DOI: 10.18178/wcse.2018.06.122

## The Maximal Kirchhoff Index of Theta Shape Graph

*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. The Kirchhoff index of a graph is a structure-descriptor based on resistance distance. The investigation on the Kirchhoff index of graph is an important topic in the theory of graph. It is difficult to implement some algorithms to compute resistance distance and Kirchhoff index in a graph from their computational complexity. Hence, it makes sense to find closed-form formulae or solve extreme problems for the Kirchhoff index. For the connected graphs whose cyclomatic number less than two, their resistance distances and the Kirchhoff indices have been described well. In this paper, we discuss the graphs with cyclomatic number two, by graph transformations the maximal Kirchhoff in

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

Jinyu Zou

School of Computer, Qinghai Normal University, CHINA

Department of Basic Research, Qinghai University, CHINA

Haizhen Ren

Department of Mathematics, Qinghai Normal University, CHINA

Cite: Jinyu Zou, Haizhen Ren, "The Maximal Kirchhoff Index of Theta Shape Graph," *Proceedings of 2018 the 8th International Workshop on Computer Science and Engineering, pp. 740-744, Bangkok, 28-30 June, 2018.*