International Journal of Mathematics and Computational Science, Vol. 1, No. 2, April 2015 Publish Date: Apr. 2, 2015 Pages: 59-69
Total Szeged Index, Vertex-Edge Wiener Index and Edge Hyper-Wiener Index of Certain Special Molecular Graphs
Yun Gao1, *, Zhiyang Jia2, Wei Gao3
1Department of Editorial, Yunnan Normal University, Kunming, China
2Tourism and culture college, Yunnan University, Lijiang, China
3School of Information Science and Technology, Yunnan Normal University, Kunming, China
Abstract
In past years, topological indices are introduced to measure the characters of chemical molecules. Thus, the study of these topological indices has raised large attention in the field of chemical science, biology science and pharmaceutical science. In this paper, by virtue of molecular structure analysis, we determine the total Szeged index, vertex-edge Wiener index and edge hyper-Wiener index of several crucial molecular graphs, such as fan molecular graph, wheel molecular graph, gear fan molecular graph, gear wheel molecular graph, and their r-corona molecular graphs. The results achieved in our paper illustrate the promising application prospects for chemical engineering.
Keywords
Chemical Graph Theory, Total Szeged Index, Vertex-Edge Wiener Index, Edge Hyper-Wiener Index, Fan Molecular Graph, Wheel Molecular Graph, Gear Fan Molecular Graph, Gear Wheel Molecular Graph, r-Corona Molecular Graph
Received: March 18, 2015
Accepted: April 1, 2015
Published online: April 2, 2015
@ 2015 The Authors. Published by the 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, PI index and Zagreb indices are introduced to reflect certain structural features of organic molecules. Several papers contributed to determine the distance-based index of special molecular graphs (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] for more detail).
Let e=uv be an edge of the molecular graph G. The number of edges and vertices of G whose distance to the vertex u is smaller than the distance to the vertex v is denoted by
. Analogously, 
is the number of edges and vertices of G whose distance to the vertex v is smaller than the distance to the vertex u.The total Szeged index of G is defined as
=
.
Manuel et. al., [11] determined the total Szeged Index of C4-Nanotubes,C4-Nanotori and Dendrimer Nanostars. It is well know that
=
+
+
,
where is edge-Szeged index, is vertex-Szeged index, and 
is edge-vertex Szeged index.
Let u be a vertex in molecular graph G. For the edges e=ab and f=xy, let
=
=
and
=
.
The vertex-edge Wiener index is introduced by Cevik and Maden [12] as follows:
=
.
Similar as Wiener index and hyper-Wiener index, we define the edge hyper-Wiener index of molecular graph G as follows
=
.
Let Pn and Cn be path and cycle with n vertices. The molecular graph Fn={v}
Pnis called a fan molecular graph and the molecular graph Wn={v}Cn is denoted as a wheel molecular graph. Molecular graph Ir(G) is called r- crown molecular 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 molecular graph Fn, the resulting molecular graph is a subdivision molecular graph called gear fan molecular graph, denote as 
. By adding one vertex in every two adjacent vertices of the wheel cycle Cn of wheel molecular graph Wn, The resulting molecular graph is a subdivision molecular graph, called gear wheel molecular graph, denoted as 
.
As we know, fan molecular graph, wheel molecular graph, gear fan molecular graph, gear wheel molecular graph and r-corona molecular graph are common molecular structure in chemical. Hence, it inspires us to study the topological indices of these important chemical molecular structures. In this paper, we present the total Szeged index, vertex-edge Wiener index and edge hyper-Wiener index of 
,
,
and 
. The rest parts of article are organized as follows: first, we compute the total Szeged index of above molecular graphs; then, the vertex-edge Wiener index of such chemical structures are presented; at last, we derive the edge hyper-Wiener index of these special molecular graphs.
2. Total Szeged Index
Theorem1. 
=
.
Proof. Let Pn=v1v2…vn and the r hanging vertices of vibe 
, 
,…, 
(1
in). Let v be a vertex in Fn beside Pn, and the r hanging vertices of v be 
, 
, …, 
. Using the definition of total Szeged index, we have
=
+
+
+
=
+
+
+
+
+
=
Corollary 1.
=
.
Theorem 2. 
=
.
Proof. Let Cn=v1v2…vn and, ,…, be the r hanging vertices of vi(1in). Let v be a vertex in Wn beside Cn, and , , …, be the r hanging vertices of v. We denote 
=
. In view of the definition of total Szeged index, we infer
=++
+
=
+
+
+
=.
Corollary 2. 
=
.
Theorem 3. 
