论文标题
摩尔 - 螺旋属的匹配级别矩阵矩阵
The Moore-Penrose Inverse of the Distance Matrix of a Helm Graph
论文作者
论文摘要
在本文中，我们为真实的对称矩阵提供了必要和足够的条件，尤其是对于Helm Graph $ h_n $的距离矩阵$ d（h_n）$，以使其摩尔 - 柔性倒置为对称laplacian类似于对称的laplacian类矩阵和一个排名一个矩阵。 As a consequence, we present a short proof of the inverse formula, given by Goel (Linear Algebra Appl. 621:86--104, 2021), for $D(H_n)$ when $n$ is even.此外，我们得出了摩尔 - 芬罗的一个公式，该公式是单数$ d（h_n）$的反向，该公式类似于$ d（h_n）^{ - 1} $的公式。确切地说，如果$ n $很奇怪，我们会发现一个对称的阳性半决赛laplacian类矩阵$ l $ of订单$ 2n-1 $和vector $ \ mathbf {w} \ in \ mathbb {r} \ begin {eqnarray*} d（h_n）\ sSymbol {2} = - \ frac {1} {2} l + \ frac {4} {3（n -1）} \ Mathbf {w} \ Mathbf {w^{w^{\ prime}}， \ end {eqnarray*} $ l $的排名为$ 2N-3 $。我们还调查了$ d（H_N）$的惯性。
In this paper, we give necessary and sufficient conditions for a real symmetric matrix, and in particular, for the distance matrix $D(H_n)$ of a helm graph $H_n$ to have their Moore-Penrose inverses as the sum of a symmetric Laplacian-like matrix and a rank one matrix. As a consequence, we present a short proof of the inverse formula, given by Goel (Linear Algebra Appl. 621:86--104, 2021), for $D(H_n)$ when $n$ is even. Further, we derive a formula for the Moore-Penrose inverse of singular $D(H_n)$ that is analogous to the formula for $D(H_n)^{-1}$. Precisely, if $n$ is odd, we find a symmetric positive semidefinite Laplacian-like matrix $L$ of order $2n-1$ and a vector $\mathbf{w}\in \mathbb{R}^{2n-1}$ such that \begin{eqnarray*} D(H_n)\ssymbol{2} = -\frac{1}{2}L + \frac{4}{3(n-1)}\mathbf{w}\mathbf{w^{\prime}}, \end{eqnarray*} where the rank of $L$ is $2n-3$. We also investigate the inertia of $D(H_n)$.
