论文标题
一个均匀数量顶点的车轮图的距离矩阵的反向公式
An inverse formula for the distance matrix of a wheel graph with even number of vertices
论文作者
论文摘要
令$ n \ geq 4 $为一个整数，$ w_n $是带有$ n $顶点的车轮图。任何两个不同的顶点$ i $和$ w_n $ $ j $之间的距离$ d_ {ij} $是连接$ i $和$ j $的最短路径的长度。令$ d $为$ n \ times n $对称矩阵，其对角线条目等于零和偏外条目等于$ d_ {ij} $。在本文中，我们发现一个积极的半finite矩阵$ \ widetilde {l} $，使得$ {\ rm strank}（\ widetilde {l}）= n-1 $，所有行总和的所有行总和$ \ widetilde {l} $等于零，等于零和排名n Matrix $ ww^t $ \ [d^{ - 1} = - \ frac {1} {2} \ widetilde {l} + \ frac {4} {n-1} {n-1} ww^t。 \ \]也证明了$ d $和$ \ widetilde {l} $之间的交织属性。
Let $n \geq 4$ be an even integer and $W_n$ be the wheel graph with $n$ vertices. The distance $d_{ij}$ between any two distinct vertices $i$ and $j$ of $W_n$ is the length of the shortest path connecting $i$ and $j$. Let $D$ be the $n \times n$ symmetric matrix with diagonal entries equal to zero and off-diagonal entries equal to $d_{ij}$. In this paper, we find a positive semidefinite matrix $\widetilde{L}$ such that ${\rm rank}(\widetilde{L})=n-1$, all row sums of $\widetilde{L}$ equal to zero and a rank one matrix $ww^T$ such that \[D^{-1}=-\frac{1}{2}\widetilde{L} + \frac{4}{n-1}ww^T. \] An interlacing property between the eigenvalues of $D$ and $\widetilde{L}$ is also proved.
