ISBN: 978-981-11-3671-9 DOI: 10.18178/wcse.2017.06.188

## The Minimal Kirchhoff Index Of Theta Shape Graph

*Xiaomin Re, Jinyu Zou, Haizhen Ren*

*Abstract*— The resistance distance between any two vertices of a connected graph G is defined as the
effective resistance between them in the electrical network constructed from G by replacing each edge of G
with unit resistor. Kirchhoff index is a structure-descriptor based on resistance distance. For the theta shape
graphs(a specified class of bicycle graphs), the ordering rations of their Kirchhoff index remain open. In this
paper, some new ordering relations are obtained by three graph transformations, and the minimal Kirchhoff
index and the corresponding graph in this class of graphs is also discussed.

*Index Terms*— electrical network, resistance distance, Kirchhoff index, theta shape graph

Xiaomin Re, Haizhen Ren

Department of Mathematics, Qinghai Normal University, CHINA

Jinyu Zou

Department of Basic Courses, Qinghai University, CHINA

**ISBN: 978-981-11-3671-9 DOI: 10.18178/wcse.2017.06.17Xsrc="http://www.wcse.org/uploadfile/2019/0823/20190823055609629.png" style="width: 120px; height: 68px;" />[Download]**

Cite: Xiaomin Re, Jinyu Zou, Haizhen Ren, "The Minimal Kirchhoff Index Of Theta Shape Graph," *Proceedings of 2017 the 7th International Workshop on Computer Science and Engineering, pp. 1084-1088, Beijing, 25-27 June, 2017.*