THE WIENER, SZEGED AND PI-INDICES OF A PHENYLAZOMETHINE DENDRIMER
M. GOLRIZa, M. R. DARAFSHEHb*, M. H. KHALIFEHb
aIslamic Azad University, Tehran South Branch, Tehran, Iran
bSchool of Mathematics, College of Science, University of Tehran, Tehran, Iran
A new type of phenylazomethine with a tetraphenylmethane core which was first investigated by Osamu Enoki et. al. is considered and its topological indices such as Wiener, Szeged and PI-index are found using a new method.
(Received August 22, 2011; accepted October 5, 2011)
1. Preliminaries
Let G be a simple graph with vertex set and edge set . The function which assigns to each pair of vertices in , the length of minimal path from to , is called the distance function between two vertices. The distance function between and edge and a vertex is where for
and. , .
The Wiener index of a graph is denoted by and is defined by .In general this kind of index is called a topological index, which is a distance based quantity assigned to a graph. This is an invariant of the graph in the sense that if a graph H is isomorphic to then . The Wiener index for the first time was introduced by H. Wiener in [11] and is related to chemical substances. The definition of the Wiener index in terms of distance between vertices of a graph for the first time was given by Hosoya in [6]. This index is extensively studied by various authors, and we may refer the reader to [5] and [1] in which a new method is found to calculate the Wiener index of a graph.
Another index that will be investigated in this paper is the Szeged index, which is defined as follows:
where for the two vertices and we define
and . The set and the quantity is defined similarly. We refer the reader to [4] to see more properties of the Szeged index of a graph.
Similar to the definition of the Szeged index we define the vertex PI-index as follows .
For more information we refer the reader to [8].
Since a bipartite graph has no cycle of odd length, for each edge we have ,
and if is not bipartite, then there is an edge such that . Hence
* Corresponding author: [email protected]
,
and it is proved in [8] that the equality holds if and only if is bipartite.
If has vertices, then
where denotes the complete bipartite graph. For more details see [2] and [10].
Another index that we will define here is the edge PI-index which was defined in[9]. In a graph two edges and are called parallel, and is written by symbol , if
.
In general the relation is not reflexive, but in a bipartite graph it is reflexive. We set ,
and , and define and similarly. Furthermore, for and edge we define
. Next we define the edge PI-index as follows:
2. Main results
In this section our aim is to compute the above indices for a new type of phenylazomethine dendrimer with a tetraphenylmethane core which is investigated in [3] whose graph is given below and is denoted by , where . In figure 1 below the graph
are drawn in such a way that .
Fig. 1. structure of the TPM dendrimer.
The graph has 4 arms, and we choose one of the arms and call it . Clearly is a subgraph of and in figure 2 the graphs are drawn.
Fig. 2. a subgraph of
With this setting the graph is obtained by joining four copies of together. Both graphs and are bipartite because they don’t include an odd cycle.
If and denote the number of vertices in and respectively then it is easy to find that
We want to compute the mentioned indices of the graph using a new method. To do this we will introduce the following concept.
Definition 1. Let be a simple graph. A subgraph of is called convex if for every two vertices and in , then contains all the edges and vertices of all the minimal paths from
to in .
Theorem 1. Let be a simple graph. If there exists a partition of the edge set of like such that is a graph with two connected components and , , and each of and is a convex graph then
Proof. For the proof we refer the reader to [7] and [12].
Theorem 2. .
Proof. It is proved in [8] that if is bipartite graph, then . Therefore to calculate it is enough to compute the number of edges of . But the number of edges of
is easily computed as
.
Hence using. .the result follows.
Theorem 3. .
Proof. In the graph if we delete an edge which is not contained in a cycle, we will obtain a graph with two components. Since each edge acts as a bridge, the two components are convex.
From the other hand in each cycle contained in , for each edge there is a different edge of which is parallel to it. If we delete and edge of and the edge parallel to it, then we again will obtain a graph with two components such that each component is a convex subgraph of .
Hence, in this way we will obtain a partition of the edge set of which satisfies the conditions of Theorem 1. Furthermore we have
It can be calculated that the number of edges in cycles contained in is equal to
, the rest of edges of , , are not
contained in any cycle. Therefore using Theorem 1 we can write
.
Theorem 4. .
Proof. Let us fix the notation used in the proof of Theorem 3. If and , then one of the components of has either or vertices, and the other component has
either or vertices respectively.
The number of such edges is or respectively, where . For these edges we can write
.
Now if , one of the components of has vertices and the other component has vertices and the number of such edges is equal to where . Therefore
Now we have 4 more edges that are not contained in , .
By Theorem 1 we can write:
,
and by substituting the values of , , , and the result will be proved.
Theorem 5. .
Proof. By Theorem 1 we can write , and
again by substituting the result follows.
Acknowledgement
This paper has been extracted from the research entitled “Computation of topological indices of phenylazomethine dendrimer” supported by the Islamic Azad University, Tehran South Branch.
Reference
[1] M. R. Darafsheh, Acta Appl. Math., 110, 1225 (2010).
[2] A. A. Dobrynin, Croat. Chemica. Acta, 70, 819 (1997).
[3] O. Enoki, H. Katoh, K. Yamamoto Organic Letters, 8(4), 569 (2006).
[4] I. Gutman, A. A. Dobrynin, Graph Theory Notes, NY 34, 37 (1998).
[5] I. Gutman and O. E. Polansky, Mathematical concepts in organic chemistry, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1986.
[6] H. Hosoya, Bull. Chem. Soc. Jpn. 4, 2332 (1971).
[7] M. H. Khalifeh, H. Yousefi-Azari, A. R. Ashrafi, Another aspect of graph invariants depending on the path metric and an application in nano-science, accepted in Computers and Mathematics with Applications.
[8] M. H. Khalifeh, H. Yosefi-Azari, A. R. Ashrafi, Disc. Appl. Math., Vol. 156 1780-1789.
[9] P. V. Khadikar, On a novel structural descriptor, PI. Natl. Acad. Scri. Lett. 23, 113 (2000) [10] M. J. Nadjafi-Arani, G. H. Fath-Tabar, A. R. Ashrafi, Appl. Math. Letter, 22, 1838 (2009).
[11] H. Wiener, Structural determination of paraffin boiling points, J. Amer. Chem. Soc. 69, 17 (1974).
[12] H. Yousefi-Azari, M. H. Khalifeh, A. R. Ashrafi, Calculating the Edge Wiener and Edge Szeged indices of graphs, submitted.
