论文标题
关于图的距离矩阵的等级
On the rank of the distance matrix of graphs
论文作者
论文摘要
令$ g $为$ v（g）= \ {v_1，\ ldots，v_n \} $的连接图。 $（i，j）$ - $ g $的距离矩阵$ d（g）的条目是$ v_i $和$ v_j $之间的距离。在本文中，使用著名的Ramsey定理，我们证明，对于每个整数$ k \ ge 2 $，都有有限数量的图表，其距离矩阵具有等级$ k $。我们展示的图表列表，距离为$ 2 $和$ 3 $的距离矩阵。此外，我们研究属于图形家族的图形距离矩阵的等级，其直径最多是两个图形，即琐碎的完美图。我们证明，对于每个$η\ ge 1 $，都有一个毫无疑问的图形，带有无效$η$。我们还表明，对于阈值图，这是琐碎的完美图家族的子家族，无效的是一个。
Let $G$ be a connected graph with $V(G)=\{v_1,\ldots,v_n\}$. The $(i,j)$-entry of the distance matrix $D(G)$ of $G$ is the distance between $v_i$ and $v_j$. In this article, using the well-known Ramsey's theorem, we prove that for each integer $k\ge 2$, there is a finite amount of graphs whose distance matrices have rank $k$. We exhibit the list of graphs with distance matrices of rank $2$ and $3$. Besides, we study the rank of the distance matrices of graphs belonging to a family of graphs with their diameters at most two, the trivially perfect graphs. We show that for each $η\ge 1$ there exists a trivially perfect graph with nullity $η$. We also show that for threshold graphs, which are a subfamily of the family of trivially perfect graphs, the nullity is bounded by one.
