论文标题
关于偏心型矩阵
On Eccentricity Matrices of Wheel Graphs
论文作者
论文摘要
简单连接的图形$ g $的偏心矩阵$ e（g）$是从距离矩阵$ d（g）$ g $获得的，通过保留每个行和列的最大距离，并将其余条目定义为零。本文重点介绍了带有$ n $ vertices的车轮图$ w_n $的偏心矩阵$ e（w_n）$。通过为$ e（W_n）$的决定因素建立公式，我们表明$ e（w_n）$在且仅当$ n \ not \ equiv 1 \ mod3 $时都是可逆的。我们通过在\ Mathbb {r}^n $中找到vector $ \ mathbf {w} \ in \ mathbb {r}^n $和一个$ n \ times n $对称的laplacian laplacian tik $ \ wideTilde {l}等级$ n-1 $ n of Cark $ n-1 $ n of Cark $ n-n-1 $ n of Cark $ n-1 e（w_n）^{ - 1} = - \ frac {1} {2} \ widetilde {l} + \ frac {6} {n-1} \ Mathbf {w} \ Mathbf {w^{\ prime}}。 \ end {eqnarray*}进一步，我们证明了摩尔 - 柔性倒数$ e（w_n）$的类似结果。我们还确定$ e（W_n）$的惯性。
The eccentricity matrix $E(G)$ of a simple connected graph $G$ is obtained from the distance matrix $D(G)$ of $G$ by retaining the largest distance in each row and column, and by defining the remaining entries to be zero. This paper focuses on the eccentricity matrix $E(W_n)$ of the wheel graph $W_n$ with $n$ vertices. By establishing a formula for the determinant of $E(W_n)$, we show that $E(W_n)$ is invertible if and only if $n \not\equiv 1\Mod3$. We derive a formula for the inverse of $E(W_n)$ by finding a vector $\mathbf{w}\in \mathbb{R}^n$ and an $n \times n$ symmetric Laplacian-like matrix $\widetilde{L}$ of rank $n-1$ such that \begin{eqnarray*} E(W_n)^{-1} = -\frac{1}{2}\widetilde{L} + \frac{6}{n-1}\mathbf{w}\mathbf{w^{\prime}}. \end{eqnarray*} Further, we prove an analogous result for the Moore-Penrose inverse of $E(W_n)$ for the singular case. We also determine the inertia of $E(W_n)$.
