# Locating and Multiplicative Locating Indices of Graphs with QSPR Analysis

Authors: Suha Wazzan; Anwar Saleh
Journal: Journal of Mathematics
 (2021)

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## 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.


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## Body

## 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 toQSPR 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 M1G=∑u∈VGdu2 and M2G=∑uv∈EGdudv, 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 LoG over a graph G (cf. [21]). Let G=V,E be a connected graph with the vertex set V=v1,v2,…,vn. A locating function of G denoted by LG is a function LG:VG⟶ℤ+∪0 such that Lvi=vi⟶=dv1,vi,dv2,vi,…,dvn,vi, where dvi,vj is the distance between the vertices vi and vj in G. The vector vi⟶ is called the locating vector corresponding to the vertex vi, where vi⟶⋅vj⟶ is actually the dot product of the vectors vi⟶ and vj⟶ in the integers space ℤ+∪0 such that vi is adjacent to vj. 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 graphG=V,E with an edge set EG and vertex set V=v1,v2,…,vn, the first and second multiplicative locating indices are defined as follows:(1)∏1ℒG=∏vi∈VGvi⟶2,(2)∏2ℒG=∏vivj∈EGvi⟶⋅vj⟶,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 Definition1, we will determine the first and second multiplicative locating indices for some special graphs such as Km, Km,n, Cm, Wm, and Pm, and also we will compute the same indices for the graph G such that G is obtained by joining two graphs G1 and G2 (notationally G≅G1+G2), where G1 and G2 are connected with diameter 2. In particular, we will assume that G as C3- and C5-free graphs.Theorem 1.
LetG≅Km be the complete graph with m≥2. Then,(1)
∏1ℒKm=m−1m(2)
∏2ℒKm=m−2mm−1Proof.
(i)
LetG≅Km be the complete graph with m≥2 and let VG=v1,v2,…,vm, and for each vertex vi∈VG, we let vi⟶ is the locating vector associated with the vertex vi. Then, vi⟶=a1,a2,…,am such that ai=0 and all the other components are equal to 1. Hence, vi⟶2=m−1. However, the total amount of vertices in G is m vertices, and so, ∏1ℒKm=m−1m.(ii)
For any arbitrary locating vectorsvi⟶ and vj⟶, where i≠j, we gain vi⟶⋅vj⟶=m−2. Therefore, ∏2ℒKm=m−2mm−1.Theorem 2.
LetG≅Km,n, where 1≤m≤n. Then,(1)
∏1ℒKm,n=4m−4+nm+4n−4+mn(2)
∏2ℒKm,n=2m+2n−4nmProof.
We identify the adjacent verticesvi and vm+j of Km,n, for all 1≤j≤n and 1≤i≤m. Then, the locating vectors vi⟶ of vi are given by(3)v1⟶=0,2,…,2︷m−1,1,1,…,1︷n,v2⟶=2,0,2,…,2︷m−2,1,1,…,1︷nv3⟶=2,2,0,2,…,2︷m−3,1,1,…,1︷n,…,vm⟶=2,…,2︷m−1,0,1,1,…,1︷n,vm+1⟶=1,…,1︷m,0,2,2,…,2︷n−1,vm+2⟶=1,…,1︷m,2,0,2,…,2︷n−2.⋮⋮⋮vm+n⟶=1,…,1︷n,2,…,2︷m−1,0.
Here, for anyi=1,2,…,m, we have vi⟶2=4m−1+n and for any i=m+1,m+2,…,m+n, therefore, vi⟶2=4n−1+m. Therefore,(4)∏1ℒKm,n=4m−1+nm+4n−1+mn=4m−4+nm+4n−4+mn.
Similarly, for any two locating verticesv⟶,u⟶ in Km,n where u adjacent to v, v⟶⋅u⟶=2m+n−2. Hence, ∏2ℒKm,n=2m+2n−4nm.Corollary 1.
LetG≅Km,m, where m≥1. Then,(5)∏1ℒKm,m=25m−4m,∏2ℒKm,n=4m−4m2.Corollary 2.
