Locating and Multiplicative Locating Indices of Graphs with QSPR Analysis

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. )is approximation can be extended into several chemical compounds.


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. ese 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. e role of molecular descriptors is well established in mathematical chemistry. ey include but are not limited to QSPR or quantitative structure-property relationship.
ere are various topological indices in the literature, and many of them have broad applications in chemistry. e 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. ey 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][2][3][4][5][6][7][8]). e 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 M 1 (G) � u∈V(G) (d(u)) 2 and M 2 (G) � uv∈E(G) d(u)d(v), respectively. Actually, several new versions of the Zagreb indices have been established for similar purposes (cf. [10][11][12][13][14][15][16][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 Lo(G) over a graph G (cf. [21]). Let G � (V, E) be a connected graph with the vertex set V � v 1 , v 2 , . . . , v n . A locating function of G denoted by L(G) is a function L(G): is the distance between the vertices v i and v j in G. e vector v i → is called the locating vector corresponding to the vertex v i , where v i → · v j → is actually the dot product of the vectors v i → and v j → in the integers space Z + ∪ 0 { } such that v i is adjacent to v j . 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 G as in the following definition. Definition 1. For a connected graph G � (V, E) with an edge set E(G) and vertex set V � v 1 , v 2 , . . . , v n , 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.

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 K m , K m,n , C m , W m , and P m , and also we will compute the same indices for the graph G such that G is obtained by joining two graphs G 1 and G 2 (notationally G � G 1 + G 2 ), where G 1 and G 2 are connected with diameter 2. In particular, we will assume that G as C 3and C 5 -free graphs.

Theorem 1.
Let G � K m be the complete graph with m ≥ 2. en, that a i � 0 and all the other components are equal to However, the total amount of vertices in G is m vertices, and so, Proof. We identify the adjacent vertices v i and v m+j of K m,n , for all 1 ≤ j ≤ n and 1 ≤ i ≤ m. en, the locating vectors Here, for any i � 1, 2, . . . , m, we have v i →2 � 4(m − 1) + n and for any i � m + 1, m + 2, . . . , m + n, therefore, v i →2 � 4(n − 1) + m. erefore, Similarly, for any two locating vertices Corollary 2. Let G be any star graph K 1,n . en, and hence v i →2 � 2( (m/2) i�1 i 2 ) − (m 2 /4). It is straightforward to see that each v i → has equivalent components but in different locations; hence, each v i →2 has the same sum as the Proof. Following the steps in the proof of eorem 4, we get and with a simple calculation, one can obtain which implies L 2 (C m ) � (m(m 2 − 1)/12) m . Further, by the symmetry, Hence, erefore, for each corresponding locating vector , by considering the same labeling as previously, Here, the permutation components in each vector v i → where i � 1, 2, . . . , m are 1, 0, 1. Hence, it is straightforward to notice that any two adjacent vertices Proof. Assume that P m is the path with (m ≥ 3) vertices. Suppose that the locating function is constructed by identifying the vertices as v 1 , v 2 , . . . , v m from left to right. Hence, the corresponding vectors for each vertex v i ∈ V(G) (i � 1, . . . , m) are given as in following: A straightforwardly calculation implies that For the other case L 2 (P m ), So, we get erefore, L 2 (P m ) is obtained as required in the statement of theorem.
In the following result, we will give our attention to the join G � G 1 + G 2 of graphs G 1 and G 2 for computing multiplicative locating indices. □ Theorem 7. Let G � G 1 + G 2 such that G 1 and G 2 are both connected graphs, where G 1 and G 2 have m 1 edges; n 1 vertices and m 2 edges n 2 vertices, respectively. en, Proof. Let G be as in the statement of theorem. Let us label the vertices of the graph G as where which is the locating vector associated with the vertex for any vertex z ∈ V(G 2 ), the locating vector z → corresponding to z is given by erefore, by the above equalities on v →2 and z →2 , we obtain L 1 (G) as required in the theorem. □ Theorem 8. Suppose that G 1 and G 2 are connected graphs having diameter 2. Let G � G 1 + G 2 such that G is a C 3 -or C 5 -free graph. Assume that G 1 has m 1 edges and n 1 vertices while G 2 has m 2 edges and n 2 vertices. en, Proof. Under the assumptions on G as in the statement of the theorem, the partition sets edges are defined by (29) For any two adjacent vertices u, v ∈ V(G 1 ) to obtain uv∈A u → · v → , we assume that the first two vertices as follows: Since G 1 and G 2 are C 3 -or C 5 -free graph, for any two vertices u and v in V(G 1 ), we can obtain Journal of Mathematics With the same way of calculation, we get As a result, we get and so u·v∈C u Hence, the result is obtained.
Theorem 10. Let G � F s,t,l (s, t, l ≥ 0) be a firefly graph of order n. en, Proof. Suppose that G � F s,t,l (s, t, l ≥ 0) is a firefly graph of n � 2s + 2t + l + 1 vertices. Let us label the vertices of the graph (see Figure 2) with clockwise direction. So, in the set where v is the center vertex of the firefly graph, v 1 , . . . v 2s is the vertices of the triangles, u 1 , . . . u l is the vertices of the pendent edges, w 1 , . . . , w t is the first vertices of the pendent paths, and z 1 , . . . , z t be the second vertices of the pendent paths. erefore, we obtain the corresponding vectors v → i for each vertex v i ∈ V(G) where i � 1, 2, . . . , 2s + 2t + l + 1 as follows:

