On minimum revised edge Szeged index of bicyclic graphs

Abstract

The revised edge Szeged index $S{z}_e^{*}(G)$ of a graph G is defined as $S{z}_e^{*}(G)=\sum _{e=uv\in E(G)}(m_u(e)+\frac{m_0(e)}{2})(m_v(e)+\frac{m_0(e)}{2}),$ where $m_u(e)$ and $m_v(e)$ are, respectively, the number of edges of G lying closer to vertex u than to vertex v and the number of edges of G lying closer to vertex v than to vertex u, and $m_0(e)$ is the number of edges equidistant to u and v. In the paper, we show the sharp lower bound of revised edge Szeged index regarding bicyclic graphs. Moreover, all extremal graphs are characterized.

Keywords:

• Wiener index
• revised Szeged index
• revised edge Szeged index
• bicyclic graphs

AMS SUBJECT CLASSIFICATION:

• 05C12
• 05C35
• 05C90
• 92E10

Acknowledgements

M. Liu is supported by National Natural Science Foundation of China (No. 11961040), Innovation ability improvement project of colleges and universities in Gansu Province (No. 2019A-037), Tianyou Youth Talent Lift Program of Lanzhou Jiaotong University. S. Ji is supported Shandong Provincial Natural Science Foundation of China (No. ZR2019MA012).

Disclosure statement

No potential conflict of interest was reported by the authors.

Data availability

The data used to support the findings of this study are available from the corresponding author upon request.

Additional information

Funding

M. Liu is supported by National Natural Science Foundation of China (No.11961040), Innovation ability improvement project of colleges and universities in Gansu Province (No.2019A-037), Tianyou Youth Talent Lift Program of Lanzhou Jiaotong University. S. Ji is supported Shandong Provincial Natural Science Foundation of China(No. ZR2019MA012).