LetG be any star graph K1,n. Then,(6)∏1ℒK1,n=n+4n+3n,∏2ℒK1,n=2n−2n.Theorem 3.
For an even integerm≥2, let G≅Cm. Then,(7)∏1ℒCm=m3+2m12m,∏2ℒCm=mm−2212m.Proof.
By identifying the vertices of the cycleCm as v1,v2,…,vm in the anticlockwise direction, we obtain(8)v1⟶=0,1,2,3,…,m2,m2−1,m2−2,…,1,v2⟶=1,0,1,2,…,m2−1,m2,m2−1,…,2,v3⟶=2,1,0,1,…,m2−2,m2−1,m2,…,3,⋮⋮vm⟶=1,2,3,…,m2,m2−1,m2−2,m2−2,…,0,and hence vi⟶2=2∑i=1m/2i2−m2/4. It is straightforward to see that each vi⟶ has equivalent components but in different locations; hence, each vi⟶2 has the same sum as the form of(9)vi⟶2=mm+1m+2−3m212.
Therefore,∏2ℒCm=m3+2m/12m. By symmetry,(10)vi⟶⋅vi+1⟶=2∑i=2m/2ii−1=2m/2m/2+1m+16−1−2m/2m/2+12−1=mm−2212,which gives ∏2ℒCm=mm−22/12m.Theorem 4.
LetG≅Cm with an odd number of vertices m≥3. Then,(11)∏1ℒCm=mm2−112m,∏2ℒCm=m−1m−2m+312m.Proof.
Following the steps in the proof of Theorem4, we get(12)v1⟶=0,1,2,3,…,m−12,m−12−1,m−12−2,…,1,v2⟶=1,0,1,2,…,m−12−1,m−12,m−12−1,…,2,v3⟶=2,1,0,1,…,m−12−2,m−12−1,m−12,…,3,⋮⋮vm⟶=1,2,3,…,m−12,m−12−1,m−12−2,…,0,and with a simple calculation, one can obtain(13)vi⟶2=2∑i=1m−1/2i2=mm2−112,which implies ∏2ℒCm=mm2−1/12m. Further, by the symmetry,(14)vi⟶⋅vi+1⟶=2∑i=2m−1/2ii−1+m−124=2m−1/2m−1/2+12m−1/2+16−1−2m−1/2m−1/2+12−1+m−124=m−1m−2m+312.
Hence,∏2ℒCm=m−1m−2m+3/12m, as required.Theorem 5.
LetG be wheel graph Wm with m+1 vertices such that m≥4. Then,(15)∏1ℒWm=m4m−9m,∏2ℒWm=8m2−38m+44m.Proof.
LetG≅Wm with m+1 vertices. Suppose that the vertices v1,v2,…,vm,vm+1∈VG are labeling in the anticlockwise direction where the center of the wheel is labeled vm+1. Hence, we get(16)v1⟶=0,1,2,2,…,2︷m−3,1,1,v2⟶=1,0,1,2,2,…,2︷m−3,1,v3⟶=2,1,0,1,2,…,2︷m−2,1,…,vm⟶=1,2,2,…,2︷m−3,1,0,1,vm+1⟶=1,1,…,1︷m,0.
Therefore, for each corresponding locating vectorvi⟶ with the vertex vii∈1,2,…,m, we have vi⟶2=4m−9 and vm+1⟶2=m. So, ∏1ℒWm=m4m−9m. For ∏2ℒWm, by considering the same labeling as previously, we get(17)v1⟶=0,1,2,2,…,2︸m−3,1,1,v2⟶=1,0,1,2,2,…,2︸m−3,1,v3⟶=2,1,0,1,2,…,2︸m−2,1,v4⟶=2,2,1,0,1,2,…,2︸m−1,1,v5⟶=2,2,2,1,0,1,2,…,2︸m,1,…,vm⟶=1,2,2,…,2︸m−3,1,0,1,vm+1⟶=1,1,…,1︸m,0.
