid hydrocarbons. The data on the physicochemical properties of twenty-two lower benzenoid hydrocarbons, including boiling point (Bp), entropy (S), octanol-water partition coefficient (logP), Kovats retention index (RI), enthalpy of formation \((\Delta H_{f})\), \(\pi\)-electronic energy (\(E_{\pi}\)), and Gibb’s energy (GE), is derived from reference [28,29]. The data in Tables 2.3 and 2.4 confirm our viewpoint.
Molecule | Bp | S | logP | RI | \(\Delta H_{f}\) | \(E_{\pi}\) | GE |
---|---|---|---|---|---|---|---|
Naphthalene | \(218\) | \(79.38\) | \(3.3\) | \(200\) | \(150.6\) | \(13.6832\) | \(252.38\) |
Phenanthrene | \(338\) | \(93.79\) | \(4.46\) | \(300\) | \(209.1\) | \(19.4483\) | \(383.08\) |
Anthracene | \(340\) | \(92.43\) | \(4.45\) | \(301.69\) | \(218.3\) | \(19.3137\) | \(383.08\) |
Chrysene | \(431\) | \(106.83\) | \(5.81\) | \(400\) | \(267.7\) | \(25.1922\) | \(513.78\) |
Tetraphene | \(425\) | \(108.22\) | \(5.76\) | \(398.5\) | \(276.9\) | \(25.1012\) | \(513.78\) |
Triphenylene | \(429\) | \(104.66\) | \(5.49\) | \(400\) | \(258.5\) | \(25.2745\) | \(513.78\) |
Naphthacene | \(440\) | \(105.47\) | \(5.76\) | \(408.3\) | \(286.1\) | \(13.6832\) | |
Benzo[a]pyrene | \(496\) | \(111.85\) | \(6.13\) | \(453.44\) | \(279.9\) | \(28.222\) | \(621.88\) |
Benzo[e]pyrene | \(493\) | \(110.46\) | \(6.44\) | \(450.73\) | \(289.1\) | \(28.3361\) | \(621.88\) |
Perylene | \(497\) | \(109.10\) | \(6.25\) | \(456.22\) | \(279.9\) | \(28.2453\) | \(621.88\) |
Anthanthrene | \(547\) | \(114.10\) | \(7.04\) | \(503.89\) | \(310.5\) | \(31.253\) | – |
Benzo[ghi]perylene | \(542\) | \(114.10\) | \(6.63\) | \(501.32\) | \(301.3\) | \(31.4251\) | \(729.98\) |
Dibenz[a,h]anthracene | \(536\) | \(119.87\) | \(6.75\) | \(495.45\) | \(335.5\) | \(30.8805\) | \(644.48\) |
Dibenz[a,j]anthracene | \(531\) | \(119.87\) | \(6.54\) | \(489.8\) | \(335.5\) | \(30.8795\) | \(644.48\) |
Picene | \(519\) | \(119.87\) | \(7.11\) | \(500\) | \(326.3\) | \(30.9432\) | \(644.48\) |
Coronene | \(590\) | \(116.36\) | \(7.64\) | \(549.67\) | \(322.7\) | \(34.5718\) | \(838.08\) |
Benzo[c]phenanthrene | \(448\) | \(113.61\) | \(5.7\) | \(391.12\) | \(280.5\) | \(25.1875\) | – |
Pyrene | \(404\) | \(96.06\) | \(4.88\) | \(351.22\) | \(230.5\) | \(22.5055\) | \(491.18\) |
Dibenzo[a,e]pyrene | \(592\) | \(124.89\) | \(7.28\) | \(551.53\) | \(338.5\) | – | – |
Dibenzo[a,h]pyrene | \(596\) | \(123.50\) | \(7.28\) | \(559.9\) | \(347.7\) | \(33.928\) | – |
Dibenzo[a,i]pyrene | \(594\) | \(123.50\) | \(7.28\) | \(556.47\) | \(347.7\) | – | – |
Dibenzo[a,l]pyrene | \(595\) | \(131.69\) | \(7.71\) | \(553\) | \(351.2\) | – | – |
Molecule | \(M_1\) | SO | R | H | SCI | ABC | ABS |
---|---|---|---|---|---|---|---|
Naphthalene | \(50\) | \(35.6354\) | \(4.9663\) | \(4.9333\) | \(5.1971\) | \(7.7377\) | \(8.1575\) |
Phenanthrene | \(76\) | \(54.1602\) | \(6.9495\) | \(6.9\) | \(7.4080\) | \(11.1924\) | \(12.0468\) |
Anthracene | \(76\) | \(54.3003\) | \(6.9327\) | \(6.8667\) | \(7.3942\) | \(11.2328\) | \(12.0724\) |
Chrysene | \(102\) | \(72.6850\) | \(8.9327\) | \(8.8667\) | \(9.6190\) | \(14.6470\) | \(15.9361\) |
Tetraphene | \(102\) | \(72.8251\) | \(8.9158\) | \(8.8333\) | \(9.6051\) | \(14.6875\) | \(15.9617\) |
Triphenylene | \(102\) | \(72.5450\) | \(8.9495\) | \(8.9\) | \(9.6328\) | \(14.6066\) | \(15.9105\) |
Naphthacene | \(102\) | \(72.9651\) | \(8.8990\) | \(8.8\) | \(9.5913\) | \(14.7279\) | \(15.9873\) |
Benzo[a]pyrene | \(120\) | \(85.5530\) | \(9.9158\) | \(9.8333\) | \(10.8299\) | \(16.6875\) | \(18.4112\) |
