Computing Gutman Connection Index of Thorn Graphs

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. (orn 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.

e 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][12][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-valencyweighted 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. e 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. e relation between the terminal Wiener indices of plerograms and kenograms was discussed in [20]. For more details about thorn graphs, see [21][22][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. Section 2 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.

Preliminaries
Here, Γ is considered as a finite connected graphs without loops and multiples edges, and let V(Γ) � v 1 , v 2 , . . . , v n be its vertex set for an n-vertex simple connected graph Γ. Consider H � (h 1 , h 2 , . . . , h n ) as an n-tuple of nonnegative integers. Since distance between any two vertices of Γ is the same in both Γ and Γ h , so we denote distance between vertices u and v with respect to both Γ and Γ h as d(u, v).

Related Graphs.
In this section, we recall the definition of caterpillar, thorn paths, thorn rods, thorn rings, and thorn stars.
Definition 1 (see [24]). For i � 1, 2, . . . , n, a thorn graph Γ h is constructed by attaching h i pendant vertices to the vertex v i of graph Γ, where |V(Γ)| � n. If V i is the set of h i thorns of the vertex v i , then V(Γ h ) � V(Γ) ∪ ∪ n i�1 V i . For more explanation, see Figure 1.
Definition 2 (see [24]). A thorn path P n,h,k is a graph formed from a path P n by attaching k neighbors to its terminal vertices and h neighbors to its nonterminal vertices. For more detail, see Figure 2.
Definition 3 (see [24]). A caterpillar (T m,n ′ ) is a thorn path obtained from path P n such that its thorn vertices (other than pendant) are of the same degree m > 2. It is clear that P n,m− 2,m− 1 � T m,n ′ , see Figure 3.
Definition 4 (see [24]). A thorn rod P n,m is a graph that is obtained by adding m − 1 pendant vertices to each terminal vertex of P n . It is clear that P n,2,m � P n,m , see Figure 4.
Definition 5 (see [24]). e thorn star S n,h 1 ,h 2 ,...,h n is obtained from the star S n by attaching h i pendant neighbors to vertex v i for i � 1, 2, . . . , n. orn star S n,h 1 ,h 2 ,...,h n defined here is shown in Figure 5.
Definition 6 (see [24]). If for each vertex of a cycle graph C n and a thorn of length m − 2 is attached, then it is called thorn ring (denoted by C n,m ). For more details, see Figure 6.

Chemical Applicability of GC Index.
is section covers the definition of Gutman connection (GC) index with its applicability.
Definition 7 (see [15]). e Gutman index of a simple connected graph Γ (denoted by Gut(Γ)) is defined as 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 graph Γ, the Gutman connection index is where τ(u/Γ) and τ(v/Γ) denote the connection number of vertices u and v, respectively, of graph Γ and d(u, v) is the distance between vertices u and v in Γ. e correlation coefficients between the values of GCI 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 (− ΔH v ), standard heat of formation (− ΔH f ), 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 ΔH f 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 as M 21

Main Development
is section covers the main results of the Gutman connection (GC) index of the thorn graphs in its general form.
Proof. Assume that τ(u/Γ h ) represents the connection numbers of u in graph Γ h and τ(u/Γ) represents the connection number of u in graph Γ. By the definition of the Gutman connection index, we have By the definition of V(Γ h ), the sum in equation (5) can be partitioned into four sums as where S 1 consists of contributions to GC(Γ h ) of pair of vertices from Γ, S 2 consists of pair of vertices from V i , for all 1 ≤ i ≤ n, S 3 is the contribution of pair of vertices one from u ∈ V(Γ) and the other one v is in V i , for all 1 ≤ i ≤ n, and S 4 is taken from all the pair of vertices such that one of them u is from V i and other vertex v from Vj. Now, and Similarly,    Journal of Mathematics By substituting the values of S 1 , S 2 , S 3 , and S 4 in equation (5), the required result is obtained. Now, using Lemma 1 and the result of eorem 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 2. Let Γ h be thorn graph of Γ with parameters
Journal of Mathematics 5

Applications
In this section, we find the GC index of the thorn path, thorn rod, and thorn ring graphs with the help of the main developed result ( eorem 1).

Theorem 2. Let n ≥ 2 and h and k be nonnegative integers and P n,h,k be a thorn graph of P n ; then,
Proof. Here, h 1 � h n � k and h i � h, for 2 ≤ i ≤ n − 1. Now, we find S 1 , S 2 , S 3 , and S 4 as derived in eorem 1.

Journal of Mathematics
Also, By substituting the values in S 1 , S 2 , S 3 , and S 4 in equation (5), we will get the required result.
For k � m − 2 and h � m − 1, thorn path P n,h,k represents a caterpillar T m,n ′ . Similarly, a thorn path P n,h,k will be thorn rod P n,m if h � 0 and k � m − 1, i.e., P n,m � P n,0,m− 1 . us, the GI index of the thorn path and thorn rod is defined in the following corollaries.
Proof. e proof is followed by eorem 1. Some special cases of eorem 3 are discussed in the following corollaries.
Now, we will discuss the GC index for the thorn ring graph. Proof. e proof is followed by eorem 1.

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) e GC index of thorn graphs is obtained in its general form (iii) e GC index of thorn paths, caterpillars, thorn rods, thorn stars, and thorn rings are also computed (iv) A descriptor M 21 (sum of connection numbers of vertices of a graph) is provided in Lemma 1 that is called as connection degree sum Now, we close this discussion that the various investigations are still needed for different (molecular) graphs or networks with the help of newly defined GC index.

Data Availability
e data used to support the findings of this study are cited at relevant places within article as references.