Bicyclic graphs with maximal edge revised Szeged index

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• 作者：Mengmeng Liu^a ; ^{liumm05@163.com" class="auth_mail" title="E-mail the corresponding author} ; Lily Chen^b ; ^{lily60612@126.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Wiener index ; Revised Szeged index ; Edge Revised Szeged index ; Bicyclic graph
• 刊名：Discrete Applied Mathematics
• 出版年：2016
• 出版时间：31 December 2016
• 年：2016
• 卷：215
• 期：Complete
• 页码：225-230
• 全文大小：369 K

文摘

The edge revised Szeged index class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303146&_mathId=si2.gif&_user=111111111&_pii=S0166218X16303146&_rdoc=1&_issn=0166218X&md5=a8131548d8913dc303e75afee65ad645">class="imgLazyJSB inlineImage" height="17" width="43" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0166218X16303146-si2.gif">class="mathContainer hidden">class="mathCode">$S{z}_e^{\ast }\left(G\right)$ is defined as class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303146&_mathId=si3.gif&_user=111111111&_pii=S0166218X16303146&_rdoc=1&_issn=0166218X&md5=9262d74f50a071f5c03cc8f12ecaaece">class="imgLazyJSB inlineImage" height="18" width="374" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0166218X16303146-si3.gif">class="mathContainer hidden">class="mathCode">$S{z}_e^{\ast }\left(G\right)={\sum }_{e=uv\in E}(m_u\left(e\right)+m_0\left(e\right)/2)(m_v\left(e\right)+m_0\left(e\right)/2)$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303146&_mathId=si4.gif&_user=111111111&_pii=S0166218X16303146&_rdoc=1&_issn=0166218X&md5=92a7f7575876b6f8b6be3f4e659077d9" title="Click to view the MathML source">m_u(e)class="mathContainer hidden">class="mathCode">$m_u\left(e\right)$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303146&_mathId=si5.gif&_user=111111111&_pii=S0166218X16303146&_rdoc=1&_issn=0166218X&md5=b9acf3e7874b963d51f8aa2c5c069e92" title="Click to view the MathML source">m_v(e)class="mathContainer hidden">class="mathCode">$m_v\left(e\right)$ are, respectively, the number of edges of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303146&_mathId=si6.gif&_user=111111111&_pii=S0166218X16303146&_rdoc=1&_issn=0166218X&md5=c1d257e7e0c1c6e751ec7c973ce29e47" title="Click to view the MathML source">Gclass="mathContainer hidden">class="mathCode">$G$ lying closer to vertex class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303146&_mathId=si7.gif&_user=111111111&_pii=S0166218X16303146&_rdoc=1&_issn=0166218X&md5=b02147f00d3fc0988c78f79fe543950a" title="Click to view the MathML source">uclass="mathContainer hidden">class="mathCode">$u$ than to vertex class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303146&_mathId=si8.gif&_user=111111111&_pii=S0166218X16303146&_rdoc=1&_issn=0166218X&md5=5bc908ae081276016c274c04dcef3977" title="Click to view the MathML source">vclass="mathContainer hidden">class="mathCode">$v$ and the number of edges of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303146&_mathId=si6.gif&_user=111111111&_pii=S0166218X16303146&_rdoc=1&_issn=0166218X&md5=c1d257e7e0c1c6e751ec7c973ce29e47" title="Click to view the MathML source">Gclass="mathContainer hidden">class="mathCode">$G$ lying closer to vertex class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303146&_mathId=si8.gif&_user=111111111&_pii=S0166218X16303146&_rdoc=1&_issn=0166218X&md5=5bc908ae081276016c274c04dcef3977" title="Click to view the MathML source">vclass="mathContainer hidden">class="mathCode">$v$ than to vertex class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303146&_mathId=si7.gif&_user=111111111&_pii=S0166218X16303146&_rdoc=1&_issn=0166218X&md5=b02147f00d3fc0988c78f79fe543950a" title="Click to view the MathML source">uclass="mathContainer hidden">class="mathCode">$u$, and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303146&_mathId=si12.gif&_user=111111111&_pii=S0166218X16303146&_rdoc=1&_issn=0166218X&md5=4b6b63bc3a1e4df0f95836858b2d1ec3" title="Click to view the MathML source">m₀(e)class="mathContainer hidden">class="mathCode">$m_0\left(e\right)$ is the number of edges equidistant to class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303146&_mathId=si7.gif&_user=111111111&_pii=S0166218X16303146&_rdoc=1&_issn=0166218X&md5=b02147f00d3fc0988c78f79fe543950a" title="Click to view the MathML source">uclass="mathContainer hidden">class="mathCode">$u$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303146&_mathId=si8.gif&_user=111111111&_pii=S0166218X16303146&_rdoc=1&_issn=0166218X&md5=5bc908ae081276016c274c04dcef3977" title="Click to view the MathML source">vclass="mathContainer hidden">class="mathCode">$v$. In this paper, we give an upper bound of the edge revised Szeged index for a connected bicyclic graphs with size class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303146&_mathId=si15.gif&_user=111111111&_pii=S0166218X16303146&_rdoc=1&_issn=0166218X&md5=6d57dbba60054e1895449a5e00bfc979" title="Click to view the MathML source">m≥5class="mathContainer hidden">class="mathCode">$m\ge 5$, that is, with equality if and only if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303146&_mathId=si6.gif&_user=111111111&_pii=S0166218X16303146&_rdoc=1&_issn=0166218X&md5=c1d257e7e0c1c6e751ec7c973ce29e47" title="Click to view the MathML source">Gclass="mathContainer hidden">class="mathCode">$G$ is the graph obtained from the cycle class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X16303146&_mathId=si18.gif&_user=111111111&_pii=S0166218X16303146&_rdoc=1&_issn=0166218X&md5=ea0c8634677981e8d5bf6620df5c717f" title="Click to view the MathML source">C_m−2class="mathContainer hidden">class="mathCode">$C_{m-2}$ by duplicating a single vertex.