The product distance matrix of a tree with matrix weights on its arcs

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• 作者：Hui Zhou^a ; ^{zhouhpku13@pku.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Qi Ding^b ; ^c ; ^{dingq2011@lzu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; ^{dingqi0@yonyou.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15A15 ; 05C05 ; 16D10
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：15 June 2016
• 年：2016
• 卷：499
• 期：Complete
• 页码：90-98
• 全文大小：278 K

文摘

Let T   be a tree with vertex set [n]={1,2,…,n}$[n]=\{1,2,\dots ,n\}$. For each i∈[n]$i\in [n]$, let m_i$m_i$ be a positive integer. An ordered pair of two adjacent vertices is called an arc. Each arc (i,j)$(i,j)$ of T   has a weight W_i,j$W_{i,j}$ which is an 13c1a503a152b" title="Click to view the MathML source">m_i×m_j$m_i\times m_j$ matrix. For two vertices i,j∈[n]$i,j\in [n]$, let the unique directed path from i to j   be P_i,j=x₀,x₁,…,x_d$P_{i,j}=x_0,x_1,\dots ,x_d$ where d⩾1$d⩾1$, x₀=i$x_0=i$ and x_d=j$x_d=j$. Define the product distance from i to j   to be the 13c1a503a152b" title="Click to view the MathML source">m_i×m_j$m_i\times m_j$ matrix M_i,j=W_x0,x1W_x1,x2⋯W_xd−1,xd$M_{i,j}=W_{x_0,x_1}{W}_{x_1,x_2}\cdots W_{x_{d-1},x_d}$. Let $N={\sum }_{i=1}^n{m}_i$. The N×N$N\times N$ product distance matrix D of T   is a partitioned matrix whose (i,j)$(i,j)$-block is the matrix M_i,j$M_{i,j}$. We give a formula for det⁡(D)$\det (\mathbf{D})$. When det⁡(D)≠0$\det (\mathbf{D})\ne 0$, the inverse of D is also obtained. These generalize known results for the product distance matrix when either the weights are real numbers, or m₁=m₂=⋯=m_n=s$m_1=m_2=\cdots =m_n=s$ and the weights 13ce5f31e5f94a97446e6cae4fd9" title="Click to view the MathML source">W_i,j=W_j,i=W_e$W_{i,j}=W_{j,i}=W_e$ for each edge e={i,j}∈E(T)$e=\{i,j\}\in E(T)$.