Computation of Topological Indices of Some Special Graphs

There are several chemical indices that have been introduced in theoretical chemistry to measure the properties of molecular topology, such as distance-based topological indices, degree-based topological indices and counting-related topological indices. Among the degree-based topological indices, the atom-bond connectivity (ABC) index and geometric–arithmetic (GA) index are the most important, because of their chemical significance. Certain physicochemical properties, such as the boiling point, stability and strain energy, of chemical compounds are correlated by these topological indices. In this paper, we study the molecular topological properties of some special graphs. The indices (ABC), (ABC4), (GA) and (GA5) of these special graphs are computed.


Introduction
Graph theory, as applied in the study of molecular structures, represents an interdisciplinary science called chemical graph theory or molecular topology.By using tools taken from graph theory, set theory and statistics, we attempt to identify structural features involved in structure-property activity relationships.Molecules and modeling unknown structures can be classified by the topological characterization of chemical structures with desired properties.Much research has been conducted in this area in the last few decades.The topological index is a numeric quantity associated with chemical constitutions purporting the correlation of chemical structures with many physicochemical properties, chemical reactivity or biological activity.Topological indices are designed on the grounds of the transformation of a molecular graph into a number that characterizes the topology of the molecular graph.We study the relationship between the structure, properties, and activity of chemical compounds in molecular modeling.Molecules and molecular compounds are often modeled by molecular graphs.A chemical graph is a model used to characterize a chemical compound.A molecular graph is a simple graph whose vertices correspond to the atoms and whose edges correspond to the bonds.It can be described in different ways, such as by a drawing, a polynomial, a sequence of numbers, a matrix or by a derived number called a topological index.The topological index is a numeric quantity associated with a graph, which characterizes the topology of the graph and is invariant under a graph automorphism.Some major types of topological indices of graphs are degree-based topological indices, distance-based topological indices and counting-related topological indices.The degree-based topological indices, the atom-bond connectivity (ABC) and geometric-arithmetic (GA) indices, are of great importance, with a significant role in chemical graph theory, particularly in chemistry.Precisely, a topological index Top(G) of a graph is a number such that, if H is isomorphic to G, then Top(H) = Top(G).It is clear that the number of edges and vertices of a graph are topological indices [1][2][3][4][5][6][7].We let G = (V, E) be a simple graph, where V(G) denotes its vertex set and E(G) denotes its edge set.For any vertex u ∈ V(G), we call the set N G (u) = {v ∈ V(G)|uv ∈ E(G)} the open neighborhood of u; we denote by d u the degree of vertex u and by S u = ∑ v∈N G (u) d(v) the degree sum of the neighbors of u.The number of vertices and number of edges of the graph G are denoted by |V(G)| and |E(G)|, respectively.A simple graph of order n in which each pair of distinct vertices is adjacent is called a complete graph and is denoted by K n .The notation in this paper is taken from the books [3,[8][9][10].
In this paper, we study the molecular topological properties of some special graphs: Cayley trees, Γ 2 n ; square lattices, SL n ; a graph G n ; and a complete bipartite graph, K m,n .Additionally, the indices (ABC), (ABC 4 ), (GA) and (GA 5 ) of these special graphs, whose definitions are discussed in the materials and methods section, are computed.Definition 1. [11] The oldest degree-based topological index, the Randi'c index, denoted by R − Definition 2. [12] For any real number α ∈ R, the general Randi'c index, R α (G), is defined as Definition 3. [13].The degree-based topological ABC index is defined as Definition 4. [2].The degree-based topological GA index is defined as Recently, several authors have introduced new versions of the ABC and GA indices, which we derive in the two definitions below.Definition 5. [14].The fourth version (ABC 4 ) of the ABC index is defined as Definition 6. [15].The fifth version (GA 5 ) of the GA index is defined as The concept of topological indices came from Wiener [16] while he was working on the boiling point of paraffin and was named the index path number.Later, the path number was renamed as the Wiener index [17].Hayat et al. [1] studied various degree-based topological indices for certain types of networks, such as silicates, hexagonals, honeycombs and oxides.Imran et al. [7] studied the molecular topological properties and determined the analytical closed formula of the ABC, ABC 4 , ABC 5 , GA, GA 4 and GA 5 indices of Sierpinski networks.M. Darafsheh [18] developed different methods to calculate the Wiener index, Szeged index and Padmakar-Ivan index for various graphs using the group of automorphisms of G.He also found the Wiener indices of a few graphs using inductive methods.A. Ayache and A. Alameri [19] provided some topological indices of m k -graphs, such as the Wiener index, the hyper-Wiener index, the Wiener polarity, Zagreb indices, Schultz and modified Schultz indices and the Wiener-type invariant.Wei Gao et al. [20] obtained certain eccentricity-version topological indices of the family of cycloalkanes C R i n .Recently, there has been extensive research into ABC and GA indices, as well as into their variants.For further studies of topological indices of various graphs and chemical structures, see [21][22][23][24][25][26].

