# Computing Gutman Connection Index of Thorn Graphs

Authors: Muhammad Javaid; Muhammad Khubab Siddique; Ebenezer Bonyah
Journal: Journal of Mathematics
 (2021)

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## Abstract

Chemical structural formula can be represented by chemical graphs in which atoms are considered as vertices and bonds between them are considered as edges. A topological index is a real value that is numerically obtained from a chemical graph to predict its various physical and chemical properties. Thorn graphs are obtained by attaching pendant vertices to the different vertices of a graph under certain conditions. In this paper, a numerical relation between the Gutman connection (GC) index of a graph and its thorn graph is established. Moreover, the obtained result is also illustrated by computing the GC index for the particular families of the thorn graphs such as thorn paths, thorn rods, thorn stars, and thorn rings.


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## Body

## 1. Introduction

LetΓ=VΓ,EΓ be a simple, finite, and connected graph with vertex set VΓ and edge set EΓ⊆VΓ×VΓ. Let Θ be a collection of such graphs; then, a topological index (TI) is a function from Θ to the set of real numbers defined under certain conditions on the vertices and edges of the graphs. Moreover, for Γ1,Γ2∈Θ, if Γ1≅Γ2, then TIΓ1=TIΓ2. The TIs are one of the graph-theoretic techniques which are widely used to study the different properties of the chemical graphs such as boiling point, melting point, flash point, temperature, pressure, tension, heat of evaporation, heat of formation, partition coefficient, retention times in chromatographic, and density [1, 2].TIs are also used in chemoinformatics which is combination of the three different subjects such as information science, chemistry, and mathematics. In chemoinformatics, on the bases of quantitative structural activity relationship (QSAR) and the qualitative structural property relationship (QSPR), the different chemical properties of a chemical graph are correlated with its structure [3, 4]. Gutman and Trinajstic [5] evaluated the total π-electron energy of the molecular structure by using the sum of square of degree (number of neighborhoods) of vertices of molecular graphs that is known by first Zagreb index nowadays. In the same paper, another descriptor appeared that is called as second Zagreb index. Furtula and Gutman [6] introduced another TI called third Zagreb index, which is also known as a forgotten index. After that, many TIs based on the degrees of vertices were established, see [7]. In 2018, Ali and Trinajstic [8] established a descriptor known as modified first Zagreb connection index. In the same paper, they also presented two more descriptors with the name first and second Zagreb connection indices. Ali et al. [9] introduced modified second and third Zagreb connection indices and compared Zagreb connection indices and modified Zagreb connection indices for T-sum graphs. Recently, Javaid et al. [10] defined the Gutman connection (GC) index with the help of connection numbers of a graph. For the various computational results, we refer [11–13].In 1947, Wiener [14] first time applied a distance-based TI to find the boiling point of paraffin. Now, it is called as the Wiener index. Gutman [15] introduced Schultz index of the second kind (Gutman index) as a type of vertex-valency-weighted sum of the distances between all pairs of vertices in a graph. In 1998, Gutman [16] introduced the idea of thorn graph with many applications in chemical graph theory. Bytautas et al. [17] developed an algorithm to find the mean Wiener terminal numbers for some thorny graphs. In 2005, Zhou [18] worked on modified Wiener indices for thorn trees. In 2011, Li [19] computed the Zagreb polynomials for thorny graphs. The study of thorn graphs provides mathematical results that relate numerical values of TIs of plerograms and kenograms. Plerograms are obtained from a molecule by expressing each atom with a vertex, but if the hydrogen atoms are not considered, then corresponding mathematical representation of a molecule is called as kenogram. The relation between the terminal Wiener indices of plerograms and kenograms was discussed in [20]. For more details about thorn graphs, see [21–23].In this study, we establish a relationship between the Gutman connection index of a simple connected graph and its thorn graph. It is also applied to evaluate the Gutman connection index of thorn paths, thorn rods, thorn rings, and thorn stars. Rest of the paper is organized as follows. Section2 contains the definitions and key concepts that are used in the remaining part of the paper. In Section 3, main and some related results are proved, and in Section 4, application of the main result is discussed for some thorn graphs.


