Locating and Multiplicative Locating Indices of Graphs with QSPR Analysis

Wazzan, Suha; Saleh, Anwar

doi:https://doi.org/10.1155/2021/5516321

Journal of Mathematics

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Topological Indices, and Applications of Graph Theory

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Research Article | Open Access

Volume 2021 | Article ID 5516321 | https://doi.org/10.1155/2021/5516321

Locating and Multiplicative Locating Indices of Graphs with QSPR Analysis

Suha Wazzan¹and Anwar Saleh²

Academic Editor: Ismail Naci Cangul

Received21 Feb 2021

Accepted26 Apr 2021

Published19 May 2021

Abstract

In this paper, by introducing a new version of locating indices called multiplicative locating indices, we compute exact values of these indices on well-known families of graphs and graphs obtained by some operations. Also, we determine the importance of locating and multiplicative locating indices of hexane and its isomers. Furthermore, we show that locating indices actually have a reasonable correlation using linear regression with physico-chemical characteristics such as enthalpy, melting point, and boiling point. This approximation can be extended into several chemical compounds.

1. Introduction

As an example of a molecular descriptor, a topological graph index is defined as a mathematical formula which is applied to any graph that models some molecular structure. These indices make analyzing mathematical values and examining certain molecules physico-chemical properties more feasible and efficient by enabling us to bypass costly and lengthy laboratory experiments. The role of molecular descriptors is well established in mathematical chemistry. They include but are not limited to or quantitative structure-property relationship. There are various topological indices in the literature, and many of them have broad applications in chemistry. The structural properties of the graphs employed in the calculations can be used to classify them. For instance, the Zagreb type indices are computed using the degrees of vertices in a graph. They helped to compare some alkane isomers boiling points and have aided in the discovery, along with other indices, of a few thousand topological graph indices enrolled in the chemical data bases. In fact there has been a rapidly increasing interest of this topic, and thus topological graph indices have been studied worldwide by both mathematicians and chemists (see [1–8]). The most widely known topological indices are the first and second Zagreb indices, which have been introduced by Gutman and Trinajstic in [9], and defined as and , respectively. Actually, several new versions of the Zagreb indices have been established for similar purposes (cf. [10–17]).

Different topological indices for some chemical compounds such as “aspirin” and the anticancer drug “carbidopa” have been studied in detail by Wazzan (cf. [18, 19]). Moreover, in a recent work, Wazzan et al. (see [20]) introduced novel topological indices called the first and second locating indices. To do that, the authors used the locating matrix over a graph (cf. [21]). Let be a connected graph with the vertex set . A locating function of denoted by is a function such that , where is the distance between the vertices and in . The vector is called the locating vector corresponding to the vertex , where is actually the dot product of the vectors and in the integers space such that is adjacent to . In the present paper, as a next step of the work in [20], we introduce the first and second multiplicative locating indices for a connected graph as in the following definition.

Definition 1. For a connected graph with an edge set and vertex set , the first and second multiplicative locating indices are defined as follows:respectively.
In this paper, we only consider simple graphs with no multiple edges. For the terminologies, we may recommend citation [22] to readers.

2. Certain Values of Multiplicative Locating Indices

In this section, by considering Definition 1, we will determine the first and second multiplicative locating indices for some special graphs such as , , , , and , and also we will compute the same indices for the graph such that is obtained by joining two graphs and (notationally ), where and are connected with diameter 2. In particular, we will assume that as - and -free graphs.

Theorem 1. Let be the complete graph with . Then,(1)(2)

Proof. (i)Let be the complete graph with and let , and for each vertex , we let is the locating vector associated with the vertex . Then, such that and all the other components are equal to 1. Hence, . However, the total amount of vertices in is vertices, and so, .(ii)For any arbitrary locating vectors and , where , we gain . Therefore, .

Theorem 2. Let , where . Then,(1)(2)

Proof. We identify the adjacent vertices and of , for all and . Then, the locating vectors of are given byHere, for any , we have and for any , therefore, . Therefore,Similarly, for any two locating vertices in where adjacent to , . Hence, .

Corollary 1. Let , where . Then,

Corollary 2. Let be any star graph . Then,

Theorem 3. For an even integer , let . Then,

Proof. By identifying the vertices of the cycle as in the anticlockwise direction, we obtainand hence . It is straightforward to see that each has equivalent components but in different locations; hence, each has the same sum as the form ofTherefore, . By symmetry,which gives .

Theorem 4. Let with an odd number of vertices . Then,

Proof. Following the steps in the proof of Theorem 4, we getand with a simple calculation, one can obtainwhich implies . Further, by the symmetry,Hence, , as required.

Theorem 5. Let be wheel graph with vertices such that . Then,

Proof. Let with vertices. Suppose that the vertices are labeling in the anticlockwise direction where the center of the wheel is labeled . Hence, we getTherefore, for each corresponding locating vector with the vertex , we have and . So, . For , by considering the same labeling as previously, we getHere, the permutation components in each vector where are . Hence, it is straightforward to notice that any two adjacent vertices and satisfy and for . Therefore, . Hence, the result is obtained.