The Cayley Tree Γ k
The Cayley tree Γ k of order k ≥ 1 is an infinite and symmetric regular tree, that is, a graph without cycles, from each vertex of which exactly k + 1 edges are issued.In this paper, we consider the Cayley tree Γ 2 n = (V, E, i) of order 2 and with n levels from the root x 0 , where V is the set of vertices of Γ 2 n , E is the set of edges of Γ 2 n , and i is the incidence function associating each edge e ∈ E with its end vertices.If i(e) = {x, y}, then x and y are adjacent vertices, and we write e =< x, y >.For any x, y ∈ V, the distance d(x, y) is defined as [27].
For the root x 0 of the Cayley tree, we have It is easy to compute the number of vertices reachable in step n or in level n starting from the root x 0 , which is , and the number of edges of , as is shown in Figure 1 below.n .Recently, there has been extensive research into ABC and GA indices, as well as into their variants.For further studies of topological indices of various graphs and chemical structures, see [21][22][23][24][25][26].

The Cayley Tree Γ k
The Cayley tree Γ k of order k ≥ 1 is an infinite and symmetric regular tree, that is, a graph without cycles, from each vertex of which exactly k + 1 edges are issued.In this paper, we consider the Cayley tree Γ 2 n = (V, E, i) of order 2 and with n levels from the root x 0 , where V is the set of vertices of Γ 2 n , E is the set of edges of Γ 2 n , and i is the incidence function associating each edge e ∈ E with its end vertices.If i(e) = {x, y}, then x and y are adjacent vertices, and we write e =< x, y >.For any x, y ∈ V, the distance d(x, y) is defined as [27].For the root x 0 of the Cayley tree, we have It is easy to compute the number of vertices reachable in step n or in level n starting from the root x 0 , which is , and the number of edges of Cayley tree Γ 2 n of order 2 with n levels, where n ≥ 1. Figure 1.Cayley tree Γ 2 n of order 2 with n levels, where n ≥ 1.

The Square Lattice Graph SL n
Lattice networks are widely used, for example, in distributed parallel computation, distributed control, wired circuits, and so forth.They are also known as grid or mesh networks.We choose a simple structure of lattice networks called a square lattice because this allows for a theoretical analysis [28].
We consider a square graph SL n (V, E) of size n × n vertices, where V denotes the set of vertices of SL n and E denotes the set of edges of SL n , such that the number of vertices is |V| = n 2 , and the number of edges is |E| = 2n(n − 1), as is shown in Figure 2  Lattice networks are widely used, for example, in distributed parallel computation, distributed control, wired circuits, and so forth.They are also known as grid or mesh networks.We choose a simple structure of lattice networks called a square lattice because this allows for a theoretical analysis [28].
We consider a square graph SL n (V, E) of size n × n vertices, where V denotes the set of vertices of SL n and E denotes the set of edges of SL n , such that the number of vertices is |V| = n 2 , and the number of edges is |E| = 2n(n − 1), as is shown in Figure 2

The Special Graph G n
Other kinds of special graphs, denoted by G n , can be obtained from other subgraphs.The structures in Figure 3 show how to obtain a graph G n .

The Special Graph G n
Other kinds of special graphs, denoted by G n , can be obtained from other subgraphs.The structures in Figure 3 show how to obtain a graph G n .Lattice networks are widely used, for example, in distributed parallel computation, distributed control, wired circuits, and so forth.They are also known as grid or mesh networks.We choose a simple structure of lattice networks called a square lattice because this allows for a theoretical analysis [28].
We consider a square graph SL n (V, E) of size n × n vertices, where V denotes the set of vertices of SL n and E denotes the set of edges of SL n , such that the number of vertices is |V| = n 2 , and the number of edges is |E| = 2n(n − 1), as is shown in Figure 2

The Special Graph G n
Other kinds of special graphs, denoted by G n , can be obtained from other subgraphs.The structures in Figure 3 show how to obtain a graph G n .In Figure 3, we have obtained the graph sequence G 1 , G 2 , G 3 , G 4 , ..., G n .We let G 1 be a complete graph of order 3 (G 1 ≡ K 3 ) and let V(G 1 ) = {v 1 , v 2 , v 3 }; we have subdivided the three edges of G 1 .The new vertices are denoted by {v 4 , v 5 , v 6 }, and It can be observed that the number of vertices of a graph G n is 3n and that the number of edges is 6n − 3 or, mathematically, |V(G n )| = 3n, and |E(G n )| = 6n − 3, respectively, where n ≥ 1.