## 2. Preliminaries

Here,Γ is considered as a finite connected graphs without loops and multiples edges, and let VΓ=v1,v2,…,vn be its vertex set for an n-vertex simple connected graph Γ. Consider H=h1,h2,…,hn as an n-tuple of nonnegative integers. Since distance between any two vertices of Γ is the same in both Γ and Γh, so we denote distance between vertices u and v with respect to both Γ and Γh as du,v.


### 2.1. Related Graphs

In this section, we recall the definition of caterpillar, thorn paths, thorn rods, thorn rings, and thorn stars.Definition 1 (see [24]).
Fori=1,2,…,n, a thorn graph Γh is constructed by attaching hi pendant vertices to the vertex vi of graph Γ, where VΓ=n. If Vi is the set of hi thorns of the vertex vi, then VΓh=VΓ∪∪i=1nVi. For more explanation, see Figure 1.Figure 1
Thorn graphΓh of the path graph Γ≅Pn with parameter h=h1,h2,h3,…,hn.Definition 2 (see [24]).
A thorn pathPn,h,k is a graph formed from a path Pn by attaching k neighbors to its terminal vertices and h neighbors to its nonterminal vertices. For more detail, see Figure 2.Figure 2
Thorn path,Pn,h,k.Definition 3 (see [24]).
A caterpillar (Tm,n′) is a thorn path obtained from path Pn such that its thorn vertices (other than pendant) are of the same degree m>2. It is clear that Pn,m−2,m−1=Tm,n′, see Figure 3.Figure 3
Caterpillar,Tm,n′.Definition 4 (see [24]).
A thorn rodPn,m is a graph that is obtained by adding m−1 pendant vertices to each terminal vertex of Pn. It is clear that Pn,2,m=Pn,m, see Figure 4.Figure 4
Thorn rod,Pn,m.Definition 5 (see [24]).
The thorn starSn,h1,h2,…,hn is obtained from the star Sn by attaching hi pendant neighbors to vertex vi for i=1,2,…,n. Thorn star Sn,h1,h2,…,hn defined here is shown in Figure 5.Figure 5
Thorn star.Definition 6 (see [24]).
If for each vertex of a cycle graphCn and a thorn of length m−2 is attached, then it is called thorn ring (denoted by Cn,m). For more details, see Figure 6.Figure 6
Thorn ring,C5,3.


### 2.2. Chemical Applicability ofGC Index

This section covers the definition of Gutman connection (GC) index with its applicability.Definition 7 (see [15]).
The Gutman index of a simple connected graphΓ (denoted by GutΓ) is defined as(1)GutΓ=∑u,v⊆ΓdΓudΓvdu,v.
In the above definition, Javaid et al. [10] replaced the vertex degree with the connection number and defined a new connection-based index known as the Gutman connective (GC) index as follows.Definition 8.
For a simple and connected graphΓ, the Gutman connection index is(2)GCΓ=∑u,v⊆ΓτuΓτvΓdu,v,where τu/Γ and τv/Γ denote the connection number of vertices u and v, respectively, of graph Γ and du,v is the distance between vertices u and v in Γ.
The correlation coefficients between the values ofGCI and eleven physicochemical properties of octane isomer boiling point (B. P), heat capacity at constant temperature (C. T), heat capacity at constant pressure (C. P), entropy (S), density (D), mean radius (Rm2), change in heat of vaporization (−ΔHv), standard heat of formation (−ΔHf), accentric factor (A. F), enthalpy of vaporization (HVAP), and standard enthalpy of vaporization (DHVAP) are shown in Table 1. It is clear that absolute value of correlation coefficient of GCI with S, A. F, HVAP, and DHVAP is above 0.9. Also, the value of its correlation coefficient with ΔHf is 0.8386. Consequently, the GC index may be a very useful index in the studies of QSPR and QSAR.
Now, before presenting the most frequent used lemma, we define some important notations asM21Γ=∑i=1nτvi/Γ and M2,j=∑i=1nτvi/Γdvi,vj, where j∈1,2,3,…,n.Table 1
Comparison of correlation coefficients between TIs and physicochemical properties of octane isomers.
GCIZC1∗M1M2ZC1ZC2B. P−0.789−0.352−0.718−0.498−0.770−0.168C. T−0.5130.0822−0.340−0.072−0.3890.165C. P0.0900.5810.294−0.4920.2670.524S−0.910−0.891−0.954−0.942−0.920−0.836D0.6160.7480.640−0.7300.5640.838Rm2−0.723−0.790−0.789−0.814−0.7590.659−ΔHf0.8390.3920.7600.5410.7840.259−ΔHv−0.171−0.071−0.222−0.129−0.2940.145A. F−0.901−0.949−0.973−0.986−0.951−0.862HVAP−0.910−0.606−0.886−0.728−0.914−0.433DHVAP−0.941−0.705−0.936−0.812−0.955−0.551Lemma 1.
LetΓ be a C3,C4-free simple and connected graph with vertex set VΓ=v1,v2,…,vn and edge set EΓ. Then, M21Γ=M1Γ−EΓ, where M1Γ is the first Zagreb index.Proof.
AsM21Γ=∑i=1nτvi/Γ=∑i=1n∑u∈NΓvidu−1, where NΓvi denotes the neighborhood of vi, where NΓvi=dvi, for all vi∈VΓ. Now, if u∈NΓvi, then vi∈NΓu. Hence, the number of neighborhoods in which vi lies is equal to the NΓvi=dvi, and then, the component of vi which contribute to τu will be dvi−1, for any u∈NΓvi. Consequently, M21Γ=∑i=1ndvidvi−1=∑i=1ndvi2−∑i=1ndvi=M1Γ−EΓ.


