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.
1. Introduction
Let be a simple, finite, and connected graph with vertex set and edge set . Let be a collection of such graphs; then, a topological index (TI) is a function from to the set of real numbers defined under certain conditions on the vertices and edges of the graphs. Moreover, for , if , then . 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 -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 -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. 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.
2. Preliminaries
Here, is considered as a finite connected graphs without loops and multiples edges, and let be its vertex set for an -vertex simple connected graph . Consider as an -tuple of nonnegative integers. Since distance between any two vertices of is the same in both and , so we denote distance between vertices and with respect to both and as .
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]). For , a thorn graph is constructed by attaching pendant vertices to the vertex of graph , where . If is the set of thorns of the vertex , then . For more explanation, see Figure 1.
Definition 2 (see [24]). A thorn path is a graph formed from a path by attaching neighbors to its terminal vertices and neighbors to its nonterminal vertices. For more detail, see Figure 2.
Definition 3 (see [24]). A caterpillar () is a thorn path obtained from path such that its thorn vertices (other than pendant) are of the same degree . It is clear that , see Figure 3.
Definition 4 (see [24]). A thorn rod is a graph that is obtained by adding pendant vertices to each terminal vertex of . It is clear that , see Figure 4.
Definition 5 (see [24]). The thorn star is obtained from the star by attaching pendant neighbors to vertex for . Thorn star defined here is shown in Figure 5.
Definition 6 (see [24]). If for each vertex of a cycle graph and a thorn of length is attached, then it is called thorn ring (denoted by ). For more details, see Figure 6.
2.2. Chemical Applicability of 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 graph (denoted by ) is defined asIn 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 , the Gutman connection index iswhere and denote the connection number of vertices and , respectively, of graph and is the distance between vertices and in .
The correlation coefficients between the values of 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 (), standard heat of formation (), 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 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 and , where .
Lemma 1. Let be a -free simple and connected graph with vertex set and edge set . Then, , where is the first Zagreb index.
Proof. As , where denotes the neighborhood of , where , for all Now, if , then . Hence, the number of neighborhoods in which lies is equal to the , and then, the component of which contribute to will be , for any . Consequently, .
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. Let be a thorn graph of the graph , where ; then,
Proof. Assume that represents the connection numbers of in graph and represents the connection number of in graph . By the definition of the Gutman connection index, we haveBy the definition of , the sum in equation (5) can be partitioned into four sums aswhere consists of contributions to of pair of vertices from , consists of pair of vertices from , for all , is the contribution of pair of vertices one from and the other one is in , for all , and is taken from all the pair of vertices such that one of them is from and other vertex from .
Now,andSimilarly,By substituting the values of , and in equation (5), the required result is obtained.
Now, using Lemma 1 and the result of Theorem 1, we obtain Corollary 1 under the condition on 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 .
Corollary 1. If is a graph free from cycles of length three and four, then
Corollary 2. Let be thorn graph of with parameters ; then,
4. Applications
In this section, we find the index of the thorn path, thorn rod, and thorn ring graphs with the help of the main developed result (Theorem 1).
Theorem 2. Let and and be nonnegative integers and be a thorn graph of ; then,
Proof. Here, and , for . Now, we find , and as derived in Theorem 1.andNow, to find , and , are required:andAlso,For the next result, we will assume :Now, we take , for . We will find out , and a general expression for for . So, and .
For ,So,Also,By substituting the values in , and in equation (5), we will get the required result.
For and , thorn path represents a caterpillar . Similarly, a thorn path will be thorn rod if and , i.e., . Thus, the index of the thorn path and thorn rod is defined in the following corollaries.
Corollary 3. For , the index of caterpillar is
Corollary 4. For , the index of thorn rod is
Now, we present the index for the thorn star. Suppose that is the number of all pendant vertices other than the pendant vertices attached to th vertex, i.e., .
Theorem 3. Let be a thorn star graph; then,
Proof. The proof is followed by Theorem 1.
Some special cases of Theorem 3 are discussed in the following corollaries.
Corollary 5. If thorn of length is attached with all the vertices other than the root vertex , for , then andwhere .
Corollary 6. If no thorn is attached with root vertex of thorn star, thenwhere .
Corollary 7. If no thorn is attached with root vertex and with other vertices a thorn of length is attached, thenwhere .
Now, we will discuss the index for the thorn ring graph.
Theorem 4. Let be a thorn graph of cycle graph with ; then,
Proof. The proof is followed by Theorem 1.
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)The index of thorn graphs is obtained in its general form(iii)The index of thorn paths, caterpillars, thorn rods, thorn stars, and thorn rings are also computed(iv)A descriptor (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 index.
Data Availability
The data used to support the findings of this study are cited at relevant places within article as references.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding this publication.