The Complete Bipartite Graph K m,n
A graph K m,n is a complete bipartite graph if its vertex set can be partitioned into two subsets X and Y, such that one of the two endpoints of each edge in X and the other in Y, as well as each vertex v ∈ X, is adjacent to all vertices of Y, as is shown in Figure 4  In Figure 3, we have obtained the graph sequence G 1 , G 2 , G 3 , G 4 , ..., G n .We let G 1 be a complete graph of order 3 (G 1 ≡ K 3 ) and let V(G 1 ) = {v 1 , v 2 , v 3 }; we have subdivided the three edges of G 1 .The new vertices are denoted by {v 4 , v 5 , v 6 }, and G 2 [v 4 , v 5 , v 6 ] ≡ K 3 .Thus, It can be observed that the number of vertices of a graph G n is 3n and that the number of edges is 6n − 3 or, mathematically, |V(G n )| = 3n, and |E(G n )| = 6n − 3, respectively, where n ≥ 1.

The Complete Bipartite Graph K m,n
A graph K m,n is a complete bipartite graph if its vertex set can be partitioned into two subsets X and Y, such that one of the two endpoints of each edge in X and the other in Y, as well as each vertex v ∈ X, is adjacent to all vertices of Y, as is shown in Figure 4 below.Clearly, if |X| = m and |Y| = n, then |V(K m,n )| = m + n and |E(K m,n )| = mn.For more details, see [10].

Results and Discussion
Prior to presenting our main results, the edge partitions of the Cayley tree, lattice square, G n and complete bipartite graph, on the basis of the degrees of end vertices of each edge and the degree sum of the neighbors of end vertices of each edge, are discussed below.
Referring to Figure 1, there are two types of edges in Γ 2 n on the basis of the degrees of the end vertices of each edge, as follows: the first type, for e = uv ∈ E(Γ 2 n ), is such that d u = 1 and In the first type, there are 3 × 2 n−1 edges, and in the other type, because edges, as is shown in Table 1.Similarly, from Figure 1, there are three types of edges in Γ 2 n on the basis of the degree sum of vertices lying at a unit distance from the end vertices of each edge, as follows: the first type, for e = uv ∈ E(Γ 2 n ), is such that S u = 3 and S v = 5; the second type, for e = uv ∈ E(Γ 2 n ), is such that S u = 5 and S v = 9; the third type, for e edges in the first, second and third types of Γ 2 n , respectively, as is shown in Table 2.

Table 1. Edge partition of Γ 2
n on the basis of degrees of end vertices of each edge.

Results and Discussion
Prior to presenting our main results, the edge partitions of the Cayley tree, lattice square, G n and complete bipartite graph, on the basis of the degrees of end vertices of each edge and the degree sum of the neighbors of end vertices of each edge, are discussed below.
Referring to Figure 1, there are two types of edges in Γ 2 n on the basis of the degrees of the end vertices of each edge, as follows: the first type, for e = uv ∈ E(Γ 2 n ), is such that d u = 1 and In the first type, there are 3 × 2 n−1 edges, and in the other type, because edges, as is shown in Table 1.Similarly, from Figure 1, there are three types of edges in Γ 2 n on the basis of the degree sum of vertices lying at a unit distance from the end vertices of each edge, as follows: the first type, for e = uv ∈ E(Γ 2 n ), is such that S u = 3 and edges in the first, second and third types of Γ 2 n , respectively, as is shown in Table 2.

Table 1. Edge partition of Γ 2
n on the basis of degrees of end vertices of each edge.
Table 2. Edge partition of Γ 2 n on the basis of degree sum of neighbors of end vertices of each edge.
(S u , S v ) Where uv ∈ E Number of Edges Referring to Figure 2, there are four types of edges in SL n on the basis of the degrees of the end vertices of each edge, as follows: the first type, for e = uv ∈ E(SL n ), is such that d u = 2 and , then there are 8, 4(n − 3), 4(n − 2), and edges in the first, second, third and fourth types of SL n , respectively, as is shown in Table 3.
Table 3. Edge partition of SL n on the basis of degrees of end vertices of each edge.
Similarly, from Figure 2, there are nine types of edges in SL n on the basis of the degree sum of vertices lying at a unit distance from the end vertices of each edge, as follows: 1.
For e = uv ∈ E(SL n ) such that S u = 6 and S v = 9, there are 8 edges.

