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Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers



Nabeel E. Arif, Roslan Hasni and Saeid Alikhani
 
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ABSTRACT

The m-order connectivity index mχ(G) of a graph G is where, runs over all paths of length m in G and di denotes the degree of vertex vi. Also is its m-sum connectivity index. A dendrimer is an artificially manufactured or synthesized molecule built up from branched units called monomers. In this study, the 4-connectivity and 4-sum connectivity indices of an infinite family of polyphenylene dendrimer are computed.

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  How to cite this article:

Nabeel E. Arif, Roslan Hasni and Saeid Alikhani, 2012. Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers. Journal of Applied Sciences, 12: 2279-2282.

DOI: 10.3923/jas.2012.2279.2282

URL: https://scialert.net/abstract/?doi=jas.2012.2279.2282
 
Received: June 24, 2012; Accepted: November 17, 2012; Published: December 03, 2012



INTRODUCTION

A graph G is a finite nonempty set V(G) of objects called vertices together with a (possibly empty) set E(G) of 2-element subsets of V(G) called edges. In a chemical graph, vertices represents atoms and edges represents bonds (Trinajstic, 1983).

A single number which characterizes the graph of a molecular is called a graph theoretical invariant or topological index. The connectivity index is one of the most popular topological indices introduced by (Randic, 1975). This index has been used in a wide spectrum of applications ranging from predicting physicochemical properties such as boiling point and solubility partition. The molecular connectivity index χ provides a quantitative assessment of branching of molecules. Randic (1975) first addressed the problem of relating the physical properties of alkanes to the degree of branching across an isomeric series. The degree of branching of a molecule was quantified using a branching index which subsequently became known as first-order molecular connectivity index χ. Kier and Hall (1986) extended this to higher orders and introduced modifications to account for heteroatoms.

Let G be a simple connected graph of order n. For an integer m = 1, the m-order connectivity index of an organic molecule whose molecule graph G is defined as:

Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers
(1)

where, i1···im+1 (for simplicity) runs over all paths of length m in G and di denote the degree of vertex vi. In particular, 4-order connectivity index is defined as follows:

Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers
(2)

Recently, a closely related variant of the Randiæ connectivity index called the sum-connectivity index was introduced by Zhou and Trinajstic (2009, 2010). For a simple connected graph G, its sum-connectivity index X(G) is defined as the sum over all edges of the graph of the terms (du+dv)-1/2, that is:

Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers
(3)

where, du and dv are the degrees of the vertices u and v, respectively. It is a graph-based molecular structure descriptor. It has been found that the sum-connectivity index correlates well with π-electronic energy of benzenoid hydrocarbons and it is frequently applied in quantitative structure property and Structure-activity studies (Kier and Hall, 1986; Todeschini and Consonni, 2000).

The m-sum connectivity index of G is defined as:

Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers
(4)

where, i1i2···im+1 runs over all paths of length n in G. In particular, 4-sum connectivity index are defined as:

Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers
(5)

Dendrimers are hyper-branched macromolecules, with a rigorously tailored architecture. They can be synthesized, in a controlled manner, either by a divergent or a convergent procedure. Dendrimers have gained a wide range of applications in supra-molecular chemistry, particularly in host guest reactions and self-assembly processes. Their applications in chemistry, biology and nano-science are unlimited. Recently, some researchers investigated m-order connectivity indices of some dendrimer nanostars, where, m = 2 and 3 (Ashrafi and Nikzad, 2009; Ahmadi and Sadeghimehr, 2009; Chen and Yang, 2011; Madanshekaf and Ghaneei, 2011; Yang et al., 2011). In this study, the 4-connectivity and 4-sum connectivity of an infinite families of polyphenylene dendrimers are computed.

FOURTH-ORDER CONNECTIVITY INDEX OF DENDRIMER

We consider polyphenylene dendrimer by construction of dendrimer generations Gn has grown n stages. We denote this graph by D4[n]. Figure 1 shows the generations G2 has grown 2 stages.

The following theorem gives the fourth-order connectivity index of polyphenylene dendrimer.

Theorem 1: Let n ∈ N. The fourth-order connectivity index of D4[n] is:

Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers

Proof: First we compute 4χ(D4[n]). Let Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers denote the number of 4-paths whose five consecutive vertices are of degree i1, i2, i3, i4, i5, respectively. In the same way, we use Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers to mean Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers in nth stages. Particularly, Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers.