## 2.1. Related Graphs

In this section, we recall the definition of caterpillar, thorn paths, thorn rods, thorn rings, and thorn stars.Definition 1 (see [24]).
Fori=1,2,…,n, a thorn graph Γh is constructed by attaching hi pendant vertices to the vertex vi of graph Γ, where VΓ=n. If Vi is the set of hi thorns of the vertex vi, then VΓh=VΓ∪∪i=1nVi. For more explanation, see Figure 1.Figure 1
Thorn graphΓh of the path graph Γ≅Pn with parameter h=h1,h2,h3,…,hn.Definition 2 (see [24]).
A thorn pathPn,h,k is a graph formed from a path Pn by attaching k neighbors to its terminal vertices and h neighbors to its nonterminal vertices. For more detail, see Figure 2.Figure 2
Thorn path,Pn,h,k.Definition 3 (see [24]).
A caterpillar (Tm,n′) is a thorn path obtained from path Pn such that its thorn vertices (other than pendant) are of the same degree m>2. It is clear that Pn,m−2,m−1=Tm,n′, see Figure 3.Figure 3
Caterpillar,Tm,n′.Definition 4 (see [24]).
A thorn rodPn,m is a graph that is obtained by adding m−1 pendant vertices to each terminal vertex of Pn. It is clear that Pn,2,m=Pn,m, see Figure 4.Figure 4
Thorn rod,Pn,m.Definition 5 (see [24]).
The thorn starSn,h1,h2,…,hn is obtained from the star Sn by attaching hi pendant neighbors to vertex vi for i=1,2,…,n. Thorn star Sn,h1,h2,…,hn defined here is shown in Figure 5.Figure 5
Thorn star.Definition 6 (see [24]).
If for each vertex of a cycle graphCn and a thorn of length m−2 is attached, then it is called thorn ring (denoted by Cn,m). For more details, see Figure 6.Figure 6
Thorn ring,C5,3.


