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 μ₁,μ₂,…,μ_n−1,μ_n=0${\mu }_1,{\mu }_2,\dots ,{\mu }_{n-1},{\mu }_n=0$, let $S_k(G)={\sum }_{i=1}^k{\mu }_i$, be the sum of k largest Laplacian eigenvalues of G  . Brouwer conjectured that $S_k(G)\le m+\left(\begin{array}{c}k+1\\ 2\end{array}\right)$, for all k=1,2,…,n$k=1,2,\dots ,n$. We obtain upper bounds for S_k(G)$S_k(G)$ 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 LE(G)$\mathit{LE}(G)$ of a graph G   is defined as $\mathit{LE}(G)={\sum }_{i=1}^n|{\mu }_i-\bar{d}|$, where $\bar{d}=\frac{2m}{n}$ is the average degree of G  . We obtain an upper bound for the Laplacian energy LE(G)$\mathit{LE}(G)$ 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.