We can see that:

Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers

Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers
Fig. 1: Polyphenylene dendrimer of generations Gn has grown 2 stages

Therefore, by Eq. 2, we obtain:

Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers

Now, we construct the relation between 4χ(D4[n]) and 4χ(D4[n-1]) for n≤2. By simple reduction, we have:

Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers

and for any:

(i1i2i3i4i5) ≠ (22222), (22223), (22232), 22322, (22332),
(22333), (23223), (23323), (23332), (23333), (32333),
(33223), (33333), (32234), (22343), (23432)

we have Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers.

Therefore, by Eq. 2, we have:

Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers

From above recursion formula, we have:

Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers

The proof is now complete.

FOURTH-SUM CONNECTIVITY INDEX OF DENDRIMER D4[N]

In this section, we will study the 4-sum connectivity index of the same family of dendrimers as shown in Fig. 1.

The following theorem gives the fourth-sum connectivity index of polyphenylene dendrimers.

Theorem 2: Let n ∈ N. The fourth-sum connectivity index of D4[n] is:

Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers

Proof: First we compute 4Sχ(D4[1]). Let Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers denote the number of 4-paths whose five consecutive vertices are of degree i1, i2, i3, i4, i5, respectively. In the same way, we use Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers to mean Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers in nth stages. Particularly, Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers.

We note that:

Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers

Therefore, by Eq. 5, we have:

Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers

Similar to that of Theorem 1, we can find the relation between 4Sχ(D4[n]) and 4Sχ(D4[n-1]) for n≤2.

We have:

Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers

and for any:

(i1i2i3i4i5) ≠ (22222), (22223), (22232), (22322), (22332),
(22333), (23223), (23323), (23332), (23333), (32333),
(33223),(33233), (33333), (32234), (22343), (23432)

we have Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers.

Therefore, by Eq. 5, we obtain:

Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers

From above recursion formula, we have:

Image for - Fourth Order and Fourth Sum Connectivity Indices of Polyphenylene Dendrimers

The proof is now complete.

CONCLUSION

In this study, the 4-connectivity and 4-sum connectivity indices for an infinite families of polyphenylene dendrimers were presented. The similar method can be extended to study of the m-connectivity and m-sum connectivity indices of other dendrimers or nano-structures.

REFERENCES
1:  Ashrafi, A.R. and P. Nikzad, 2009. Connectivity index of the family of dendrimer nanostars. Digest J. Nanomater. Biostruct., 4: 269-273.
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2:  Ahmadi, M.B. and M. Sadeghimehr, 2009. Second-order connectivity index of an infinite class of dendrimer nanostars. Digest J. Nanomater. Biostruct., 4: 639-643.
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3:  Chen, S. and J. Yang, 2011. Second-order and third-order connectivity indices of dendrimer nanostars. Int. Mathe. Forum, 6: 223-225.
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4:  Kier, L.B. and L.H. Hall, 1986. Molecular Connectivity in Structure Activity Analysis. John Wiley, London.

5:  Madanshekaf, A. and M. Ghaneei, 2011. The second-order connectivity index of dendrimer nanostars. Digest J. Nanomater. Biostruct., 6: 433-437.
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6:  Randic, M., 1975. On characterization of molecular branching. J. Am. Chem. Soc., 97: 6609-6615.
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7:  Todeschini, R. and V. Consonni, 2000. Handbook of Molecular Descriptors. 1st Edn., Wiley-VCH, Weinheim.

8:  Yang, J., F. Xia and S. Chen, 2011. Second-order and third-order connectivity indices of an infinite family of dendrimer nanostars. Int. J. Contemp. Math. Sci., 6: 215-220.
Direct Link  |  

9:  Zhou, B. and N. Trinajstic, 2010. On general sum-connectivity index. J. Math. Chem., 47: 210-218.
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10:  Zhou, B. and N. Trinajstic, 2009. On a novel connectivity index. J. Math. Chem., 46: 1252-1270.
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11:  Trinajstic, N., 1983. Chemical Graph Theory. Vol. 1 and 2, CRC Press, Boca Raton, FL., USA.

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