Topological aspects of some dendrimer structures

1 Introduction

The branch of mathematics, precisely mathematical chemistry, in which graph theory is applied to solve molecular problems is called chemical graph theory (CGT). In CGT, a molecule is represented by a graph (molecular graph) in which atoms are taken as vertices and atomic bounds are taken as edges. The main objective of CGT is to associate a numerical number to the molecular graph, which correlates the chemical and physical properties of the molecule.

Dendrimers are repetitively branched molecules. A dendrimer is typically symmetric around the core, and often adopts a spherical three-dimensional morphology. The first dendrimers were made by divergent synthesis approaches by Vögtle in 1978 [1]. This molecule has wide applications in nanosciences, biology, and chemistry. A lot of research papers on computation of topological indices of dendrimers have been published [2], [3], [4], [5], [6].

A molecular graph G=(V,E) is a simple graph with the vertex set V(G) and the edge set E(G). The vertices represent nonhydrogen atoms and the edges represents covalent bonds between the corresponding atoms. The set of all the vertices adjacent to the vertex v is called the neighborhood, N_G(v), of the vertex v. In graph G, d(v)=|N_G(v)| represents the degree of v and S(v)=∑u∈NG(u)d(u). We used the standard notations from graph theory mostly taken from books [7], [8].

In 1975, Randić [9] proposed the first degree based topological index and defined as R(G)=1d(u)d(v). After that, Gutman and Tirnajstić [10] introduced the first Zagreb index and the second Zagreb index for a simply connected graph. Estrada et al. [11] defined the well-known topological index named as atom-bond connectivity (ABC) index. This topological index can be used to model the enthalpy of formation of alkanes and is defined as:

The fourth version of ABC index was proposed by Ghorbani and Hosseinzadeh [12]:

Vukičević and Furtula [13] introduced another well-known topological index called geometric-arithmetic (GA) index. The predictive power of the GA index is better than R− 12 for many physicochemical properties, for example entropy, boiling point, vaporization, enthalpy, acentric factor, and enthalpy of formation. For a simply connected graph G, the GA index is defined as:

Graovac et al. [14] introduced the fifth version of GA index and defined as:

For further history and results on topological indices, see [12], [15], [16], [17], [18], [19], [20], [21], [22].

In this paper, we compute the fourth ABC (ABC₄) and fifth GA (GA₅) indices for porphyrin D_nP_n, zinc porphyrin DPZ_n and poly propyl ether imine (PETIM) dendrimer structure.

2 Discussion and main results

This section contains computational results of some dendrimers. We give close formulae of GA₅ and ABC₄ indices of some dendrimers. We start with the PETIM dendrimer of generation having growth stages n. Figure 1 shows the growth of PETIM dendrimer. In the molecular structure of PETIM dendrimer, there are 24×2ⁿ−23 atoms and 24×2ⁿ−24 bonds.

Figure 1:

The molecular structure of PETIM dendrimer with nth growth.

In the following result, we computed the ABC₄ index of PETIM dendrimer.

Theorem 1.Let G be the molecular graph of PETIM dendrimer, then the ABC₄ index of G is

ABC4(G)=14.44×2n−3×7.6117.

Proof. For the molecular graph G of PETIM dendrimer, |V(G)|=24×2ⁿ−23 and |E(G)|=24×2ⁿ−24. We divide the edge set based on the sum of degree of neighbors of end-vertices of every edge. This partition is shown in Table 1.

Now from Table 1,

ABC4(PETIM)=∑uv∈E(PETIM)S(u)+S(v)−2S(u)S(v)=2×2n2+3−22×3+2×2n3+4−23×4+(8×2n−12)4+4−24×4+(6×2n−6)4+5−24×5+(6×2n−6)5+6−25×6

After some simplification, we obtain

ABC4(G)=14.44×2n−3×7.6117.

The following result gives the GA₅ index of PETIM dendrimer.

Theorem 2.Let G be the molecular graph of PETIM dendrimer, then the GA₅ index of G is

GA5(G)=2.9846×2n+3−23.938.

Proof. For the molecular graph G of PETIM dendrimer. From the definition of GA₅, we have

GA5(G)=∑uv∈E(PETIM)2S(u)S(v)S(u)+S(v)=2×2n22×32+3+2×2n23×43+4+(8×2n−12)24×44+4+(6×2n−6)24×54+5+(6×2n−6)25×65+6.

After some simplification, we get

GA5(G)=2.9846×2n+3−23.938.

Now, we compute the ABC₄ and GA₅ indices of zinc porphyrin dendrimer DPZ_n. The molecular graph of DPZ_n has 96n−10 atoms and 105n−11 bonds. The molecular graph of zinc porphyrin dendrimer DPZ_n is shown in Figure 2. In Table 2, the edge-set partition of DPZ_n is given.

Figure 2:

Molecular graph of zinc porphyrin dendrimer DPZ_n.

The following theorem is about the ABC₄ index for zinc porphyrin dendrimer DPZ_n.

