Edge Wiener Index of Gear Related Molecular Graphs and Their r-Corona Molecular Graphs
Yun Gao1, *, Sainan Zhou2, Wei Gao3
1Department of Editorial, Yunnan Normal University, Kunming, China
2Zhejiang Zhenyuan Pharmaceutical Co., Ltd, Shaoxing, China
3School of Information Science and Technology, Yunnan Normal University, Kunming, China
Abstract
Chemical compounds and drugs are often modelled as graphs (for example, Polyhex Nanotubes and Dendrimer Nanostar) where each vertex represents an atom of molecule and covalent bounds between atoms are represented by edges between the corresponding vertices. This graph derived from a chemical compounds is often called its molecular graph and can be different structures. The edge Wiener index of a graph is defined as the sum of the distances between all pairs of edges, and it has been found extensive applications in chemistry. In this paper, we determine the edge Wiener index of gear fan graph, gear wheel graph and their r-corona graphs.
Keywords
Organic Molecules, Edge Wiener Index, Fan Graph, Wheel Graph, Gear Fan Graph, Gear Wheel Graph, r-Corona Graph
Received: March 23, 2015
Accepted: April 5, 2015
Published online: April 10, 2015
@ 2015 The Authors. Published by American Institute of Science. This Open Access article is under the CC BY-NC license. http://creativecommons.org/licenses/by-nc/4.0/
1. Introduction
Wiener index, Hyper-Wiener index and edge Wiener index are introduced to reflect certain structural features of organic molecules. We denote Pn and Cn are path and cycle with n vertices. The graph Fn={v}Pn is called a fan graph and the graph Wn={v}
Cn is called a wheel graph. Graph Ir(G) is called r- crown graph of G which splicing r hang edges for every vertex in G. By adding one vertex in every two adjacent vertices of the fan path Pn of fan graph Fn, the resulting graph is a subdivision graph called gear fan graph, denote as
. By adding one vertex in every two adjacent vertices of the wheel cycle Cn of wheel graph Wn, The resulting graph is a subdivision graph, called gear wheel graph, denoted as
.
The (molecular) graphs considered in this paper are simple and connected. The vertex and edge sets of G are denoted by V(G) and E(G), respectively. The Wiener index is defined as the sum of distances between all unordered pair of vertices of a graph G, i.e.,
=
,
where is the distance between u and v in G. The Hyper-wiener index is defined as
=
.
Several papers contributed to determine the Wiener index and Hyper-wiener index of special graphs. Yan et al., [1] presented the graphs which minimize the Hyper-Wiener index among all graphs with given chromatic number and clique number and the graphs which maximum the Hyper-Wiener index among all graphs with given chromatic number and clique number. More results see Yan et al., [1-2], Gao et al., [3-4], Gao and Shi [5], Gao and Wang [6], Xi and Gao [7-8], Xi et al., [9], Gao et al., [10].
Then the edge-Wiener index of G is defined as the sum of the distances (in the line graph) between all pairs of edges of G, i.e.,
=
,
where the distance between two edges is the distance between the corresponding vertices in the line graph of G.
Buckley [11] proved that =
for a tree with order n. Gutman [12] presented that if G is a connected graph of order n and size q, then
. Gutman and Pavlovic [13] showed that if G is a connected unicyclic graph of order n, then
, with equality if and only if G
Cn. Recently, Dankelmanna et al., [14] verified that the asymptotically sharp upper bound
for graphs of order n.
In this paper, we present the edge wiener index of and
first; then, the edge wiener index of gear fan graph and gear wheel graph are determined; at last, the edge wiener index of
and
are derived.
2. Main Results and Proof
Theorem 1. =
+
+
.
Proof. Let Pn=v1v2…vn and the r hanging vertices of vi be ,
,…,
(1
i
n). Let v be a vertex in Fn beside Pn, and the r hanging vertices of v be
,
, …,
.
By the definition of edge Wiener index, we have
=
+
+
+
++
+
+
+
++
=+
+
+
+
+
+
+
+
++
=+
+
.
Corollary 1. =
.
Theorem 2. =
+
+
.
Proof. Let Cn=v1v2…vn and ,
,…,
be the r hanging vertices of vi (1
i
n). Let v be a vertex in Wn beside Cn, and
,
, …,
be the r hanging vertices of v. We denote
=
.
By the definition of edge Wiener index, we have
=
+
+
++
+
+
+
++
=+
+
+
+
+
+
+
+
+
+
=+
+
.
Corollary 2. =
.
Theorem 3. =
.
Proof. Let Pn=v1v2…vn and be the adding vertex between vi and vi+1. Let v be a vertex in Fn beside Pn. By virtue of the definition of edge Wiener index, we get
=
+
+
+
++
=+
+
+
+
+
=.
Theorem 4. =
.
Proof. Let Cn=v1v2…vn and v be a vertex in Wn beside Cn. Let be the adding vertex between vi and vi+1. Let
=
,
=
. In view of the definition of edge Wiener index, we deduce
=
+
+
+
++
=+
+
+
+
+
=.
Theorem 5. =
+
+
.
Proof. Let Pn=v1v2…vn and be the adding vertex between vi and vi+1. Let
,
,…,
be the r hanging vertices of vi (1
i
n). Let
,
,…,
be the r hanging vertices of
(1
i
n-1). Let v be a vertex in Fn beside Pn, and the r hanging vertices of v be
,
, …,
.
By virtue of the definition of edge Wiener index, we get
={
+
+
+
++
+
++
+
+
++
+
+
++
}+{
+
++
+
+
}
={+
+
+
+
+
+
+
++
+
+
+
+
+
+
+}+{
+
+
+
+
+
}
=+
+
.
Theorem 6. =
+
+
.
Proof. Let Cn=v1v2…vn and v be a vertex in Wn beside Cn. be the adding vertex between vi and vi+1. Let
,
, …,
. be the r hanging vertices of v and
,
,…,
be the r hanging vertices of vi (1
i
n). Let
=
and
,
,…,
be the r hanging vertices of
(1
i
n). Let
=
,
=
. In view of the definition of edge Wiener index, we deduce
={
+
+
+
++
+
++
+
+
++
+
+
++
}+{
+
++
+
+
}
={+
+
+
+
+
+
+
+
+
+
++
+
+
+
+
}+{
+
++
+
+
}
=+
+
.
3. Conclusion
Chemical compounds and drugs are often modeled as graphs where each vertex represents an atom of molecule, and covalent bounds between atoms are represented by edges between the corresponding vertices. This graph derived from a chemical compounds is often called its molecular graph, and can be different structures. An indicator defined over this molecular graph, the edge Wiener index, has been shown to be strongly correlated to various chemical properties of the compounds. Fan graph, wheel graph, gear fan graph, gear wheel graph and their r-corona graphs are common structural features of organic molecules. The contributions of our paper are determining the edge Wiener index of these special structural features of organic molecules.
Acknowledgements
First, we thank the reviewers for their constructive comments in improving the quality of this paper. This work was supported in part by the PhD start funding of the third author. We also would like to thank the anonymous referees for providing us with constructive comments and suggestions.
References