A note on the sum of the largest signless Laplacian eigenvalues

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• 作者：Bruno Amaro¹ ; ^{bruno.amaro@ufms.br" class="auth_mail" title="E-mail the corresponding author}
• 关键词：signless Laplacian ; sum of eigenvalues ; upper bound
• 刊名：Electronic Notes in Discrete Mathematics
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：54
• 期：Complete
• 页码：175-180
• 全文大小：188 K

文摘

Let G be a graph with n   vertices and e(G)$e(G)$ edges. The signless Laplacian of G  , denoted by Q(G)$Q(G)$, is given by Q(G)=D(G)+A(G)$Q(G)=D(G)+A(G)$, where 33d4a774c21a5b0f82b2bc402aea926" title="Click to view the MathML source">D(G)$D(G)$ and A(G)$A(G)$ are the diagonal matrix of its vertex degree and A(G)$A(G)$ is the adjacency matrix. Let q₁(G),…,q_n(G)$q_1(G),\dots ,q_n(G)$ be the eigenvalues of Q(G)$Q(G)$ in non-increasing order and let $T_k(G)={\sum }_{i=1}^k{q}_i(G)$ be the sum of the k largest signless Laplacian eigenvalues of G  . In this paper, we obtain an upper bound to 33d60" title="Click to view the MathML source">T_k(H)$T_k(H)$, when H   is the P₃$P_3$-join   graph isomorphic to P₃[(n−k−1)K₁,K_k−1,K₂]$P_3[(n-k-1)K_1,K_{k-1},K_2]$ for d4541" title="Click to view the MathML source">3≤k≤n−2$3\le k\le n-2$. Also, we conjecture that T_k(G)$T_k(G)$ is bounded above by 33d60" title="Click to view the MathML source">T_k(H)$T_k(H)$ for any G with n vertices.