On the sum of the Laplacian eigenvalues of a graph and Brouwer's conjecture

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• 作者：Hilal A. Ganie^a ; ^{hilahmad1119kt@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Ahmad M. Alghamdi^b ; ^{amghamdi@uqu.edu.sa" class="auth_mail" title="E-mail the corresponding author} ; S. Pirzada^a ; ^{pirzadasd@kashmiruniversity.ac.in" class="auth_mail" title="E-mail the corresponding author}
• 关键词：05C30 ; 05C50
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：15 July 2016
• 年：2016
• 卷：501
• 期：Complete
• 页码：376-389
• 全文大小：343 K

文摘

For a simple graph G with n-vertices, m   edges and having Laplacian eigenvalues class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300453&_mathId=si1.gif&_user=111111111&_pii=S0024379516300453&_rdoc=1&_issn=00243795&md5=f0aba3bfc27ce19fc7852f428e892510" title="Click to view the MathML source">μ₁,μ₂,…,μ_n−1,μ_n=0class="mathContainer hidden">class="mathCode">${\mu }_1,{\mu }_2,\dots ,{\mu }_{n-1},{\mu }_n=0$, let class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300453&_mathId=si2.gif&_user=111111111&_pii=S0024379516300453&_rdoc=1&_issn=00243795&md5=4464fee4c238a3a65a98607a8b1dabff">class="imgLazyJSB inlineImage" height="21" width="122" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379516300453-si2.gif">class="mathContainer hidden">class="mathCode">$S_k(G)={\sum }_{i=1}^k{\mu }_i$, be the sum of k largest Laplacian eigenvalues of G  . Brouwer conjectured that class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300453&_mathId=si3.gif&_user=111111111&_pii=S0024379516300453&_rdoc=1&_issn=00243795&md5=5fb652af8b0b634e6ed1bc6e4eaf9d35">class="imgLazyJSB inlineImage" height="21" width="137" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379516300453-si3.gif">class="mathContainer hidden">class="mathCode">$S_k(G)\le m+\left(\begin{array}{c}k+1\\ 2\end{array}\right)$, for all class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300453&_mathId=si28.gif&_user=111111111&_pii=S0024379516300453&_rdoc=1&_issn=00243795&md5=a52d09e5d9f4b46856826700f13a3bbe" title="Click to view the MathML source">k=1,2,…,nclass="mathContainer hidden">class="mathCode">$k=1,2,\dots ,n$. We obtain upper bounds for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300453&_mathId=si5.gif&_user=111111111&_pii=S0024379516300453&_rdoc=1&_issn=00243795&md5=dd877d3da487be3f2d42eb3559c6160a" title="Click to view the MathML source">S_k(G)class="mathContainer hidden">class="mathCode">$S_k(G)$ in terms of the clique number ω, the vertex covering number τ and the diameter d of a graph G  . We show that Brouwer's conjecture holds for certain classes of graphs. The Laplacian energy class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300453&_mathId=si6.gif&_user=111111111&_pii=S0024379516300453&_rdoc=1&_issn=00243795&md5=89c39c6ff28ccc1ed20522cc8f5b3538" title="Click to view the MathML source">LE(G)class="mathContainer hidden">class="mathCode">$\mathit{LE}(G)$ of a graph G   is defined as class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300453&_mathId=si7.gif&_user=111111111&_pii=S0024379516300453&_rdoc=1&_issn=00243795&md5=3c4f254bf253da3be26034251e4142d5">class="imgLazyJSB inlineImage" height="19" width="162" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379516300453-si7.gif">class="mathContainer hidden">class="mathCode">$\mathit{LE}(G)={\sum }_{i=1}^n|{\mu }_i-\bar{d}|$, where class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300453&_mathId=si176.gif&_user=111111111&_pii=S0024379516300453&_rdoc=1&_issn=00243795&md5=0bb6a580f2eb9655056152e92fc158bf">class="imgLazyJSB inlineImage" height="20" width="52" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379516300453-si176.gif">class="mathContainer hidden">class="mathCode">$\bar{d}=\frac{2m}{n}$ is the average degree of G  . We obtain an upper bound for the Laplacian energy class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516300453&_mathId=si6.gif&_user=111111111&_pii=S0024379516300453&_rdoc=1&_issn=00243795&md5=89c39c6ff28ccc1ed20522cc8f5b3538" title="Click to view the MathML source">LE(G)class="mathContainer hidden">class="mathCode">$\mathit{LE}(G)$ of a graph G in terms of the number of vertices n, the number of edges m, the vertex covering number τ and the clique number ω of the graph.