IJCRR - 9(15), August, 2017
Pages: 01-07
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Atom Bond Connectivity Indices of Kragujevac Trees (Kgd,k)
Author: Keerthi G. Mirajkar, Bhagyashri R. Doddamani, Priyanka Y.B.
Category: General Sciences
Abstract:Aim: The aim of this article is to determine first four types of atom bond connectivity indices of Kragujevac trees with isomorphic branches and increased ordered branches.
Methodology: The method applied to obtain the goal of this article is analytic.
Results: The results we constructed here are for first four types of atom bond connectivity indices of Kragujevac tree with all branches of tree are mutually isomorphic to each other and increasing ordered branches.
Conclusion: The first four types of atom bond connectivity indices on kragujevac trees with isomorphic branches and increasing ordered branches are determined. Also, atom bond connectivity indices can be computed for other class of graphs.
Keywords: Atom bond connectivity indices, Kragujevac tree
DOI: 10.7324/IJCRR.2017.9151
Full Text:
Introduction:
Mathematical chemistry is a branch of theoretical chemistry using mathematical methods to discuss and predict molecular properties without necessarily referring to quantum mechanics [1,15,22]. Chemical graph theory is a branch of mathematical chemistry which applies graph theory in mathematical modeling of chemical phenomena [8]. This theory has an important effect on the development of the chemical sciences.
A graph G = (V, E) is a collection of points and lines connecting them. The points and lines of a graph are also called vertices and edges respectively. If e is an edge of G, connecting to the vertices u and v, then we write e = uv. A connected graph is a graph such that there exists a path between all pairs of vertices. The distance d(u, v) = dG(u, v) between two vertices u and v is the length of the shortest path between u and v in G.
A
molecular graph is a simple graph such that its vertices correspond to the atoms and edges corresponds to the bonds. According to the IUPAC terminology, a topological index is a numerical value associated with chemical constitution, which can be then used for correlation of chemical structure with various physical and chemical properties, chemical reactivity and biological activity [9,10,12,17,19,20,21,24].
In mathematical chemistry, numbers encoding certain structural features of organic molecules and derived from the corresponding molecular graph, are called graph invariants or more commonly called as topological indices.
Among topological descriptors, connectivity indices are very important and they have a prominent role in chemistry. In other words, if G be the connected graph, then we can introduce many connectivity topological indices for it, by distinct and different definition.
One of the best known and widely used is the connectivity index, introduced in 2009, Furtule[4] proposed the first atom bond connectivity index of a graph G as:
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1511934454028.png)
Where du denotes degree of vertex u and dv denotes degree of vertex v in G.The second atom bond connectivity index is introduced by A. Graovac[7]. It is defined as follows:
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1511934960184.png)
Where nu denotes the number of vertices of G whose distance to the vertex u is smaller than distance to the vertex v.Farahani[3] proposed a third atom bond connectivity index of G as follows:
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1511935151871.png)
where mu is the number of edges of G lying closer to u than to v and mv is the number of edges of G lying closer to v than to u
Ghorbani [6] defines a new version of atom bond connectivity index. It is named as fourth atom bond connectivity index and defined as:
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1511937981476.png)
where su denotes the sum of degrees of all neighbor of vertex u in G. Reader can find the history and results on this family of indices in [23,25-27].
All graphs considered here are connected, finite without multiple edges and loops. For undefined terminologies, we refer to [16].
Motivated by [3,4,6,7], In this article we study and compute the above mentioned four types of Atom Bond Connectivity Indices on the special class of graph called the Kragujevac tree with isomorphic branches and increasing ordered branches.
Definition 2.1. [11] Let P3 be the 3 vertex tree rooted at one of its terminal vertices, see Figure 1. For k = 2, 3,… construct the rooted tree Bk by identifying the roots of k copies of P3 .The vertex obtained by identifying the roots of P3 trees is the root of Bk .
Examples illustrating the structure of the rooted tree Bk are depicted in Figure.1.