2.
For e = uv ∈ E(SL n ) such that S u = 9 and S v = 10, there are 8 edges.
For e = uv ∈ E(SL n ) such that S u = 9 and S v = 14, there are 8 edges. 5.
For e = uv ∈ E(SL n ) such that S u = 10 and S v = 15, there are 4(n − 4) edges.

6.
For e = uv ∈ E(SL n ) such that S u = 14 and S v = 15, there are 8 edges.7.
For e = uv ∈ E(SL n ) such that S u = Table 4 shows the edge partition of the square lattice SL n on the basis of the degree sum of vertices lying at a unit distance from the end vertices of each edge.Referring to Figure 3, for the edge partition of G n on the basis of the degrees of the end vertices of each edge, we have two cases for n = 1 and n ≥ 2, as follows: For n = 1, because G 1 ≡ K 3 , then there are 3 edges such that d u = d v = 2, as is shown in Table 5.
In the first type, there are 6 edges, and in the other type, because |E(G n )| = 6n − 3, then there are (6n − 3) − 6 = 6n − 9 edges, as is shown in Table 6.Similarly, from Figure 3, for the edge partition of G n on the basis of the degree sum of vertices lying at a unit distance from the end vertices of each edge, we have three cases for n = 1, n = 2 and n ≥ 3, as follows: For n = 1, because G 1 ≡ K 3 , then there are 3 edges such that S u = S v = 4, as is shown in Table 7.For n = 2, there are two types of edges in G 2 , as follows: the first type, for e = uv ∈ E(G 2 ), is such that S u = 8 and S v = 12; the other type, for e = uv ∈ E(G 2 ), is such that S u = S v = 12.There are 6 edges in the first type of G 2 and 3 edges in the second type of G 2 , as is shown in Table 8.For n ≥ 3, there are three types of edges in G n , as follows: the first type, for e = uv ∈ E(G n ), is such that S u = 8 and S v = 12; the second type, for e = uv ∈ E(G n ), is such that S u = 12 and S v = 16; the third type, for e = uv ∈ E(G n ), is such that S u = S v = 16.There are 6 edges in both the first type and the second type of G n , and because |E(G n )| = 6n − 3, then there are (6n − 3) − 6 − 6 = 6n − 15 edges, as is shown in Table 9. Proof.
1. From Table 3, by using the edge partition of SL n on the basis of the degrees of the end vertices of each edge, and because . After an easy simplification, we obtain 4 .2. By using the edge partition of SL n on the basis of the degree sum of the neighbors of the end vertices of each edge shown in Table 4, and because  1) and (2), we complete the proof of the result.Theorem 4. For the square lattice graph SL n the GA index and the fifth version (GA 5 ) of the GA index are equal to the following, respectively:

Proof.
1. From Table 3, by using the edge partition of SL n on the basis of the degrees of the end vertices of each edge, and because this gives that After an easy simplification, we obtain 2. By using the edge partition of SL n on the basis of the degree sum of the neighbors of the end vertices of each edge shown in Table 4, and because Proof.We prove this by using Tables 7-9.We use the edge partition of G n on the basis of the degree sum of the neighbors of the end vertices of each edge.Tables 7-9 explain such a partition for the graph G n for n = 1, n = 2 and n ≥ 3, respectively.Now by using the partitions given in Tables 7-9, we can apply the formula of the ABC 4 index to compute this index for the graph G n .Because we have

Table 4 .
Edge partition of SL n on the basis of degree sum of neighbors of end vertices of each edge.

Table 5 .
Edge partition of G n (n = 1) on the basis of degrees of end vertices of each edge.For n ≥ 2, there are two types of edges in G n , as follows: the first type, for e = uv ∈ E(G n ), is such that d u = 2 and d v = 4; the other type, for e

Table 6 .
Edge partition of G n (n ≥ 2) on the basis of degrees of end vertices of each edge. (d

Table 7 .
Edge partition of G n (n = 1) on the basis of degree sum of neighbors of end vertices of each edge.
(S u , S

Table 8 .
Edge partition of G n (n = 2) on the basis of degree sum of neighbors of end vertices of each edge. (S