INTRODUCTION
A graph G is a finite nonempty set V(G) of objects called vertices together
with a (possibly empty) set E(G) of 2element 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 firstorder
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 morder
connectivity index of an organic molecule whose molecule graph G is defined
as:
where, i_{1}···i_{m+1} (for simplicity)
runs over all paths of length m in G and d_{i} denote the degree of
vertex v_{i}. In particular, 4order connectivity index is defined as
follows:
Recently, a closely related variant of the Randiæ connectivity index
called the sumconnectivity index was introduced by Zhou
and Trinajstic (2009, 2010). For a simple connected
graph G, its sumconnectivity index X(G) is defined as the sum over all edges
of the graph of the terms (d_{u}+d_{v})^{1/2}, that
is:
where, d_{u} and d_{v} are the degrees of the vertices u and
v, respectively. It is a graphbased molecular structure descriptor. It has
been found that the sumconnectivity index correlates well with πelectronic
energy of benzenoid hydrocarbons and it is frequently applied in quantitative
structure property and Structureactivity studies (Kier and
Hall, 1986; Todeschini and Consonni, 2000).
The msum connectivity index of G is defined as:
where, i_{1}i_{2}···i_{m+1} runs
over all paths of length n in G. In particular, 4sum connectivity index are
defined as:
Dendrimers are hyperbranched 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
supramolecular chemistry, particularly in host guest reactions and selfassembly
processes. Their applications in chemistry, biology and nanoscience are unlimited.
Recently, some researchers investigated morder 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 4connectivity and 4sum connectivity
of an infinite families of polyphenylene dendrimers are computed.
FOURTHORDER CONNECTIVITY INDEX OF DENDRIMER
We consider polyphenylene dendrimer by construction of dendrimer generations
G_{n} has grown n stages. We denote this graph by D_{4}[n].
Figure 1 shows the generations G_{2} has grown 2 stages.
The following theorem gives the fourthorder connectivity index of polyphenylene
dendrimer.
Theorem 1: Let n ∈ N. The fourthorder connectivity index of D_{4}[n]
is:
Proof: First we compute ^{4}χ(D_{4}[n]). Let
denote the number of 4paths whose five consecutive vertices are of degree i_{1},
i_{2}, i_{3}, i_{4}, i_{5}, respectively. In
the same way, we use
to mean
in nth stages. Particularly, .
We can see that:

Fig. 1: 
Polyphenylene dendrimer of generations G_{n }has grown
2 stages 
Therefore, by Eq. 2, we obtain:
Now, we construct the relation between ^{4}χ(D_{4}[n])
and ^{4}χ(D_{4}[n1]) for n≤2. By simple reduction,
we have:
and for any:
(i_{1}i_{2}i_{3}i_{4}i_{5})
≠ (22222), (22223), (22232), 22322, (22332),
(22333), (23223), (23323), (23332), (23333), (32333),
(33223), (33333), (32234), (22343), (23432) 
we have .
Therefore, by Eq. 2, we have:
From above recursion formula, we have:
The proof is now complete.
FOURTHSUM CONNECTIVITY INDEX OF DENDRIMER D_{4}[N]
In this section, we will study the 4sum connectivity index of the same family
of dendrimers as shown in Fig. 1.
The following theorem gives the fourthsum connectivity index of polyphenylene
dendrimers.
Theorem 2: Let n ∈ N. The fourthsum connectivity index of D_{4}[n]
is:
Proof: First we compute ^{4S}χ(D_{4}[1]). Let
denote the number of 4paths whose five consecutive vertices are of degree i_{1},
i_{2}, i_{3}, i_{4}, i_{5}, respectively. In
the same way, we use
to mean
in nth stages. Particularly, .
We note that:
Therefore, by Eq. 5, we have:
Similar to that of Theorem 1, we can find the relation between ^{4S}χ(D_{4}[n])
and ^{4S}χ(D_{4}[n1]) for n≤2.
We have:
and for any:
(i_{1}i_{2}i_{3}i_{4}i_{5})
≠ (22222), (22223), (22232), (22322), (22332),
(22333), (23223), (23323), (23332), (23333), (32333),
(33223),(33233), (33333), (32234), (22343), (23432) 
we have .
Therefore, by Eq. 5, we obtain:
From above recursion formula, we have:
The proof is now complete.
CONCLUSION
In this study, the 4connectivity and 4sum connectivity indices for an infinite
families of polyphenylene dendrimers were presented. The similar method can
be extended to study of the mconnectivity and msum connectivity indices of
other dendrimers or nanostructures.