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1511942722579.png)
A Kragujevac tree of degree d = 5, in which β1, β2, β3≅B2, β4≅B3,β5≅B5. The branches Bk has 2k + 1 vertices and 2k edges. A typical Kragujevac tree is denoted by Kgd,k where d ≥ 2 is the degree of central vertex and k ≥ 2.
Methadology:
In this article, we determine the first four types of atom bond connectivity indices on a special class of graph called as Kragujevac tree. To find these four types of atom bond connectivity indices on Kragujevac tree we have applied analytic method and hence established the following results.
Results:
Theorem 3.1. Let Kgd,k be the Kragujevac tree of degree d ≥ 2 with all branches Bk of tree are mutually isomorphic to each other. Then the first atom bond connectivity of Kgd,k is
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1511943522880.png)
Proof. By the definition of kragujevac tree Kgd,k of degree d ≥ 2, with all isomorphic branches Bk ,∀
k ≥ 2. Further each branch Bk of Kgd,k contains k pendent vertices. Then the kragujevac tree contains [k (2k + 1) + 1] vertices and (2k + d) edges.
We consider the following cases to compute the proof, which are depending upon the degree of the vertices in Kgd,k.
Case i. Each branch Bk has 2k edges, where k edges are incident with pendent vertices and the vertices of degree 2. Remaining k edges are incident with vertices of degree 2 and vertex of degree (k + 1). Since there are d number of branches present in Kgd,k, then
From equation (1),
The ABC1 for d number of Bk branches of Kgd,k is
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1511943689846.png)
Case ii. Now consider the d edges incident with central vertex of degree d and the vertices of degree (k + 1).
From equation (1),
The ABC1 for d edges of Kgd,k is
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1511943855473.png)
From equation (5) and (6),
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1511943971746.png)
Corollary 3.2. The first atom bond connectivity of Kragujevac tree Kgd,k , when d = k is
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1511944121696.png)
Proof. The proof follows from the theorem 3.1 and replacing d by k
Theorem 3.3. Let Kgd,k be the Kragujevac tree of degree d ≥2 with all branches Bk of tree are mutually isomorphic to each other, then the second atom bond connectivity of Kgd,k is
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1511944247696.png)
Proof. Here we consider the Kragujevac tree Kgd,k of degree d ≥2, with all isomorphic branches Bk ,∀
k ≥2. We compute the ABC2 by considering the following
cases depending upon the distance between of the vertices in Kgd,k.
Case i. Each branch Bk has 2k edges, where k edges are incident with pendent vertices and the vertices of degree 2. Here one vertex of degree 2 is closer to the pendent vertices and [d (2k + 1)] vertices are closer to the vertices of degree 2. The remaining k edges are incident with vertices of degree 2 and vertex of degree (k + 1). Hence 2 vertices are closer to vertices of degree 2 and [d (2k + 1) - 1] edges are closer to the vertex of degree (k + 1). Since there are d number of branches present in Kgd,k, then
From equation (2),
The ABC2 for d number of Bk branches of Kgd,k is
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1511944926383.png)
Case ii. Now consider the d edges incident with central vertex of degree d and the vertices of degree (k + 1). Here the (2k + 1) vertices are closer to the vertices of degree (k + 1) and [d (2k + 1) – 2k] vertices are closer to the vertex of degree d.
From equation (2), The ABC2 for d edges of Kgd,k is
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1511945049471.png)
Corollary 3.4. The second atom bond connectivity of Kragujevac tree Kgd,k , when d = k is
Proof. The proof follows from the theorem 3.3 and replacing d by k.
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1511945235450.png)
Theorem 3.5. Let Kgd,k be the Kragujevac tree of degree d ≥ 2 with all branches Bk of tree are mutually isomorphic to each other. Then the third atom bond connectivity of Kgd,k is
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1511945554419.png)
Proof. Here we consider the Kragujevac tree of degree d ≥2, with all isomorphic branches Bk, ∀
k ≥ 2. We consider the following cases to compute the proof, which are depending upon the distance between of the edges and vertices in Kgd,k.