## 2.2. Chemical Applicability ofGC Index

This section covers the definition of Gutman connection (GC) index with its applicability.Definition 7 (see [15]).
The Gutman index of a simple connected graphΓ (denoted by GutΓ) is defined as(1)GutΓ=∑u,v⊆ΓdΓudΓvdu,v.
In the above definition, Javaid et al. [10] replaced the vertex degree with the connection number and defined a new connection-based index known as the Gutman connective (GC) index as follows.Definition 8.
For a simple and connected graphΓ, the Gutman connection index is(2)GCΓ=∑u,v⊆ΓτuΓτvΓdu,v,where τu/Γ and τv/Γ denote the connection number of vertices u and v, respectively, of graph Γ and du,v is the distance between vertices u and v in Γ.
The correlation coefficients between the values ofGCI and eleven physicochemical properties of octane isomer boiling point (B. P), heat capacity at constant temperature (C. T), heat capacity at constant pressure (C. P), entropy (S), density (D), mean radius (Rm2), change in heat of vaporization (−ΔHv), standard heat of formation (−ΔHf), accentric factor (A. F), enthalpy of vaporization (HVAP), and standard enthalpy of vaporization (DHVAP) are shown in Table 1. It is clear that absolute value of correlation coefficient of GCI with S, A. F, HVAP, and DHVAP is above 0.9. Also, the value of its correlation coefficient with ΔHf is 0.8386. Consequently, the GC index may be a very useful index in the studies of QSPR and QSAR.
Now, before presenting the most frequent used lemma, we define some important notations asM21Γ=∑i=1nτvi/Γ and M2,j=∑i=1nτvi/Γdvi,vj, where j∈1,2,3,…,n.Table 1
Comparison of correlation coefficients between TIs and physicochemical properties of octane isomers.
GCIZC1∗M1M2ZC1ZC2B. P−0.789−0.352−0.718−0.498−0.770−0.168C. T−0.5130.0822−0.340−0.072−0.3890.165C. P0.0900.5810.294−0.4920.2670.524S−0.910−0.891−0.954−0.942−0.920−0.836D0.6160.7480.640−0.7300.5640.838Rm2−0.723−0.790−0.789−0.814−0.7590.659−ΔHf0.8390.3920.7600.5410.7840.259−ΔHv−0.171−0.071−0.222−0.129−0.2940.145A. F−0.901−0.949−0.973−0.986−0.951−0.862HVAP−0.910−0.606−0.886−0.728−0.914−0.433DHVAP−0.941−0.705−0.936−0.812−0.955−0.551Lemma 1.
LetΓ be a C3,C4-free simple and connected graph with vertex set VΓ=v1,v2,…,vn and edge set EΓ. Then, M21Γ=M1Γ−EΓ, where M1Γ is the first Zagreb index.Proof.
AsM21Γ=∑i=1nτvi/Γ=∑i=1n∑u∈NΓvidu−1, where NΓvi denotes the neighborhood of vi, where NΓvi=dvi, for all vi∈VΓ. Now, if u∈NΓvi, then vi∈NΓu. Hence, the number of neighborhoods in which vi lies is equal to the NΓvi=dvi, and then, the component of vi which contribute to τu will be dvi−1, for any u∈NΓvi. Consequently, M21Γ=∑i=1ndvidvi−1=∑i=1ndvi2−∑i=1ndvi=M1Γ−EΓ.


