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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003236&_mathId=si111.gif&_user=111111111&_pii=S0166218X15003236&_rdoc=1&_issn=0166218X&md5=fc0e58e003ae70bc40db694e7523f347" title="Click to view the MathML source">Gclass="mathContainer hidden">class="mathCode">$G$ is defined as the network effective resistance between them if each edge of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003236&_mathId=si111.gif&_user=111111111&_pii=S0166218X15003236&_rdoc=1&_issn=0166218X&md5=fc0e58e003ae70bc40db694e7523f347" title="Click to view the MathML source">Gclass="mathContainer hidden">class="mathCode">$G$ is replaced by a unit resistor. The Kirchhoff index class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003236&_mathId=si191.gif&_user=111111111&_pii=S0166218X15003236&_rdoc=1&_issn=0166218X&md5=44babbea4efbe627987e7e803d93a7b3" title="Click to view the MathML source">K(G)class="mathContainer hidden">class="mathCode">$K\left(G\right)$ is the sum of the resistance distances between all the pairs of vertices in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003236&_mathId=si111.gif&_user=111111111&_pii=S0166218X15003236&_rdoc=1&_issn=0166218X&md5=fc0e58e003ae70bc40db694e7523f347" title="Click to view the MathML source">Gclass="mathContainer hidden">class="mathCode">$G$. 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X15003236&_mathId=si7.gif&_user=111111111&_pii=S0166218X15003236&_rdoc=1&_issn=0166218X&md5=dfae5e76ce7aa75009d42874e2daf579" title="Click to view the MathML source">n≥4class="mathContainer hidden">class="mathCode">$n\ge 4$ with minimal Kirchhoff index and determine bounds on the Kirchhoff index of bicyclic graphs. This improves and extends some earlier results.