The minimal degree Kirchhoff index of bicyclic graphs

Yinzhen Mei; Chengxiao Guo; Yinzhen Mei; Chengxiao Guo

doi:10.3934/math.2024968

AIMS Mathematics

2024, Volume 9, Issue 7: 19822-19842. doi: 10.3934/math.2024968

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Research article

The minimal degree Kirchhoff index of bicyclic graphs

Yinzhen Mei ^,,
Chengxiao Guo

School of Mathematics, North University of China, Taiyuan, Shanxi 030051, China

Received: 22 April 2024 Revised: 22 May 2024 Accepted: 31 May 2024 Published: 18 June 2024
MSC : 05C09

The degree Kirchhoff index of graph $ G $ is defined as $ Kf^{*}(G) = \sum\limits_{{u, v}\subseteq V(G)}d(u)d(v)r_{G}(u, v) $, where $ d(u) $ is the degree of vertex $ u $ and $ r_{G}(u, v) $ is the resistance distance between the vertices $ u $ and $ v $. In this paper, we characterize bicyclic graphs with exactly two cycles having the minimum degree Kirchhoff index of order $ n\geq5 $. Moreover, we obtain the minimum degree Kirchhoff index on bicyclic graphs of order $ n\geq4 $ with exactly three cycles, and all bicyclic graphs of order $ n\geq4 $ where the minimum degree Kirchhoff index has been obtained.
- bicyclic graph,
- degree Kirchhoff index,
- resistance distance,
- $ \Theta $-graph
Citation: Yinzhen Mei, Chengxiao Guo. The minimal degree Kirchhoff index of bicyclic graphs[J]. AIMS Mathematics, 2024, 9(7): 19822-19842. doi: 10.3934/math.2024968

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Abstract

The degree Kirchhoff index of graph $ G $ is defined as $ Kf^{*}(G) = \sum\limits_{{u, v}\subseteq V(G)}d(u)d(v)r_{G}(u, v) $, where $ d(u) $ is the degree of vertex $ u $ and $ r_{G}(u, v) $ is the resistance distance between the vertices $ u $ and $ v $. In this paper, we characterize bicyclic graphs with exactly two cycles having the minimum degree Kirchhoff index of order $ n\geq5 $. Moreover, we obtain the minimum degree Kirchhoff index on bicyclic graphs of order $ n\geq4 $ with exactly three cycles, and all bicyclic graphs of order $ n\geq4 $ where the minimum degree Kirchhoff index has been obtained.

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