Here, the permutation components in each vectorvi⟶ where i=1,2,…,m are 1,0,1. Hence, it is straightforward to notice that any two adjacent vertices vi and vji,j∈1,2,…,m satisfy vi⟶⋅vj⟶=4m−11 and vi⟶⋅vm+1⟶=2m−4 for i=1,2,…,m. Therefore, ∏2ℒWm=8m2−38m+44m. Hence, the result is obtained.Theorem 6.
For any pathPm with m≥3 vertices,(18)∏1ℒPm=m−jm−j+12m−2j+1+jj−12j−16,∏2ℒPm=∏j=1m−1m−jm−j+1m−j−13.Proof.
Assume thatPm is the path with m≥3 vertices. Suppose that the locating function is constructed by identifying the vertices as v1,v2,…,vm from left to right. Hence, the corresponding vectors for each vertex vi∈VGi=1,…,m are given as in following:(19)v1⟶=0,1,2,3,…,m−1,v2⟶=1,0,1,2,…,m−2,…,vm−1⟶=m−2,m−1,…,0,1,vm⟶=m−1,m−2,m−3,…,0.
A straightforwardly calculation implies that(20)∏1ℒPm=∏j=1m∑i=1m−ji2+∑i=1j−1i2=∏j=1mm−jm−j+12m−2j+1+jj−12j−16.
For the other case∏2ℒPm,(21)v⟶1⋅v⟶2=0⋅1+1⋅0+⋯+m−1m−2=∑i=1m−1ii−1,v⟶2⋅v⟶3=1⋅2+0⋅1+⋯+m−2m−3=∑i=1m−1ii−1,v⟶3⋅v⟶4=∑i=1m−1ii−1.
So, we get(22)∏2ℒPm=∏j=1m−1∑i=1m−jii−1=∏j=1m−1∑i=1m−ji2−∑i=1m−ji.
Therefore,∏2ℒPm is obtained as required in the statement of theorem.
In the following result, we will give our attention to the joinG≅G1+G2 of graphs G1 and G2 for computing multiplicative locating indices.Theorem 7.
LetG≅G1+G2 such that G1 and G2 are both connected graphs, where G1 and G2 have m1 edges; n1 vertices and m2 edges n2 vertices, respectively. Then,(23)∏1ℒG=∏i=1n1n2+4n1−4−3degG1vi∏j=n1+1n1+n2n1+4n2−4−3degG2vj.Proof.
LetG be as in the statement of theorem. Let us label the vertices of the graph G as(24)v1,v2,…,vn1,vn1+1,vn1+2,…,vn1+n2,where v1,v2,…,vn1∈VG1 and vn1+1,vn1+2,…,vn1+n2∈VG2. In addition, suppose that(25)v⟶=0,1,…,1︸degG1v,2,…,2︸n1−degG1v−1,1,…,1︸n2,which is the locating vector associated with the vertex v∈VG1. Then, v⟶2=n2+4n1−4−3degG1v. Similarly, for any vertex z∈VG2, the locating vector z⟶ corresponding to z is given by(26)z⟶=1,…,1︸n1,0,1,…,1︸degG2z,2,…,2︸n2−degG2z−1.
So,z⟶2=n1+4n2−4−3degG2z. Therefore, by the above equalities on v⟶2 and z⟶2, we obtain ∏1ℒG as required in the theorem.Theorem 8.
Suppose thatG1 and G2 are connected graphs having diameter 2. Let G≅G1+G2 such that G is a C3- or C5-free graph. Assume that G1 has m1 edges and n1 vertices while G2 has m2 edges and n2 vertices. Then,(27)∏2ℒG=αβ,where(28)α=n2+2n1−4m1n1+2n2−4m2,β=∏u∈VG1,v∈VG2,uv∈EG2n1+n2−4−degG1u−degG2vn1n2.Proof.
Under the assumptions onG as in the statement of the theorem, the partition sets edges are defined by(29)A=uv:u,v∈VG1,B=uv:u,v∈VG2,C=uv:u∈VG1,v∈VG2.