## 3. Main Development

This section covers the main results of the Gutman connection (GC) index of the thorn graphs in its general form.Theorem 1.
LetΓh be a thorn graph of the graph Γ, where VΓ=n; then,(3)GCΓh=GCΓΓh+∑i=1ndviΓ+hi−12hihi−1+∑j=1ndviΓ+hi−1hjM2,jΓ+M21Γ+∑i=1n∑p∈Nvihpdvi,vj+1+∑1≤i<j≤nndviΓ+hi−1dviΓ+hi−1dvi,vj+2hihj.Proof.
Assume thatτu/Γh represents the connection numbers of u in graph Γh and τu/Γ represents the connection number of u in graph Γ. By the definition of the Gutman connection index, we have(4)GCΓh=∑u,v⊆VΓhτuΓhτvΓhdu,v.
By the definition ofVΓh, the sum in equation (5) can be partitioned into four sums as(5)∑u,v⊆VΓhτuΓhτvΓhdu,v=S1+S2+S3+S4,where S1 consists of contributions to GCΓh of pair of vertices from Γ, S2 consists of pair of vertices from Vi, for all 1≤i≤n, S3 is the contribution of pair of vertices one from u∈VΓ and the other one v is in Vi, for all 1≤i≤n, and S4 is taken from all the pair of vertices such that one of them u is from Vi and other vertex v from Vj.
Now,(6)S1=∑u,v⊆VΓτuΓhτvΓhdu,v=GCΓΓhand(7)S2=∑i=1n∑u,v⊆ViτuΓhτvΓhdu,v=∑i=1n∑u,v⊆VidviΓ+hi−1dviΓ+hi−1.2=∑i=1ndviΓ+hi−12.hihi−1.
Similarly,(8)S3=∑i=1n∑j=1n∑u=vi∑v∈VjτuΓhτvΓhdu,v=∑i=1n∑j=1n∑u=vi∑v∈VjτuΓhdvΓ+hj−1dvi,vj+1=∑i=1nτviΓh∑j=1ndvjΓ+hj−1hjdvi,vj+1=∑j=1ndvjΓ+hj−1hj∑i=1nτviΓhdvi,vj+1=∑j=1ndvjΓ+hj−1hj∑i=1nτviΓ+∑k∈Nvihkdvi,vj+1=∑j=1ndvjΓ+hj−1hjM2,jΓ+M21Γ+∑i=1n∑p∈Nvihpdvi,vj+1,(9)S4=∑1≤i<j≤nn∑u∈Vi∑v∈VjτuΓhτvΓhdu,v=∑1≤i<j≤nn∑u∈Vi∑v∈VjdviΓ+hi−1dvjΓ+hj−1dvi,vj+2=∑1≤i<j≤nndviΓ+hi−1dvjΓ+hj−1dvi,vj+2∑u∈Vi1∑v∈Vj1=∑1≤i<j≤nndviΓ+hi−1dvjΓ+hj−1dvi,vj+2hihj.
By substituting the values ofS1,S2,S3, and S4 in equation (5), the required result is obtained.
Now, using Lemma1 and the result of Theorem 1, we obtain Corollary 1 under the condition on Γ that it is free from cycles of length three and four. Moreover, Corollary 2 is obtained by attaching the same number of pendant vertices to each vertex of Γ.Corollary 1.
IfΓ is a graph free from cycles of length three and four, then(10)GCΓh=GCΓΓh+∑i=1ndviΓ+hi−12.hihi−1+∑j=1ndvjΓ+hj−1hjM2,jΓ+M1Γ−EΓ+∑i=1n∑p∈Nvihpdvi,vj+1+∑1≤i<j≤nndviΓ+hi−1dvjΓ+hj−1dvi,vj+2hihj.Corollary 2.
LetΓh be thorn graph of Γ with parameters h1=h2=⋯=hn=h; then,(11)GCΓh=GCΓΓh+hh−1M1Γ+h−12.n−1+2h−1EΓ∑j=1ndvjΓ+h−1hM2,jΓ+M1Γ−EΓ+h∑i=1n∑p∈Nvidvi,vj+1+∑1≤i<j≤nndvjΓ+h−1dvjΓ+h−1dvi,vj+2h2.