Case i. Each branch Bk has 2k edges, where k edges are incident with pendent vertices and the vertices of degree 2. Here one edge is closer to the pendent vertices and [d (2k + 1)] edges are closer to the vertices of degree 2. The remaining k edges are incident with vertices of degree 2 and vertex of degree (k + 1). Hence 2 edges are closer to vertices of degree 2 and [d (2k + 1) - 1] edges are closer to the vertex of degree (k + 1). Since there are d number of branches present in Kgd,k, then
From equation (3),
The ABC3 for d number of Bk branches of Kgd,k is
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1511945731634.png)
Case ii. Now consider the d edges incident with central vertex of degree d and the vertices of degree (k + 1). Here (2k + 1) edges are closer to the vertices of degree (k + 1) and [d (2k + 1) – 2k] edges are closer to the vertex of degree d.
From equation (3),
The ABC3 for d edges of Kgd,k is
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1511946284435.png)
Corollary 3.6. The third atom bond connectivity of Kragujevac tree Kgd,k , when d = k is
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1511946438390.png)
Proof. The proof follows from the theorem 3.5 and replacing d by k.
Theorem 3.7. Let Kgd,k be the Kragujevac tree of degree d ≥ 2, with all branches Bk f tree are mutually isomorphic to each other, then the fourth atom bond connectivity of Kgd,k is
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1511946698243.png)
Proof. Let Kgd,k be the Kragujevac tree of degree d ≥ 2, with all isomorphic branches Bk,∀
k ≥ 2. We consider the following cases to compute the proof, which are depending upon the degree of neighbor vertices of Kgd,k..
Case i. Each branch Bk has 2k edges, where k edges are incident with pendent vertices and the vertices of degree 2. For the pendent vertices the neighbor vertices are of degree 2 and for the vertices of degree 2 the neighbor vertices are of degree (k + 2). The remaining k edges are incident with vertices of degree 2 and vertex of degree (k + 1). For the vertices of degree 2 the neighbor vertices are of degree (k + 2) and for the vertices of degree (k + 1) the neighbor vertices are of degree (2k + d). Since there are d number of branches present in Kgd,k, then
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1511946954413.png)
Case ii. Now consider the d edges of Kgd,k which are incident with central vertex of degree d and the vertices of degree (k + 1). For the vertex of degree d the neighbor vertices are of degree [d (k + 1)] and for the vertices of degree (k + 1) the neighbor vertices are of degree (2k + d).
From equation (4),
The ABC4 for d edges of Kgd,k is
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1511947123065.png)
Corollary 3.8. The fourth atom bond connectivity of Kragujevac tree Kgd,k
, when d = k is
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1514621463256.png)
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1521872350376.png)
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1521872375463.png)
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1521872395014.png)
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1521872413023.png)
![](https://ijcrr.com/admin/public/uploads/1/myfiles/1_1521872426012.png)
DISCUSSION
The original atom bond connectivity index was introduced in 1990’s and is defined in [2]. Atom bond connectivity Indices of Kragujevac trees emerged in several studies addressed to solve the problem of characterizing the tree with minimal atom bond connectivity index [5,13,14]. Let G be a simple graph on n vertices. By uv we denote the edge connecting the vertices u and v. A vertex of degree one is referred to as a pendent vertex. An edge whose one end vertex is pendent is referred to as a pendent edge. The formal definition of a Kragujevac tree was introduced in [18]. Hence by using degree of vertices, distances between the vertices and edges of Kragujevac tree, we established our results for atom bond connectivity indices of Kragujevac trees.
CONCLUSION
In this article, we studied the first four types of atom bond connectivity indices and have calculated the first four types of atom bond connectivity indices for Kragujevac trees with isomorphic branches and increased ordered branches. Also in this article we observe the second and third atom bond connectivity indices are same for Kragujevac trees for both isomorphic and increasing ordered branches. Nevertheless, there are still many other class of graphs that are not covered here. For further research, the atom bond connectivity indices for other class of graphs can be computed.
ACKNOWLEDGEMENT
Authors acknowledge the immense help received from the scholars whose articles are cited and included in references of this manuscript. The authors are also grateful to authors / editors / publishers of all those articles, journals and books from where the literature for this article has been reviewed and discussed. Conflict of Interest: Nil Source of Funding: Nil
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