Abstract
Energy of a molecule plays an important role in physics, chemistry and biology. In mathematics, the concept of energy is used in graph theory to help other subjects such as chemistry and physics. In graph theory, nullity is the number of zeros extracted from the characteristic polynomials obtained from the adjacency matrix, and inertia represents the positive and negative eigenvalues of the adjacency matrix. Energy is the sum of the absolute eigenvalues of its adjacency matrix. In this study, the inertia, nullity and signature of the aforementioned structures have been discussed.
1 Introduction
A molecular graph is a mathematical object defined as
The eigenvalues play an important role in the field of mathematics, but these values are also very important in other fields such as chemistry, economics and many more. As far as our study concerns about eigenvalues, these values interpret in chemistry not only as the form of energy but also as different physicochemical properties of a chemical compound. We need to understand the relationship between mathematics and chemistry. The positive eigenvalues are linked with antibonding level, negative eigenvalues are linked with bonding levels and zero eigenvalues are linked with nonbonding level [1–3]. The number of positive eigenvalues
Phenylene belongs to the special class of conjugated hydrocarbons which plays an important role in the field of chemistry. The geometrical construction of phenylene is as follow: its shape is based on two mathematical objects, namely, hexagon and square. These objects are connected in such a way that a square is connected with two hexagons, which show that every two hexagons are totally separated (see Figure 1).
The molecular graph of anthracene consists of three fused benzene rings (see Figure 2). Actually, it is a part of coal tar which is a very useful component in daily life. Coal tar, which contains around 1.5% anthracene, remains a major source of this material. Its chemical formula is
Quantitative structure activity relationship (QSAR) is a regression and classification model is used in chemical and biological sciences and for many other purposes, but the most important one is to characterize the chemical structures of different compounds in terms of a single real number. This number represents the correlation between chemical structures of different compounds and the related chemical and biological activities or properties [4,5]. There are a lot of categorical QSAR models (categorical models are not correlation driven) but the topological index is the popular one because of its effectiveness [1,6].
For further study of energy, see refs. [7–9]. A graph spectrum-based invariant, put forward by Estrada, is defined as
In this study, we have observed the following inequalities:
2 Method
In this section, we discuss the process of finding the energy and Estrada index of the phenylene and anthracene by using the computational methods. In the first step, we use HyperChem to draw the molecule of our chemical structures [22]. In the second step, we construct an adjacency matrix of the graph by using TopoCluj [23]. In the third step, we find the energy with the help of MATLAB which is represented in Table 1. In the fourth step, we find the suitable curve (polynomial) of degree two with the help of “cftoolbox” of MATLAB [24,25]. The results obtained from this data are displayed in Table 2. In Tables 3 and 4, we found the errors between the exact and the estimated values extracted from the curve for energy and Estrada index of phenylene. We also calculated the inertia index, nullity and signature of the phenylene and are displayed in Table 5. Similar process has been used for anthracene. In Tables 6 and 7, we found the polynomials by using the abovementioned techniques, and then in Tables 8 and 9, we calculated the errors of the exact and estimated values of the energy and Estrada index of anthracene. Finally, in Table 10, we displayed the inertia index, nullity and signature of anthracene.
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Estrada index
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1 |
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Error |
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84.51 | 84.21 | 0.30 |
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172.99 | 172.69 | 0.30 |
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261.40 | 261.17 | 0.23 |
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349.96 | 349.66 | 0.30 |
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438.61 | 438.15 | 0.46 |
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527.16 | 526.66 | 0.50 |
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615.43 | 615.17 | 0.26 |
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704.11 | 703.69 | 0.42 |
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792.63 | 792.22 | 0.41 |
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881.11 | 880.74 | 0.37 |
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Error |
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176.5041 | 176.5141 | 0.0099 |
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363.8355 | 363.8555 | 0.0200 |
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551.1702 | 551.2074 | 0.0372 |
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738.5094 | 738.5593 | 0.0499 |
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925.8424 | 925.9112 | 0.0687 |
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1113.1775 | 1113.2631 | 0.0851 |
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1300.5126 | 1300.6150 | 0.1023 |
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1484.8476 | 1487.9669 | 0.1189 |