## 4. Applications

In this section, we find theGC index of the thorn path, thorn rod, and thorn ring graphs with the help of the main developed result (Theorem 1).Theorem 2.
Letn≥2 and h and k be nonnegative integers and Pn,h,k be a thorn graph of Pn; then,(12)GCPn,h,k=16h4n3−136h4n+3h4+h3n3−2h3n2−4h3n+8h3+h2k2n2+h2k2n−6h2k2+h2kn2−3h2kn+2h2k+136h2n3−8h2n2+596h2n−7h2+3hk2n2−hk2n−10hk2+3hkn2−13hk+16kn+18hk+2hn3−10hn2+23hn−26h+k4n+3k4+2k3n+2k2n2−k2n−7k2+2kn2−10kn+23n3−4n2+343n−14.Proof.
Here,h1=hn=k and hi=h, for 2≤i≤n−1. Now, we find S1,S2,S3, and S4 as derived in Theorem 1.(13)S1=GCΓ=∑u,v⊆ΓτuΓhτvΓhdu,v=τv1Γh∑j=3n−2τvjΓhdv1,vj+τv2Γh∑j=3n−2τvjΓhdv2,vj+τvn−1Γh∑j=2n−2τvnΓhdvn−1,vj+τvnΓh∑j=3n−2τvnΓhdvn,vj+τv1Γhτv2Γhdv1,v2+τv1Γhτvn−1Γhdv1,vn−1+τv1ΓhτvnΓhdv1,vn+τv2Γhτvn−1Γhdv2,vn−1+τv2ΓhτvnΓhdv2,vn+τvn−1ΓhτvnΓhdvn−1,vn+∑u,v⊆VΓ/v1,v2,vn−1,vnτuΓhτvΓhdu,v=h+1∑j=3n−22h+2j−1+h+k+1∑j=3n−22h+2j−2+h+k+1∑j=3n−22h+2n−1−j+h+1∑j=3n−22h+2n−j+h+1h+k+1+h+1h+k+1n−2+h+12n−1+h+k+12n−3+h+1h+k+1n−2+h+1h+k+1+∑u,v⊆VΓ/v1,v2,vn−1,vn2h+2.2h+2du,v=∑j=3n−22h+12n−1+2h+1h+k+1n−3+2h+1h+k+1n−1+h+12n−1+h+k+12n−3+4h+12∑u,v⊆VΓ/v1,v2,vn−1,vndu,v=2h+12n−1+2h+1h+k+1n−3∑j=3n−21+2h+1h+k+1n−1+h+12n−1+h+k+12n−3+4h+12∑j=1n−5jj+12=23h2n3−4h2n2+343h2n−14h2+2hkn2−10hkn+16hk+43hn3−8hn2+683hn−28h+k2n−3k2+2kn2−10kn+16k+23n3−4n2+343n−14s.and(14)S2=∑i=1ndviΓ+hi−12hihi−1=dv1Γ+h1−12h1h1−1+∑i=2n−1dviΓ+hi−12hihi−1+dvnΓ+hn−12hnhn−1=1+k−12kk−1+∑i=2n−12+h−12hh−1+1+k−12kk−1=2k3k−1+hh+12h−1n−2.
Now, to findS3, M21 and M2,j,j=1,2,…,n, are required:(15)M21Γ=∑i=1nτvi=1+1+∑i=3n−22+1+1=2+2n−4+2=2n−2=2n−4,(16)M2,1=∑i=1nτviΓdvi,v1=∑i=3n−2τviΓdvi,v1+τv2Γdv2,v1+τvn−1Γdvn−1,v1+τvnΓdvn,v1=2∑i=3n−2dvi,v1+1+n−2+n−1=n−2n−1=n2−3n+2,(17)M2,2=∑i=1nτviΓdvi,v2=∑i=3n−2τviΓdvi,v2+τv1Γdv1,v2+τvn−1Γdvn−1,v2+τvnΓdvn,v2=2∑i=3n−2dvi,v2+1+n−3+n−2=n−3n−4+2n−2=n2−5n+8,and(18)M2,n−1=∑i=1nτviΓdvi,vn−1=∑i=3n−2τviΓdvi,vn−1+τv1Γdv1,vn−1+τv2Γdv2,vn−1+τvnΓdvn,vn−1=2∑i=3n−2dvi,vn−1+n−2+n−3+1=n−3n−4+2n−2=n2−5n+8.
Also,(19)M2,n=∑i=1nτviΓdvi,vn=∑i=3n−2τviΓdvi,vn+τv1Γdv1,vn+τv2Γdv2,vn+τvn−1Γdvn−1,vn=2∑i=3n−2dvi,vn+n−1+n−2+1=n−4n−1+2n−1=n−1n−2.
For the next result, we will assume3≤j≤n−2:(20)M2,j=∑i=1nτviΓdvi,vj=τv1Γdv1,vj+τv2Γdv2,vj+τvn−1Γdvn−1,vj+τvnΓdvn,vj+∑i=3n−2τviΓdvi,vj=j−1+j−2+n−1−j+n−j+2∑i=3j−1j−i+∑i=j+1n−2i−j=2j2−2jn−2j+n2−n+4.
