论文标题
齿轮图距离矩阵的逆公式
Inverse formula for distance matrices of gear graphs
论文作者
论文摘要
在\ cite {jak}中研究了一些恒星的距离矩阵。这些图是树，是恒星，车轮图，辅助图和齿轮图。除了上面的恒星列表中的齿轮图（如图）外，文献中还有精确的公式可用于计算其距离矩阵的倒数/摩尔 - 柔性。这些公式说明，如果$ d $是$ g $的距离矩阵，则$ d^\ dagger = - \ frac {1} {2} {2} l+uu' $，其中$ l $是一个类似laplacian的矩阵，是一个正式矩阵，是一个正半数的半际半决赛，所有行总成等于零。矩阵$ l $和向量$ u $仅取决于$ g $中的顶点和数量，因此可以直接从$ g $写入。最早获得的公式是用于格雷厄姆和洛瓦斯\ cite {gl}的树木的距离矩阵。在本文中，我们获得了这种优雅的公式，以计算齿轮图的距离矩阵的摩尔 - 柔性。
Distance matrices of some star like graphs are investigated in \cite{JAK}. These graphs are trees which are stars, wheel graphs, helm graphs and gear graphs. Except for gear graphs in the above list of star like graphs, there are precise formulas available in the literature to compute the inverse/Moore-Penrose inverse of their distance matrices. These formulas tell that if $D$ is the distance matrix of $G$, then $D^\dagger = -\frac{1}{2}L+uu'$, where $L$ is a Laplacian-like matrix which is positive semidefinite and all row sums equal to zero. The matrix $L$ and the vector $u$ depend only on the degree and number of vertices in $G$ and hence, can be written directly from $G$. The earliest formula obtained is for distance matrices of trees in Graham and Lovász \cite{GL}. In this paper, we obtain an elegant formula of this kind to compute the Moore-Penrose inverse of the distance matrix of a gear graph.