Hence,∏2ℒG is expressed as(30)∏uv∈Au⟶⋅v⟶∏uv∈Bu⟶⋅v⟶∏uv∈Cu⟶⋅v⟶.
For any two adjacent verticesu,v∈VG1 to obtain ∏uv∈Au⟶⋅v⟶, we assume that the first two vertices as follows:(31)u⟶=0,1,…,1︸degG1u,2,…,2︸n1−degG1u−1,1,…,1︸n2,v⟶=1,0,2,…,2︸degG1u−1,1,…,1︸n1−degG1u−1,1,…,1︸n2.
SinceG1 and G2 are C3- or C5-free graph, for any two vertices u and v in VG1, we can obtain(32)u⟶⋅v⟶=2degG1u−1+2n1−degG1u−1+n2=n2+2n1−4,which implies ∏u⋅v∈Au⟶⋅v⟶=n2+2n1−4m1.
With the same way of calculation, we get∏u⋅v∈Bu⟶⋅v⟶=n1+2n2−4m2. Now, to achieve the computation of ∏uv∈Cu⟶⋅v⟶, let us take u∈VG1 and v∈VG2. Thus,(33)u⟶=0,1,…,1︸degG1u,2,…,2︸n1−degG1u−1,1,…,1︸n2.v⟶=1,…,1︸degG1u+1,1,…,1︸n1−degG1u−1,0,1,…,1︸degG2v,2,…,2︸n2−degG2v−1.
As a result, we get(34)u⟶⋅v⟶=degG1u+2n1−degG2u−1+degG2v+2n2−degG2v−1=2n1+n2−4−degG1u−degG2v,and so ∏u⋅v∈Cu⟶⋅v⟶=∏u∈VG1,v∈VG2,uv∈EG2n1+n2−4−degG1u−degG2vn1n2. Then, by all above calculations, we finally get ∏2ℒG=αβ, where(35)α=n2+2n1−4m1n1+2n2−4m2,β=∏u∈VG1,v∈VG2,uv∈EG2n1+n2−4−degG1u−degG2vn1n2.
Hence, the result is obtained.Theorem 9.
LetG be a book graph Bt=P2×St with 2t+2 vertices. Then,(36)∏1ℒG=5t+1213t−72t,∏2ℒG=4t12t−88t+42t.Proof.
Let us label the verticesVG=v,u,v1,v2,….vt,u1,u2,…,ut as in Figure 1. So, we have(37)v⟶=0,1,1,1,…,1︷t,2,2,…,2︷t,u⟶=1,0,2,2,…,2︷t,1,1,…,1︷tv1⟶=1,2,0,2,…,2︷t−1,1,3,…,3︷t−1,u1⟶=2,1,1,3,…,3︷t−1,0,2,…,2︷t−1v2⟶=1,2,2,0,2,…,2︷t−2,3,1,3,…,3︷t−2,u2⟶=2,1,3,1,3,…,3︷t−2,2,0,2,…,2︷t−2⋮,⋮vt⟶=1,2,2,…,2︷t−1,0,3,…,3︷t−1,1,ut⟶=2,1,3,…,3︷t−1,1,2,…,2︷t−1,0.
Considering the components of the locating vectors of the book graph, we getv⟶2=u⟶2=t+1+4t=5t+1, and for i=1,2,3,…,t, we have vi⟶2=ui⟶2=13t−7. Hence, ∏1ℒG=5t+1213t−72t. Similarly, we have v⟶⋅u⟶=4t, and for any i=1,2,3,…,t, vi⟶.ui⟶=4+6t−1+6t−1=12t−8, and in the same way, v⟶.vi⟶=u⟶.ui⟶=8t−4. Hence, ∏2ℒG=4t12t−88t−42t.Figure 1
A book graphBt.