Theorem 3.Let DPZ_n be the molecular graph of zinc porphyrin dendrimer. Then

ABC4(DPZn)=4.9771×2n+3−10.1770.

Proof. Consider the graph of DPZ_n dendrimer. The graphical structure of DPZ_n contains V|DPZ_n|=56×2ⁿ−7 vertices and E|DPZ_n|=64×2ⁿ−4. We find the edge set partition based on the sum of degrees of neighbors of end-vertices of every edge for the zinc porphyrin dendrimer. Table 2 shows the edge partition of DPZ_n depending upon sum of degree of neighbors of end-vertices of every edge. Now, by using definition and Table 2, we have

ABC4(G)=∑uv∈E(G)S(u)+S(v)−2S(u)S(v).

So,

ABC4(DPZn)=∑uv∈E(DPZn)S(u)+S(v)−2S(u)S(v) =8×2n4+4−24×4+84+5−24×5 +(16×2n−12)5+5−25×5+(8×2n−12)5+6−25×6 +(16×2n−12)5+7−25×7+85+8−25×8 +(4×2n)6+6−26×6+46+7−26×7 +(4×2n−4)6+8−26×8+(8×2n−8)7+8−27×8 +47+9−27×9+88+9−28×9+88+10−28×10 +410+12−210×12.

After some simplification, we obtain

ABC4(DPZn)=4.9771×2n+3−10.1770.

The following theorem gives the information about the computation of the GA₅ of porphyrin dendrimer DPZ_n.

Theorem 4.Let DPZ_n be the molecular graph of zinc porphyrin dendrimer. Then

GA5(DPZn)=3.9802×2n+4−0.1135.

Proof. From the definition and Table 2, we have

GA5(DPZn)=∑uv∈E(DPZn)2S(u)S(v)S(u)+S(v) =8×2n24×44+4+824×54+5+(16×2n−12)25×55+5 +(8×2n−12)25×65+6+(16×2n−12)25×75+7 +825×85+8+(4×2n)26×66+6+426×76+7 +(4×2n−4)26×86+8+(8×2n−8)27×87+8 +427×97+9+828×98+9+828×108+10 +4210×1210+12.

After some simplification, we obtain

GA5(DPZn)=3.9802×2n+4−0.1135.

Now, we shall compute the ABC₄ and GA₅ topological indices of porphyrin dendrimers D_nP_n for n>0. Note that n=2^m, where m≥2. The number of vertices and the number of edge in the structure of porphyrin dendrimers are 96n− 10 and 105n−11, respectively. The n growth stages of porphyrin dendrimer is shown in Figure 3. Now, we compute the ABC₄ and GA₅ indices of porphyrin dendrimer D_nP_n.

Figure 3:

Molecular graph of porphyrin dendrimer D_nP_n.

Theorem 5.Consider the graph of porphyrin dendrimer D_nP_n. Then,

ABC4(DnPn)=53.85n+4.1951

Proof. Consider the graph of D_nP_n dendrimer. The molecular graph of D_nP_n dendrimer has V|D_nP_n|=96×2ⁿ−10 vertices and E|D_nP_n|=105×2ⁿ−11. The edge set partition of D_nP_n dendrimer based on the sum of degrees of neighbors of end-vertices of every edge is shown in Table 3. With the help of Table 3, we have

ABC4(G)=∑uv∈E(G)S(u)+S(v)−2S(u)S(v).

So,

ABC4(DnPn)=∑uv∈E(DnPn)S(u)+S(v)−2S(u)S(v) =2n3+5−23×5+(n+1)4+4−24×4+(8n−6)4+5−24×5 +24n4+6−24×6+4n5+5−25×5 +(4n−6)5+6−25×6+6n5+7−25×7+18n6+7−26×7 +25n6+8−26×8+n7+8−27×8 +11n7+9−27×9+n8+9−28×9 =53.85n+4.1951.

The following theorem tells about the closed formula of GA₅ for porphyrin dendrimer D_nP_n.

Theorem 6.Consider the molecular graph of porphyrin dendrimer D_nP_n dendrimer. Then, the GA₅index of D_nP_n dendrimer is equal to

GA5(DnPn)=103.9015n−10.938

Proof. From the definition of ABC₄(G) and Table 3, we have

GA5(DnPn)=∑uv∈E(DnPn)2S(u)×S(v)S(u)+S(v)=2n23×53+5+(n+1)24×44+4+(8n−6)24×54+5+24n24×64+6+4n25×55+5+(4n−6)25×65+6+6n25×75+7+18n26×76+7+25n26×86+8+n27×87+8+11n27×97+9+n28×98+9.

After some simplification, we obtain the result:

GA5(DnPn)=103.9015n−10.938.

3 Conclusion

In this paper, we dealt with zinc porphyrin, poly propyl ether imine, and zinc porphyrin dendrimer structures and studied their topological indices. We have computed the fourth version of ABC index and fifth version of the GA index for these families of dendrimers.