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arXiv 2006.02841math.COmath.FA
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顶点数为偶数的轮图的距离矩阵的求逆公式

An inverse formula for the distance matrix of a wheel graph with even number of vertices

R. Balaji, R. B. Bapat, Shivani Goel

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摘要：

设$n \geq 4$为偶数，$W_n$为具有$n$顶点的轮图. $W_n$的任何两个不同顶点$i$和$j$之间的距离$d_{ij}$是连接$i$和$j$的最短路径的长度。令$D$为对角元素等于零且非对角元素等于$d_{ij}$的$n \times n$对称矩阵。本文找到了一个半正定矩阵$\widetilde{L}$，使得${\rm rank}(\widetilde{L})=n-1$，$\widetilde{L}$的所有行和均为零，以及一个秩为1的矩阵$ww^T$，使得\[D^{-1}=-\frac{1}{2}\widetilde{L} + \frac{4}{n-1}ww^T. \]，并证明了$D$和$\widetilde{L}$的特征值之间的交错性.

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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