## 3. Multiplicative Locating Indices of Firefly Graphs

A firefly graphFs,t,n−2s−2t−1 (s≥0, t≥0, and n−2s−2t−1≥0) is a graph of order n that consists of s triangles, t pendant paths of length 2, and n−2s−2t−1 pendent edges that are sharing a common vertex [23]. Let ℱn be the set of all firefly graphs Fs,t,n−2s−2t−1. Note that ℱn contains the stars Sn≅F0,0,n−1, stretched stars ≅F0,t,n−2t−1, friendship graphs ≅Fn−1/2,0,0, and butterfly graphs ≅Fs,0,n−2s−1.In the following result, the first and second multiplicative locating indices for the firefly graph are calculated. To simplify the calculations, let us denoten−2s−2t−1 by l.Theorem 10.
LetG≅Fs,t,l (s,t,l≥0) be a firefly graph of order n. Then,(38)∏1ℒG=2s+5t+l28s+13t+4l−62s8s+13t+4l−3l8s+13t+4l−11t18s+25t+9l−20t,(39)∏2ℒG=8s+13t+4l−72s−14s+8t+2l−32s4s+8t+2l−2l4s+8t+2l−6t12s+8t+6l−16t.Proof.
Suppose thatG≅Fs,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(40)VG=v,v1,v2,…v2s,u1,u2,…,ul,w1,w2,…,wt,z1,z2,…,zt,where v is the center vertex of the firefly graph, v1,…v2s is the vertices of the triangles, u1,…ul is the vertices of the pendent edges, w1,…,wt is the first vertices of the pendent paths, and z1,…,zt be the second vertices of the pendent paths. Therefore, we obtain the corresponding vectors v⟶i for each vertex vi∈VG where i=1,2,…,2s+2t+l+1 as follows:(41)v⟶=0,1,1,…,1︸2s,1,1,…,1︸l,1,1,…,1︸t,2,2,…,2︸t,v⟶1=1,0,1,2,2,…,2︸2s−2,2,2,…,2︸l,2,2,…,2︸t,3,3,…,3︸t,v⟶2=1,1,0,2,2,…,2︸2s−2,2,2,…,2︸l,2,2,…,2︸t,3,3,…,3︸t,v⟶3=1,2,2,0,1,2,2,…,2︸2s−4,2,2,…,2︸l,2,2,…,2︸t,3,3,…,3︸t,v⟶2s−1=1,2,2,…,2︸2s−2,0,1,2,2,…,2︸l,2,2,…,2︸t,3,3,…,3︸t,v⟶2s=1,2,2,…,2︸2s−2,1,0,2,2,…,2︸l,2,2,…,2︸t,3,3,…,3︸t,⋮u⟶1=1,2,2,…,2︸2s,0,2,2,…,2︸l−1,2,2,…,2︸t,3,3,…,3︸t,u⟶2=1,2,2,…,2︸2s,2,0,2,2,…,2︸l−2,2,2,…,2︸t,3,3,…,3︸t,⋮u⟶l=1,2,2,…,2︸2s,2,2,…,2︸l−1,0,2,2,…,2︸t,3,3,…,3︸t,w⟶1=1,2,2,…,2︸2s,2,2,…,2︸l,0,2,2,…,2︸t−1,1,3,3,…,3︸t−1,w⟶2=1,2,2,…,2︸2s,2,2,…,2︸l,2,0,2,2,…,2︸t−2,3,1,3,3,…,3︸t−2,⋮w⟶t=1,2,2,…,2︸2s,2,2,…,2︸l,2,2,…,2︸t−1,0,3,3,…,3︸t−1,1,z⟶1=2,3,3,…,3︸2s,3,3,…,3︸l,1,3,3,…,3︸t−1,0,4,4,…,4︸t−1,z⟶2=2,3,3,…,3︸2s,3,3,…,3︸l,3,1,3,3,…,3︸t−2,4,0,4,4,…,4︸t−2,⋮z⟶t=2,3,3,…,3︸2s,3,3,…,3︸l,3,3,…,3︸t−1,1,4,4,…,4︸t−1,0.
Obviously,(42)∏1ℒG=v⟶2∏i=12sv⟶i2∏i=1lu⟶i2∏i=1tw⟶i2∏i=1tz⟶i2.
