On the distance and distance Laplacian eigenvalues of graphs

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• 作者：Huiqiu Lin^a ; ^{huiqiulin@126.com" class="auth_mail" title="E-mail the corresponding author} ; Baoyindureng Wu^b ; Yingying Chen^c ; Jinlong Shu^c
• 关键词：05C50
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 March 2016
• 年：2016
• 卷：492
• 期：Complete
• 页码：128-135
• 全文大小：271 K

文摘

Let G=(V,E)$G=(V,E)$ be a simple graph with vertex set V(G)={v₁,v₂,…,v_n}$V(G)=\{v_1,v_2,\dots ,v_n\}$ and edge set i3" class="mathmlsrc">i3.gif&_user=111111111&_pii=S0024379515006771&_rdoc=1&_issn=00243795&md5=01f78a8d28698c58f131b1a6ff787278" title="Click to view the MathML source">E(G). Let D(G)$D(G)$ be the distance matrix of G. For a given nonnegative integer k, when n is sufficiently large with respect to k  , we show that λ_n−k(D)≤−1${\lambda }_{n-k}(D)\le -1$, thereby solving a problem proposed by Lin et al. (2014) [8]. The distance Laplacian spectral radius of a connected graph G is the spectral radius of the distance Laplacian matrix of G, defined as where Tr(G)$\mathit{Tr}(G)$ is the diagonal matrix of vertex transmissions of G. Aouchiche and Hansen (2014) [3] conjectured that m(λ₁(D^L))≤n−2$m({\lambda }_1(D^L))\le n-2$ when G≇K_n$G\ncong K_n$, and the equality holds if and only if either G≅K_1,n−1$G\cong K_{1,n-1}$ or lineImage" height="18" width="74" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379515006771-si11.gif">$G\cong K_{\frac{n}{2},\frac{n}{2}}$. In this paper, we confirm the conjecture.