Characterization of extremal graphs from distance signless Laplacian eigenvalues

详细信息    查看全文

• 作者：Huiqiu Lin^a ; ^{huiqiulin@126.com" class="auth_mail" title="E-mail the corresponding author} ; Kinkar Ch. Das^b ; ^{kinkardas2003@googlemail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：05C50
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 July 2016
• 年：2016
• 卷：500
• 期：Complete
• 页码：77-87
• 全文大小：290 K

文摘

Let lineImage" height="16" width="81" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379516300039-si1.gif">$G=(V,E)$ be a connected graph with vertex set lineImage" height="17" width="172" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379516300039-si17.gif">$V(G)=\{v_1,v_2,\dots ,v_n\}$ and edge set i3" class="mathmlsrc">i3.gif&_user=111111111&_pii=S0024379516300039&_rdoc=1&_issn=00243795&md5=09c148656da51061c0553faab9af3850" title="Click to view the MathML source">E(G). The transmission Tr(v_i)$\mathit{Tr}(v_i)$ of vertex v_i$v_i$ is defined to be the sum of distances from v_i$v_i$ to all other vertices. Let Tr(G)$\mathit{Tr}(G)$ be the n×n$n\times n$ diagonal matrix with its lineImage" height="16" width="33" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0024379516300039-si8.gif">$(i,i)$-entry equal to Tr_G(v_i)${\mathit{Tr}}_G(v_i)$. The distance signless Laplacian is defined as D^Q(G)=Tr(G)+D(G)$D^Q(G)=\mathit{Tr}(G)+D(G)$, where D(G)$D(G)$ is the distance matrix of G  . Let ∂₁(G)≥∂₂(G)≥⋯≥∂_n(G)${\partial }_1(G)\ge {\partial }_2(G)\ge \cdots \ge {\partial }_n(G)$ denote the eigenvalues of distance signless Laplacian matrix of G  . In this paper, we first characterize all graphs with ∂_n(G)=n−2${\partial }_n(G)=n-2$. Secondly, we characterize all graphs with ∂₂(G)∈[n−2,n]${\partial }_2(G)\in [n-2,n]$ when n≥11$n\ge 11$. Furthermore, we give the lower bound on ∂₂(G)${\partial }_2(G)$ with independence number α and the extremal graph is also characterized.