Theorem 6. For any path with vertices,

Proof. Assume that is the path with vertices. Suppose that the locating function is constructed by identifying the vertices as from left to right. Hence, the corresponding vectors for each vertex are given as in following:A straightforwardly calculation implies thatFor the other case ,So, we getTherefore, is obtained as required in the statement of theorem.
In the following result, we will give our attention to the join of graphs and for computing multiplicative locating indices.

Theorem 7. Let such that and are both connected graphs, where and have edges; vertices and edges vertices, respectively. Then,

Proof. Let be as in the statement of theorem. Let us label the vertices of the graph aswhere and . In addition, suppose thatwhich is the locating vector associated with the vertex . Then, . Similarly, for any vertex , the locating vector corresponding to is given bySo, . Therefore, by the above equalities on and , we obtain as required in the theorem.

Theorem 8. Suppose that and are connected graphs having diameter 2. Let such that is a - or -free graph. Assume that has edges and vertices while has edges and vertices. Then,where

Proof. Under the assumptions on as in the statement of the theorem, the partition sets edges are defined byHence, is expressed asFor any two adjacent vertices to obtain , we assume that the first two vertices as follows:Since and are - or -free graph, for any two vertices and in , we can obtainwhich implies .
With the same way of calculation, we get . Now, to achieve the computation of , let us take and . Thus,As a result, we getand so . Then, by all above calculations, we finally get , whereHence, the result is obtained.

Theorem 9. Let be a book graph with vertices. Then,

Proof. Let us label the vertices as in Figure 1. So, we haveConsidering the components of the locating vectors of the book graph, we get , and for , we have . Hence, . Similarly, we have , and for any , , and in the same way, . Hence, .

3. Multiplicative Locating Indices of Firefly Graphs

A firefly graph (, , and ) is a graph of order that consists of triangles, pendant paths of length 2, and pendent edges that are sharing a common vertex [23]. Let be the set of all firefly graphs . Note that contains the stars , stretched stars , friendship graphs , and butterfly graphs .

In the following result, the first and second multiplicative locating indices for the firefly graph are calculated. To simplify the calculations, let us denote by .

Theorem 10. Let () be a firefly graph of order . Then,

Proof. Suppose that () is a firefly graph of vertices. Let us label the vertices of the graph (see Figure 2) with clockwise direction.
So, in the setwhere is the center vertex of the firefly graph, is the vertices of the triangles, is the vertices of the pendent edges, is the first vertices of the pendent paths, and be the second vertices of the pendent paths. Therefore, we obtain the corresponding vectors for each vertex where as follows:Obviously,Hence, we obtain the equality in (38).Similarly, as in the above process, sincewe get the equality in (39) as required.

Corollary 3. (1)For any friendship graph of vertices,(2)For any butterfly graph of vertices,

4. Locating and Multiplicative Locating Indices of Hexane and Its Isomers

In this section, we will compute some first and second locating and multiplicative indices for hexane and its isomers. Recall that the first and second locating indices [20] are defined as follows:

Hexane and its four structural isomers, namely, 2-methylpentane, 3-methylpentane, 2,2-dimethylbutane, and 2,3-dimethylbutane, were fully optimized free of any structural constrains using the one of the well-known functional of the density functional theory (DFT), i.e., B3LYP. B3LYP stands for the Lee–Yang–Parr correlation functional (B3LYP) (see [24, 25]). This functional was combined with a quite large basis set, i.e., stands for a split-valance triple zeta () enlarged with two diffuse basis functionals (), one is sp-orbitals added for the carbon atoms and s-orbital added to all hydrogen atoms. Additionally, larger polarization functionals, and orbitals added for the carbon and hydrogen atoms, respectively, were included. The frequency calculations were performed on all optimized geometries, and the absence of negative frequencies implies that the geometries are all minima points. Optimization and frequency calculations were performed using Gaussian 09 (see [5]), and data were visualized using GaussView (version 5.0.8) (see [26]) programs. The chemical structures, optimized geometries, the distributions, and energies of the highest occupied molecular orbitals (HOMOs) and the lowest unoccupied molecular orbitals (LUMOs) and also the total densities mapped with electrostatic potentials (ESPMs) at isovalue = 0.2 a.u. are all included in Figure 3.

The ESPM is referring to a three-dimensional plot of the total electronic densities mapped with electrostatic potentials. Therefore, it helps in visualizing the electron density distribution around each atom/region of the molecule. The five isomers energies are all large negative values which confirm on the suitability of the applied level of theory. The five isomers can be arranged according to their total electronic energies and thus to their stability as follows: 2,3-dimethylbutane 3-methylpentane 2,2-dimethylbutane 2-methylpentane hexane. By Figure 3, the energies of highest occupied molecular orbitals (HOMOs) and the lowest unoccupied molecular orbitals (LUMOs) are all negative and arranged the isomers in terms of their ability to donate/accept electrons during a chemical reaction.