Hence, we obtain the equality in (38).(43)∏1ℒG=2s+5t+l28s+13t+4l−62s8s+13t+4l−3l8s+13t+4l−11t18s+25t+9l−20t.
Similarly, as in the above process, since(44)∏2ℒG=∏i=12s−1vi⟶⋅vi+1⟶∏i=12sv⟶⋅vi⟶∏i=1lv⟶⋅ui⟶∏i=1tv⟶⋅wi⟶∏i=1twi⟶⋅zi⟶,we get the equality in (39) as required.Figure 2
Firefly graphFs,t,l.Corollary 3.
(1)
For any friendship graph ofn vertices,(45)∏1ℒG=n−124n−10n−1,∏2ℒG=4n−11n−22n−5n−1.(2)
For any butterfly graphFs,0,n−2s−1 of n vertices,(46)∏1ℒG=n−124n−102s4n−7n−2s−1,∏2ℒG=4n−92s−12n−52s2n−4n−2s−1.


## 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:(47)M1ℒG=∑vi∈Vv⟶i2,(48)M2ℒG=∑vivj∈Evi⟶⋅vj⟶.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., 6−311++G2d,2p.6−311++G2d,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, 2d− and 2p− 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.Figure 3
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. of (a) hexane, (b) 2-methylpentane, (c) 3-methylpentane, (d) 2,2-dimethylbutane, and (e) 2,3-dimethylbutane.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 ofhexane 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.Figure 4
Molecular graphs of hexane and its isomers: (a) hexane; (b) 2-methylpentane; (c) 3-methylpentane; (d) 2,2-dimethylbutane; (e) 2,3-dimethylbutane.
(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 Figure4(a), let us first compute v⟶i for each vi∈V (hexane). Thus, we have(49)v1⟶=0,1,2,3,4,5,v2⟶=1,0,1,2,3,4,v3⟶=2,1,0,1,2,3,v4⟶=3,2,1,0,1,2,v5⟶=4,3,2,1,0,1,v6⟶=5,4,3,2,1,0.
Then, by using equations (47) and (1), the first locating and multiplicative indices of hexane are presented by(50)M1ℒhexane=∑vi∈Vv⟶i2=55+31+31+19+31+55=222,∏1ℒhexane=∏vi∈Vv⟶i2=55×31×31×19×31×55=1712237725.
On the other hand, by using equations (48) and (2), the second locating and multiplicative indices of hexane are presented by(51)M2ℒhexane=∑vivj∈Evi⟶⋅vj⟶=v1⟶⋅v2⟶+v2⟶⋅v3⟶+v3⟶⋅v4⟶+v4⟶⋅v5⟶+v5⟶⋅v6⟶=40+22+16+22+40=140,∏2ℒhexane=∏vivj∈Evi⟶⋅vj⟶=v1⟶⋅v2⟶×v2⟶⋅v3⟶×v3⟶⋅v4⟶×v4⟶⋅v5⟶×v5⟶⋅v6⟶=12390400.
Hence, the result is obtained.
In the following results, although we will follow completely the same way as in the proof of Theorem11, 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 Figure4(b), let us calculate v⟶i for each vi∈V (2−methylpentane). So, we have(52)v1⟶=0,1,2,3,4,4,v2⟶=1,0,1,2,3,3,v3⟶=2,1,0,1,2,2,v4⟶=3,2,1,0,1,1,v5⟶=4,3,2,1,0,2,v6⟶=4,3,2,1,2,0.
By equations (47) and (1), the first locating and multiplicative indices of 2-methylpentane are given by(53)M1ℒ2−methylpentane=∑vi∈Vv⟶i2=46+24+14+16+34+34=168,∏1ℒ2−methylpentane=∏vi∈Vv⟶i2=46×24×14×16×34×34=285874176.