=
.
Proof. Let Pn=v1v2…vn and 
be the adding vertex between vi and vi+1. Let , ,…, be the r hanging vertices of vi (1in). Let 
, 
,…, 
be the r hanging vertices of (1in-1). Let v be a vertex in Fn beside Pn, and the r hanging vertices of v be , , …, .
By virtue of the definition of total Szeged index, we yield
=+++
+
+
=
+
+
+
+
+
+
=.
Corollary3. 
=
.
Theorem 4. 
=
.
Proof. Let Cn=v1v2…vn and v be a vertex in Wn beside Cn, andbe the adding vertex between vi and vi+1. Let , , …, be the r hanging vertices of v and , ,…, be the r hanging vertices of vi (1in). Let 
=
and , ,…, be the r hanging vertices of (1in). Let 
=
, 
=
. In view of the definition of total Szeged index, we deduce
=+++
+
+
=
+
+
+
++
=.
Corollary 4. 
=
.
3. Vertex-Edge Wiener Index
The notations and terminologies for particular molecular graphs presented in this section can refer to section 2.
Theorem 5. 
=
.
Proof. Using the definition of vertex-edge Wiener index, we have
=
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
=
+
+
+
+0++
+
+
+
+
+
+
++
+
=.
Corollary 5. 
=
.
Theorem 6. 
=
.
Proof. In view of the definition of vertex-edge Wiener index, we infer
=+++++
+++
+
+
+
++++
=++++0+++
+
+
+
+
++
+
+
=.
Corollary 6. 
=
.
Theorem 7. 
=
.
Proof. By virtue of the definition of vertex-edge Wiener index, we yield
=++++
+
++++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+++
+
+
+
+
+
+
+
+
+
=+++++
+0++++
+
+++
+
+
+
++++++++
+
+
+
+
+++
++
+
=.
Corollary7. 
=
.
Theorem 8. 
=
.
Proof. In view of the definition of vertex-edge Wiener index, we deduce
=++++
+
++++
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+++
++
+
+
+
+
+
+
+
=+++++
+0++++
+
++
+
+
+
+
+++++
+++++++
+++
++
+ 
=.
Corollary 8. 
=
.
4. Edge Hyper-Wiener Index
The notations and terminologies for special molecular graphs used in this section can refer to section 2.
Theorem 9. 
=
+
+
.
Proof. By the definition of edge hyper-Wiener index, we have
=
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
=
+
+
+++
++
+
+
+
+{
++
+
++
+
+
++
+
}}
=
+
+
+{
+
+
}}
=++.
Corollary 9.
=
.
Theorem 10. 
=
+
+
.
Proof. By the definition of edge hyper-Wiener index, we have
=++
++
++
+
+
+
++
+++
++
++
+
=++
+++
+++
+
+
+{++
+++
+
+
++
+
}}
=
+
+
+{
+
+
=++.
Corollary 10. 
=
.
Theorem 11. 
=
+
+
.
Proof. By virtue of the definition of edge hyper-Wiener index, we get
=+++
+
++
+
++
+
++
+
+
+
+
++
+
+
+
+
++++
+
++
+
++
+
++
+
+
+
+
++
+
+
+
+
=++
+
+
+
+
+
+++++
++
+
+++
++
++
+{++
++
+
+
+
+++++
++
+
+++
++
+
+
=
+
+
+{
+
+
=++.
Corollary11. 
=
.
Theorem 12. 
=
+
+
.
Proof. In view of the definition of edge hyper-Wiener index, we deduce
=+++
+
++
+
++
+
++
+
+
+
+
++
+
+
+
+
++++
+
++
+
++
+
++
+
+
+
+
++
+
+
+
+
=++
++
++
+
++
+++++
++++
++
+
+
+{+++
+
++
+
+++++
+
+
+
+++
++
+
+
=
+
+
+{
+
+
=++.
Corollary12. 
=
.
5. Future Work
Let 
be the eccentricity of a vertex v in a connected molecular graph G, n=
be a number of vertices, and m=
be a number of edges. Cevik and Maden [12] defined following average class of molecular graph invariants:
• Average eccentricity:
=
=
.
• average edge eccentricity: 
=
.
• average vertex-edge eccentricity: 
=
.
• weighted average eccentricity: 
=
.
Hence, these average indices of special structural molecules graphs can be consider as subjects for future research.
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 National Natural Science Foundation of China (61262071), and the Key Science and Technology Research Project of Education Ministry (210210). We also would like to thank the anonymous referees for providing us with constructive comments and suggestions.
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