Now, we takeBj=∑i=1n∑p∈Nvihpdvi,vj+1, for j=1,2,…,n. We will find out B1, B2,andBn−1 and a general expression for Bj for j=3,4,…,n−2. So,(21)B1=∑i=1n∑p∈Nvihpdvi,v1+1=∑p∈Nv1hpdv1,v1+1+∑p∈Nv2hpdv2,v1+1+∑p∈Nvn−1hpdvn−1,v1+1+∑p∈Nvnhpdvn,v1+1+∑i=3n−2∑p∈Nvihpdvi,v1+1=h1+h+k2+h+kn−1+hn+2h∑i=3n−2j−1+1=n+12h+k+hn−4n+1=n+12h+k+hn−4=n+1k+n−2h,(22)B2=∑i=1n∑p∈Nvihpdvi,v2+1=∑p∈Nv1hpdv1,v2+1+∑p∈Nv2hpdv2,v2+1+∑p∈Nvn−1hpdvn−1,v2+1+∑p∈Nvnhpdvn,v2+1+∑i=3n−2∑p∈Nvihpdvi,v2+1=h2+h+k1+h+kn−2+hn−1+2h∑i=3n−2i−2+1=2nh+kn−1+hn−4n−1.
Bn−1=n+12h+k+hn−4n+1 and Bn=2nh+kn−1+hn−4n−1.
For3≤j≤n−2,(23)Bj=∑i=1n∑p∈Nvihpdvi,vj+1=∑p∈Nv1hpdv1,vj+1+∑p∈Nv2hpdv2,vj+1+∑p∈Nvn−1hpdvn−1,vj+1+∑p∈Nvnhpdvi,vj+1+∑i=3n−2∑p∈Nvihpdvi,vj+1=hj−1+1+h+kj−2+1+h+kn−1−j+1+hn−j+1+∑i=3n−2h+hdvi,vj+1=2nh+kn−1+2h∑i=3jdvi,vj+1+∑i=j+1n−2dvi,vj+1=2nh+kn−1+2h∑i=3jj−i+1+∑i=j+1n−2i−j+1=2nh+kn−1+hj−1j−2+n−j−2n−j+1.
So,(24)S3=∑j=1ndvjΓhj−1hjM2,jΓ+M21Γ+Bj=dv1Γh1−1h1M2,1Γ+M21Γ+B1+dv2Γ+h2−1h2M2,2Γ+M21Γ+B2+dvn−1Γhn−1−1h1M2,n−1Γ+M21Γ+Bn−1+dvnΓhn−1hnM2,nΓ+M21Γ+Bn+∑j=3n−2dvjΓhj−1hjM2,jΓ+M21Γ+Bj=1+k−1kn2−3n+2+2n−4+n+1k+n−2h+2+h−1hn2−5n+8+2n−4+2nh+kn−1+hn−4n−1+2+h−1hn2−5n+8+2n−4+hn−4n−1+1+k−1kn2−3n+2+2n−4+n+1k+n−2h+∑j=3n−22+h−1hM2,jΓ+2n−4+2nh+kn−1+hj−1j−2+n−j−2n−j+1=2k2n2−n−2+n+1k+n−2h+2hh+1n2−3n+4+2nh+kn−1+hn−4n−1+hh+1∑j=3n−22j2−2jn−2j+n2−n+4+2n−4+2nh+kn−1+hj−1j−2+n−j−2n−j+1=2k2n2−n−2+n+1k+n−2h+2hh+1n2−3n+4+2nh+kn−1+hn−4n−1+hh+14k−8h+223n−4hn2+23hn3+kn2+223hn−5kn−4n2+23n3−8.
Also,(25)S4=∑1≤i<j≤ndviΓ+hi−1dvjΓ+hj−1dvi,vj+2hihj=dv1Γ+h1−1dvnΓ+hn−1dv1,vn+2h1hn+∑j=2n−1dv1Γ+h1−1dvjΓ+hj−1dv1,vj+2h1hj+∑j=2n−1dvnΓ+hn−1dvjΓ+hj−1dvn,vj+2hnhj+∑2≤i<j≤n−1dviΓ+hi−1dvjΓ+hj−1dvi,vj+2hihj=1+k−11+k−1n+1k.k+∑j=2n−11+k−12+h−1j−1+2kh+∑j=2n−11+k−12+h−1n−j+2kh+∑2≤i<j≤n−12+h−12+h−1j−i+2hh=k4n+1+k2hh+1∑j=2n−1j+1+k2hh+1∑j=2n−1n−j+2+h2h+12∑2≤i<j≤n−1j−i+2=k4n+1+k2hh+1n−2n+3+h2h+12n−3n−2n+56.
By substituting the values inS1,S2,S3, and S4 in equation (5), we will get the required result.
Fork=m−2 and h=m−1, thorn path Pn,h,k represents a caterpillar Tm,n′. Similarly, a thorn path Pn,h,k will be thorn rod Pn,m if h=0 and k=m−1, i.e., Pn,m=Pn,0,m−1. Thus, the GI index of the thorn path and thorn rod is defined in the following corollaries.Corollary 3.
Forn,m≥2, the GC index of caterpillar is(26)GCT′m,n=16m4n3+m4n2−16m4n+13m3n3−4m3n2−343m3n+16m2n3−3m2n2+2216m2n+34m2+2mn2−10mn−104m+4n+36.Corollary 4.
