Complete characterization of bicyclic graphs with minimal Kirchhoff index

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• 作者：Jia-Bao Liu^a ; ^b ; ^{liujiabaoad@163.com" class="auth_mail" title="E-mail the corresponding author} ; Xiang-Feng Pan^a ; ^{xfpan@ustc.edu" class="auth_mail" title="E-mail the corresponding author} ; Lei Yu^a ; ^c ; ^{yu1lei@mail.ustc.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Dong Li^a ; ^{djldong@126.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Resistance distance ; Kirchhoff index ; Bicyclic graphs ; Extremal graphs ; ΘΘ-type graphs ; Wiener index
• 刊名：Discrete Applied Mathematics
• 出版年：2016
• 出版时间：19 February 2016
• 年：2016
• 卷：200
• 期：Complete
• 页码：95-107
• 全文大小：531 K

文摘

The resistance distance between any two vertices of a graph le="Click to view the MathML source">G$G$ is defined as the network effective resistance between them if each edge of le="Click to view the MathML source">G$G$ is replaced by a unit resistor. The Kirchhoff index le="Click to view the MathML source">K(G)$K\left(G\right)$ is the sum of the resistance distances between all the pairs of vertices in le="Click to view the MathML source">G$G$. A bicyclic graph is a connected graph whose number of edges is exactly one more than its number of vertices. In this paper, we completely characterize the bicyclic graphs of order le="Click to view the MathML source">n≥4$n\ge 4$ with minimal Kirchhoff index and determine bounds on the Kirchhoff index of bicyclic graphs. This improves and extends some earlier results.