论文标题
在带有奇数顶点的车轮图的距离矩阵上
On distance matrices of wheel graphs with odd number of vertices
论文作者
论文摘要
令$ w_n $表示带有$ n $ vertices的车轮图。如果$ i $和$ j $是$ w_n $的任何两个顶点，则定义\ [d_ {ij}：= \ begin {cases} 0＆\ mbox {if} 〜i = j \\ 1＆\ mbox {if} 〜i〜i〜 i〜 \ mbox {and} 〜j〜j〜j〜j〜j〜 \ \ mbox {mbox} \ end {cases} \]让$ d $为$ n \ times n $矩阵，带有$（i，j）^{\ rm th} $输入等于$ d_ {ij {ij} $。矩阵$ d $称为$ w_n $的距离矩阵。假设$ n \ geq 5 $是一个奇数。在本文中，我们推断出一种公式来计算摩尔 - 芬罗的倒数$ D $。更准确地说，我们获得了一个$ n \ times n $矩阵$ \ widetilde {l} $和一个排名一个矩阵$ ww'u $，这样就可以\ [d^\ dagger = - \ frac {1} {2} {2} {2} {2} \ wideteLde {l} $ \ widetilde {l} $是正半芬矿，$ {\ rm strank}（\ widetilde {l}）= n-2 $，所有行总和等于零。
Let $W_n$ denote the wheel graph having $n$-vertices. If $i$ and $j$ are any two vertices of $W_n$, define \[d_{ij}:= \begin{cases} 0 & \mbox{if}~i=j \\ 1 & \mbox{if}~i~ \mbox{and} ~j~ \mbox{are adjacent} \\ 2 & \mbox{else}. \end{cases}\] Let $D$ be the $n \times n$ matrix with $(i,j)^{\rm th}$ entry equal to $d_{ij}$. The matrix $D$ is called the distance matrix of $W_n$. Suppose $n \geq 5$ is an odd integer. In this paper, we deduce a formula to compute the Moore-Penrose inverse of $D$. More precisely, we obtain an $n\times n$ matrix $\widetilde{L}$ and a rank one matrix $ww'$ such that \[D^\dagger = -\frac{1}{2} \widetilde{L}+\frac{4}{n-1}ww'.\] Here, $\widetilde{L}$ is positive semidefinite, ${\rm rank}(\widetilde{L})=n-2$ and all row sums are equal to zero.