Forn,m≥2, the GC index of thorn rod is(27)GCPn,m=m4n+3m4−2m3n−12m3+2m2n2−m2n+11m2−2mn2−6mn+18m+23n3−4n2+583n−34.Now, we present theGC index for the thorn star. Suppose that L is the number of all pendant vertices other than the pendant vertices attached to nth vertex, i.e., L=∑p=1n−1hp.Theorem 3.
LetSn,h1,h2,…,hn be a thorn star graph; then,(28)GCSn,h1,h2,…,hn=n−1n−2+hnL+n−2n−2+hn+∑i=1n−1hi3hi−1+n+hn−22hnhn−1+n+hn−2hn2n−1n−2+hn+L+3n−5n+hn−2+2L∑j=1n−1hj2+3hn+n−2hn∑j=1n−1hj2+4∑1≤i<j≤n−1hi2hj2.Proof.
The proof is followed by Theorem1.
Some special cases of Theorem3 are discussed in the following corollaries.Corollary 5.
If thorn of lengthh is attached with all the vertices other than the root vertex hi=h, for i≤n−1, then L=hnn−1/2 and(29)GCΓh=n−1n−2+hnL+n−2n−2+hn+h3h−1n−1+n+hn−22hnhn−1+n+hn−2hn2n−1n−2+hn+L+3n−5n+hn−2+hnn−1hnn−12+3hn+n−2hnhnn−12+2h4n−1n−2,where Γh=Sn,h,…,h,hn.Corollary 6.
If no thorn is attached with root vertex of thorn star, then(30)GCΓh=n−1n−2L+n−22+∑i=1n−1hi3hi−1+3n−5n−2+2L∑j=1n−1hj2+4∑1≤i<j≤n−1hi2hj2,where Γh=Sn,h1,h2⋯,hn−1,0.Corollary 7.
If no thorn is attached with root vertex and with other vertices a thorn of lengthh is attached, then(31)GCΓh=n−1n−2hn−1+n−22+h3h−1n−1+3n−5n−2+2hn−1h2n−1+2h4n−1n−2,where Γh=Sn,h,…,h,0.Now, we will discuss theGC index for the thorn ring graph.Theorem 4.
LetCm,n be a thorn graph of cycle graph Cn with n≥5; then,(32)GCCm,n=m−12nn2−12+m−12m−2m−3n+m−1m−2nn2−12+2n+m−2n2+4n−12+m−12m−22nn−1n+98,ifnis odd,m−12n32+m−12m−2m−3n+m−1m−2nn22+2n+m−2nn+42+m−12m−22nn2+8n−88,ifnis even.Proof.
The proof is followed by Theorem1.


## 5. Conclusion

In this section, we conclude our study as follows:(i)
Chemical applicability of GCI for several octane isomers is discussed, and it is found that it has high correlations with entropy, accentric factor, enthalpy of vaporization, standard enthalpy of vaporization, and standard heat of formation(ii)
TheGC index of thorn graphs is obtained in its general form(iii)
TheGC index of thorn paths, caterpillars, thorn rods, thorn stars, and thorn rings are also computed(iv)
A descriptorM21 (sum of connection numbers of vertices of a graph) is provided in Lemma 1 that is called as connection degree sumNow, we close this discussion that the various investigations are still needed for different (molecular) graphs or networks with the help of newly definedGC index.



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