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1675.1827 | 1675.3188 | 0.1358 |
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1862.5178 | 1862.6707 | 0.1527 |
P = Phenylene | p (P) | n (P) | η (P) | s (P) |
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(8, 1) | 24 | 24 | 0 | 0 |
(8, 2) | 48 | 48 | 0 | 0 |
(8, 3) | 72 | 72 | 0 | 0 |
(8, 4) | 96 | 96 | 0 | 0 |
(8, 5) | 120 | 120 | 0 | 0 |
(8, 6) | 144 | 144 | 0 | 0 |
(8, 7) | 168 | 168 | 0 | 0 |
(8, 8) | 192 | 192 | 0 | 0 |
(8, 9) | 216 | 216 | 0 | 0 |
(8, 10) | 240 | 240 | 0 | 0 |
(8, 11) | 264 | 264 | 0 | 0 |
(8, 12) | 288 | 288 | 0 | 0 |
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Error |
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218.7247 | 218.6827 | 0.0420 |
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453.8533 | 453.7937 | 0.0596 |
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688.9118 | 688.9047 | 0.0071 |
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924.0977 | 924.0157 | 0.0820 |
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1159.1496 | 1159.1268 | 0.0229 |
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1394.2470 | 1394.2377 | 0.0093 |
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1629.3590 | 1629.3487 | 0.0103 |
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1864.4720 | 1864.4597 | 0.0123 |
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2099.5850 | 2099.5707 | 0.0143 |
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2334.6970 | 2334.6818 | 0.0152 |
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Error |
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622.1380 | 622.1370 | 0.0010 |
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1606.9220 | 1606.9200 | 0.0020 |
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2591.7070 | 2591.7030 | 0.0040 |
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3576.4930 | 3576.4860 | 0.0070 |
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4561.2790 | 4561.2690 | 0.0100 |
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5546.0650 | 5546.0520 | 0.0130 |
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6530.8500 | 6530.8350 | 0.0150 |
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7515.6360 | 7515.6180 | 0.0180 |
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8500.4220 | 8500.4010 | 0.0210 |
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9485.2070 | 9485.1840 | 0.0230 |
G = Anthracene | p (G) | n (G) | η (G) | s (G) |
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(8, 1) | 35 | 35 | 0 | 0 |
(8, 2) | 70 | 70 | 0 | 0 |
(8, 3) | 105 | 105 | 0 | 0 |
(8, 4) | 140 | 140 | 0 | 0 |
(8, 5) | 175 | 175 | 0 | 0 |
(8, 6) | 210 | 210 | 0 | 0 |
(8, 7) | 280 | 280 | 0 | 0 |
(8, 8) | 315 | 315 | 0 | 0 |
(8, 9) | 350 | 350 | 0 | 0 |
(8, 10) | 385 | 385 | 0 | 0 |
(8, 11) | 420 | 420 | 0 | 0 |
(8, 12) | 455 | 455 | 0 | 0 |
2.1 Results of phenylene
In this section, we will study the energy, Estrada index, nullity, inertia and signature of phenylene. We will also show the comparison between exact and estimated values of energy and Estrada index of phenylene.
Proposition 2.1
Let
The two curves that are representing the energy and Estrada index of the graph are given by equations (1) and (2), respectively. We observed that the exact value of the energy of phenylene is always greater than the estimated values of the energy of phenylene, i.e.,
Similarly, we have observed that the exact value of Estrada index of phenylene is always less than the estimated values of Estrada index of phenylene, i.e.,
Also the errors between the exact values and the estimated values of energy and Estrada index are summarized in Tables 3 and 4.
2.2 Results of anthracene
In this section, we will study the energy, Estrada index, nullity, inertia and signature of anthracene. We will also show the comparison between the exact and estimated values of energy and Estrada index of anthracene.
Proposition 2.2
Let
The errors of energy and Estrada index are as given in Tables 8 and 9.
We observed that exact value of the energy of anthracene is always greater than the estimated values of the energy of anthracene, i.e.,
3 Conclusion
We studied Estrada index, energy, inertia, nullity and signature of phenylene and anthracene. The following inequalities
Also, since the nullity of phenylene and anthracene is zero, the molecule of these structures are stable and closed shell. At the end, we give a graphical representation of these parameters (Figures 3–6).
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Author contributions: Z. A. analyzed the data curation. Z. S. F. contributed to visualization, investigation and wrote the initial draft of the study. M. F. N. contributed to conceptualization, visualization and methodology. H. S. contributed to designing the experiments, validation. H. M. A. S. was in charge of formal analyzing of experiments, resources and some computations. All authors read and approved the final version of the study.
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Funding information: The authors received no financial support for the research, authorship, and/or publication of this article.
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Conflict of interest: The authors have no conflicts of interest.
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Ethical approval: The conducted research is not related to either human or animal use.
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Data availability statement: All data generated or analyzed during this study are included in this published article.
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