Journal of Mathematics
Obviously, Hence, we obtain the equality in (38). 8

Journal of Mathematics
Similarly, as in the above process, since we get the equality in (39) as required. (45) (2) For any butterfly graph F s,0,n− 2s− 1 of n vertices,

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.
is functional was combined with a quite large basis set, i.e., 6 − 311 + +G(2 d, 2p). 6 − 311 + +G(2 d, 2p) stands for a split-valance triple zeta (ξ) 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, 2 d− and 2p− orbitals added for the carbon and hydrogen atoms, respectively, were included. e 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. e 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. e ESPM is referring to a three-dimensional plot of the total electronic densities mapped with electrostatic potentials. erefore, it helps in visualizing the electron density distribution around each atom/region of the molecule. e five isomers energies are all large negative values which confirm on the suitability of the applied level of theory. e five isomers can be arranged according to their total electronic energies and thus to their stability as follows: 2,3dimethylbutane < 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. e molecular graph of hexane 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.

Theorem 11.
e first locating and multiplicative indices of Hexane are 222 and 1712237725, respectively. e second locating and multiplicative indices of Hexane are 140 and 12390400, respectively.
Proof. By taking into account Figure 4(a), let us first 0, 1, 2, 3, 4),     Figure 3: e 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. of (a) hexane, (b) 2-methylpentane, (c) 3-methylpentane, (d) 2,2-dimethylbutane, and (e) 2,3-dimethylbutane.  Journal of Mathematics On the other hand, by using equations (48) and (2), the second locating and multiplicative indices of hexane are presented by Hence, the result is obtained. In the following results, although we will follow completely the same way as in the proof of eorem 11, we prefer to write some of those proofs again separately since the classification structural isomers is so important. Proof. Considering Figure 4 6 → � (4, 3, 2, 1, 2, 0).

(52)
By equations (47) and (1), the first locating and multiplicative indices of 2-methylpentane are given by Similarly as previous proofs, by equations (48) and (2), the second locating and multiplicative indices of 2-methylpentane are given by Similarly as previous proofs, by equations (47) and (1), the first locating and multiplicative indices of 3-methylpentane are given by By equations (48) and (2), the second locating and multiplicative indices of 3-methylpentane are given by   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 physiochemical properties, i.e., the five investigated isomers. e degree of correlation between any two data sets is measured by the value of the correlation coefficient (R 2 ). When the value of R 2 becomes close to unity, two data sets are more correlated. We can also note from Figure 5 that R 2 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 R 2 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 R 2 equals 0.538 and 0.324 for these two plots, respectively. e values of R 2 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 R 2 in these two plots are less than 0.2. e 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 R 2 � 0.369. erefore, the former plot represents a poor correlation and the later can be considered as a better correlation.

Conclusion
is study combined pure data from the chemistry textbooks and a mathematical effort to find new topological indices of five well-known chemical compounds. e 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 e 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
e authors declare that they have no conflicts of interest.