Benzo[e]pyrene | \(120\) | \(85.4130\) | \(9.9327\) | \(9.8667\) | \(10.8437\) | \(16.6470\) | \(18.3856\) |
Perylene | \(120\) | \(85.4130\) | \(9.9327\) | \(9.8667\) | \(10.8437\) | \(16.6470\) | \(18.3856\) |
Anthanthrene | \(138\) | \(98.4209\) | \(10.8990\) | \(10.8\) | \(12.0408\) | \(18.7279\) | \(20.8863\) |
Benzo[ghi]perylene | \(138\) | \(98.2809\) | \(10.9158\) | \(10.8333\) | \(12.0546\) | \(18.6875\) | \(20.8607\) |
Dibenz[a,h]anthracene | \(128\) | \(91.3499\) | \(10.8990\) | \(10.8\) | \(11.8161\) | \(18.1421\) | \(19.8510\) |
Dibenz[a,j]anthracene | \(128\) | \(91.3499\) | \(10.8990\) | \(10.8\) | \(11.8161\) | \(18.1421\) | \(19.8510\) |
Picene | \(128\) | \(91.2098\) | \(10.9158\) | \(10.8333\) | \(11.8299\) | \(18.1017\) | \(19.8254\) |
Coronene | \(156\) | \(111.1489\) | \(11.8990\) | \(11.8\) | \(13.2655\) | \(20.7279\) | \(23.3358\) |
Benzo[c]phenanthrene | \(102\) | \(72.6850\) | \(8.9327\) | \(8.8667\) | \(9.6190\) | \(14.6470\) | \(15.8942\) |
Pyrene | \(94\) | \(67.0282\) | \(8.9327\) | \(7.8667\) | \(7.9327\) | \(13.2328\) | \(14.5219\) |
Dibenzo[a,e]pyrene | \(146\) | \(103.9377\) | \(11.9158\) | \(11.8333\) | \(13.0546\) | \(20.1017\) | \(22.2749\) |
Dibenzo[a,h]pyrene | \(146\) | \(104.0778\) | \(11.8990\) | \(11.8\) | \(13.0408\) | \(20.1421\) | \(22.3005\) |
Dibenzo[a,i]pyrene | \(146\) | \(104.0778\) | \(11.8990\) | \(11.8\) | \(13.0408\) | \(20.1421\) | \(22.3005\) |
Dibenzo[a,l]pyrene | \(146\) | \(103.9378\) | \(11.9158\) | \(11.8333\) | \(13.0546\) | \(20.1017\) | \(22.2749\) |
Physico-chemical property | \(M_1\) | \(SO\) | R | H | SCI | ABC | ABS |
---|---|---|---|---|---|---|---|
Bp | \(0.9924\) | \(0.9925\) | \(0.9958\) | \(0.9957\) | \(0.9965\) | \(0.9964\) | \(0.9951\) |
S | \(0.9236\) | \(0.9238\) | \(0.9591\) | \(0.9593\) | \(0.9507\) | \(0.9443\) | \(0.9349\) |
logP | \(0.9828\) | \(0.9829\) | \(0.9890\) | \(0.9890\) | \(0.9889\) | \(0.9881\) | \(0.9862\) |
RI | \(0.9922\) | \(0.9923\) | \(0.9988\) | \(0.9987\) | \(0.9986\) | \(0.9979\) | \(0.9959\) |
\(\Delta H_{f}\) | \(0.9410\) | \(0.9418\) | \(0.9731\) | \(0.9724\) | \(0.9656\) | \(0.9614\) | \(0.9524\) |
\(E_{\pi}\) | \(0.9069\) | \(0.9057\) | \(0.9039\) | \(0.9055\) | \(0.9066\) | \(0.9046\) | \(0.9062\) |
GE | \(0.9963\) | \(0.9962\) | \(0.9737\) | \(0.9738\) | \(0.9822\) | \(0.9865\) | \(0.9918\) |
\(M_1\) | \(SO\) | R | H | SCI | ABC | ABS | |
---|---|---|---|---|---|---|---|
\(M_1\) | \(1\) | \(1\) | \(0.9916\) | \(0.9916\) | \(0.9957\) | \(0.9975\) | \(0.9993\) |
\(SO\) | \(1\) | \(1\) | \(0.9917\) | \(0.9917\) | \(0.9958\) | \(0.9976\) | \(0.9993\) |
R | \(0.9916\) | \(0.9917\) | \(1\) | \(1\) | \(0.9993\) | \(0.9981\) | \(0.9957\) |
H | \(0.9916\) | \(0.9917\) | \(1\) | \(1\) | \(0.9993\) | \(0.9980\) | \(0.9956\) |
SCI | \(0.9957\) | \(0.9958\) | \(0.9993\) | \(0.9993\) | \(1\) | \(0.9997\) | \(0.9984\) |
ABC | \(0.9975\) | \(0.9976\) | \(0.9981\) | \(0.9980\) | \(0.9997\) | \(1\) | \(0.9995\) |
ABS | \(0.9993\) | \(0.9993\) | \(0.9957\) | \(0.9956\) | \(0.9984\) | \(0.9995\) | \(1\) |
In this paper, we discussed how to achieve almost identical physicochemical property predictions with fewer connectivity indices in QSPR analysis of benzenoid hydrocarbons. The following question is of interest and further research:
Question 1. How to achieve almost identical physicochemical property predictions with fewer topological indices in QSPR analysis of the similar compounds.
The authors are grateful to the anonymous referee for careful reading and valuable comments which result in an improvement of the original manuscript. This work was supported by the Qinghai Normal University Youth Science Fund.