Similarly as previous proofs, by equations (48) and (2), the second locating and multiplicative indices of 2-methylpentane are given by(54)M2ℒ2−methylpentane=∑vivj∈Evi⟶⋅vj⟶=v1⟶⋅v2⟶+v2⟶⋅v3⟶+v3⟶⋅v4⟶+v4⟶⋅v5⟶+v4⟶⋅v6⟶=22+16+12+22+22=94,∏2ℒ2−methylpentane=∏vivj∈Evi⟶⋅vj⟶=v1⟶⋅v2⟶×v2⟶⋅v3⟶×v3⟶⋅v4⟶×v4⟶⋅v5⟶×v4⟶⋅v6⟶=22×16×12×22×22=2044416.
These 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 Figure4(c), we compute v⟶i for each vi∈V (3-methylpentane), then we get(55)v1⟶=0,1,2,3,3,4,v2⟶=1,0,1,2,2,3,v3⟶=2,1,0,1,1,2,v4⟶=3,2,1,0,2,3,v5⟶=3,2,1,2,0,1,v6⟶=4,3,2,3,1,0.
Similarly as previous proofs, by equations (47) and (1), the first locating and multiplicative indices of 3-methylpentane are given by(56)M1ℒ3−methylpentane=∑vi∈Vv⟶i2=39+19+11+27+19+39=118,∏1ℒ3−methylpentane=∏vi∈Vv⟶i2=39×19×11×27×19×39=163077057.
By equations (48) and (2), the second locating and multiplicative indices of 3-methylpentane are given by(57)M2ℒ3−methylpentane=∑vivj∈Evi⟶⋅vj⟶=v1⟶⋅v2⟶+v2⟶⋅v3⟶+v3⟶⋅v4⟶+v3⟶⋅v5⟶+v5⟶⋅v6⟶=26+12+16+12+26=92,∏2ℒ3−methylpentane=∏vivj∈Evi⟶⋅vj⟶=v1⟶⋅v2⟶×v2⟶⋅v3⟶×v3⟶⋅v4⟶×v3⟶⋅v5⟶×v5⟶⋅v6⟶=26×12×16×12×26=1557504.
These 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 Figure4(d), for 1≤i≤3, the vectors v⟶i for each vi∈V (2,2-dimethylbutane) can be obtained as(58)v1⟶=0,1,2,3,3,3,v2⟶=1,0,1,2,2,2,v3⟶=2,1,0,1,1,1,v4⟶=3,2,1,0,2,2,v5⟶=3,2,1,2,0,2,v6⟶=3,2,1,2,2,0.
Thus, 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 Figure4(e). Then, the vectors v⟶i for each vi∈V (2,3-dimethylbutane) can be obtained as(59)v1⟶=0,1,2,2,3,3,v2⟶=1,0,1,1,2,2,v3⟶=2,1,0,2,3,3,v4⟶=2,1,2,0,1,1,v5⟶=3,2,3,1,0,2,v6⟶=3,2,3,1,2,0,with 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.
Table1 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.).
Figure5 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 (R2). When the value of R2 becomes close to unity, two data sets are more correlated. We can also note from Figure 5 that R2 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 R2 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 R2 equals 0.538 and 0.324 for these two plots, respectively. The values of R2 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 R2 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 R2  = 0.369. Therefore, the former plot represents a poor correlation and the later can be considered as a better correlation.Table 1
Hexane and its isomers with their physico-chemical properties and topological indices values.
Isomer nameF.L.IS.L.IF.M.IS.M.IB.P.F°M.P.F°E.C./eVF.P.Hexane222140171223772512390400155.7−139.5−6448.22247.22-Methylpentane168942858741762044416140.5−244.7−6448.212663-Methylpentane118921630770571557504145.9−180.4−6448.162662,2-Dimethylbutane1206538162432290304121.5−147.6−6448.202442,3-Dimethylbutane1307264304361524288136.4−199.5−6448.15244Figure 5
Graphical relationships between the boiling points (B.P.), melting points (M.P.), flash points (F.P.), enthalpy change (E.G.), and the calculated topological indices of hexane and its four isomers, where R2 represents the correlation coefficient.


## 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.



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