Total Szeged Index, Vertex-Edge Wiener Index and Edge Hyper-Wiener Index of Certain Special Molecular Graphs

Yun Gao^{1}^{, *}, Zhiyang Jia^{2}, Wei Gao^{3}

^{1}Department of Editorial, Yunnan Normal University, Kunming, China

^{2}Tourism and culture college, Yunnan University, Lijiang, China

^{3}School 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 *P _{n}* and

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

*Theorem**1. *=.

*Proof. *Let *P _{n}*=

=+++

=+

++

++

=

*Corollary 1.*=.

*Theorem **2. *=.

*Proof. *Let *C _{n}*=

=+++

=

+

++

=.

*Corollary 2. *=.

*Theorem **3. *=.

*Proof. *Let P_{n}=v_{1}v_{2}…v_{n} 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 P_{n}, and the r hanging vertices of v be , , …, .

By virtue of the definition of total Szeged index, we yield

=+++

++

=

+

+

+

+

+

+

=.

*Corollary3. *=.

*Theorem **4. *=.

*Proof. *Let C_{n}=v_{1}v_{2}…v_{n} and v be a vertex in W_{n} beside C_{n}, andbe the adding vertex between v_{i} and v_{i+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.

References

- L. Yan, Y. Li, W. Gao, J. S. Li, On the extremal hyper-wiener index of graphs, Journal of Chemical and Pharmaceutical Research, 2014, 6(3): 477-481.
- L. Yan, W. Gao, J. S. Li, General harmonic index and general sum connectivity index of polyominochains and nanotubes, Journal ofComputational and Theoretical Nanoscience, In press.
- W. Gao, L. Liang, Y. Gao, Some results on wiener related index and shultz index of molecular graphs, Energy Education Science and Technology: Part A, 2014, 32(6): 8961-8970.
- W. Gao, L. Liang, Y. Gao, Total eccentricity, adjacent eccentric distance sum and Gutmanindex of certain special molecular graphs, The Biotechnology: An Indian Journal, 2014, 10(9): 3837-3845.
- W. Gao, L. Shi, Wiener index of gear fan graph and gear wheel graph, Asian Journal of Chemistry, 2014, 26(11): 3397-3400.
- W. Gao, W. F. Wang, Second atom-bond connectivity index of special chemical molecular structures, Journal of Chemistry, Volume 2014, Article ID 906254, 8 pages, http://dx.doi.org/10.1155/2014/906254.
- W. F. Xi, W. Gao, Geometric-arithmetic index and Zagreb indices of certain special molecular graphs, Journal of Advances in Chemistry, 2014, 10(2): 2254-2261.
- W. F. Xi, W. Gao, -Modified extremal hyper-Wiener index of molecular graphs, Journal of Applied Computer Science & Mathematics, 2014, 18 (8):43-46.
- W. F. Xi, W. Gao, Y. Li, Three indices calculation of certain crown molecular graphs, Journal of Advances in Mathematics, 2014, 9(6): 2696-2304.
- Y. Gao, W. Gao, L. Liang, Revised Szeged index and revised edge Szeged index of certain special molecular graphs, International Journal of Applied Physics and Mathematics, 2014, 4(6): 417-425.
- P. Manuel, I.Rajasingh, M.Arockiaraj, Total-Szeged Index of C4-Nanotubes,C4-Nanotori and DendrimerNanostars, Journal of Computational and Theoretical Nanoscience, 2013, 10:405–411.
- A. S. Cevik, A. D.Maden, New distance-based graph invariants and relations among them, Applied Mathematics and Computation, 2013, 219: 11171-11177.

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