The molecular graph of and its isomers is shown in Figure 4. In this figure, while the vertices represent the atoms, the edges represent the chemical bond. We should note that the hydrogen atom is omitted.

(a)

(b)

(c)

(d)

(e)

Theorem 11. The first locating and multiplicative indices of Hexane are 222 and 1712237725, respectively. The second locating and multiplicative indices of Hexane are 140 and 12390400, respectively.

Proof. By taking into account Figure 4(a), let us first compute for each (). Thus, we haveThen, by using equations (47) and (1), the first locating and multiplicative indices of hexane are presented byOn the other hand, by using equations (48) and (2), the second locating and multiplicative indices of hexane are presented byHence, the result is obtained.
In the following results, although we will follow completely the same way as in the proof of Theorem 11, we prefer to write some of those proofs again separately since the classification structural isomers is so important.

Theorem 12. The first locating and multiplicative indices of 2-methylpentane are 168 and 285874176, respectively. The second locating and multiplicative indices of 2-methylpentane are 94 and 2044416, respectively.

Proof. Considering Figure 4(b), let us calculate for each (). So, we haveBy equations (47) and (1), the first locating and multiplicative indices of 2-methylpentane are given bySimilarly as previous proofs, by equations (48) and (2), the second locating and multiplicative indices of 2-methylpentane are given byThese all above progresses complete the proof.

Theorem 13. The first locating and multiplicative indices of 3-methylpentane are 118 and 163077057, respectively. The second locating and multiplicative indices of 3-methylpentane are 92 and 1557504, respectively.

Proof. By taking into account Figure 4(c), we compute for each (3-methylpentane), then we getSimilarly as previous proofs, by equations (47) and (1), the first locating and multiplicative indices of 3-methylpentane are given byBy equations (48) and (2), the second locating and multiplicative indices of 3-methylpentane are given byThese all above progresses complete the proof.

Theorem 14. The first locating and multiplicative indices of 2,2-dimethylbutane are 120 and 38162432, respectively. The second locating and multiplicative indices of 2,2-dimethylbutane are 65 and 290304, respectively.

Proof. By Figure 4(d), for , the vectors for each (2,2-dimethylbutane) can be obtained asThus, as previously, by equations (47) and (1) and equations (48) and (2), we obtain the required results on the first and second locating and multiplicative locating indices for 2,2-dimethylbutane.
Finally, let us consider Figure 4(e). Then, the vectors for each (2,3-dimethylbutane) can be obtained aswith the same approach as before, by equations (47) and (1) and equations (48) and (2), we get the next final theorem for the first and second locating and multiplicative indices over 2,3-dimethylbutane.

Theorem 15. The first locating and multiplicative indices of 2,3-dimethylbutane are 130 and 64304361, respectively. The second locating and multiplicative indices of 2,3-dimethylbutane are 72 and 524288, respectively.
Table 1 indicates the exact values of first and second locating and multiplicative locating indices of hexane and its isomers with their physico-chemical properties such as boiling point (B.P.), melting point (M.P.), enthalpy change (E.C.), and flash point (F.P.).
Figure 5 indicates how much the obtained topological indices are correlated with the well-known physio-chemical properties, i.e., the five investigated isomers. The degree of correlation between any two data sets is measured by the value of the correlation coefficient (). When the value of becomes close to unity, two data sets are more correlated. We can also note from Figure 5 that of the plot between F.L.I and boiling points (B.P.) equals 0.458 while it is equal to 0.781 for the plot between S.L.I and boiling points. In fact these two obtained values of for these two plots are quite satisfactory. Similar conclusion can be obtained for the plots among F.L.I and S.L.I data, and the enthalpy changes values since equals 0.538 and 0.324 for these two plots, respectively. The values of are not big enough but still indicates good correlations between these two data sets. However, the achieved correlation coefficients between two topological indices and the melting points of five isomers are too small and so should be indicated a poor correlation between them since the values of in these two plots are less than 0.2. The plots between F.L.I and the flash points (F.P.) are equal to 0.108 while a better correlation is obtained between S.L.I and F.P. as the value = 0.369. Therefore, the former plot represents a poor correlation and the later can be considered as a better correlation.

5. Conclusion

This study combined pure data from the chemistry textbooks and a mathematical effort to find new topological indices of five well-known chemical compounds. The cases in which good correlations were obtained suggested the validity of the calculated topological indices to be further used to predict the physio-chemical properties of much complicated chemical compounds.

Data Availability

The chemical data used in this paper are strictly personal since most of those are obtained with some payments in a computing center after the theoretical parts obtained. However, the reader may contact the corresponding author for more details and special permissions of data.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. G: 218-247-1442. The authors, therefore, acknowledge DSR technical and financial support. The authors acknowledge Prof. Nuha Wazzan from Chemistry department at King Abdulaziz University for her contribution with the DFT calculations and King Abdulaziz University’s High-Performance Computing Centre (Aziz Supercomputer) (http://hpc.kau.edu.sa) for supporting the computation for the work described in this paper.

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Copyright © 2021 Suha Wazzan and Anwar Saleh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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