ABSTRACT

The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose $i{j}^{th}$ entry (for $i\ne j$) is nonzero whenever $(i,j)$ is an edge in G and is zero otherwise. The sum of the minimum rank of a graph and its maximum nullity (similarly defined) is always the number of vertices in G. This article compares the minimum rank with the clique covering number of G and the Boolean rank of its adjacency matrix. It does the same analysis for bipartite graphs. Finally, we investigate the linear operators on the set of graphs on n vertices that preserve the minimum rank.

中文翻译：

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图的最小秩和最大空值及其线性保护器

摘要

简单图G的最小秩被定义为所有对称实矩阵上的最小秩，其$\mathrm{一世}{j}^{吨H}$进入（对于$\mathrm{一世}\ne j$) 是非零的$(\mathrm{一世},j)$是 G 中的一条边，否则为零。一个图的最小秩和它的最大空值（类似定义）的总和总是G中的顶点数。本文将最小秩与G的团覆盖数及其邻接矩阵的布尔秩进行比较。它对二分图进行相同的分析。最后，我们研究了保持最小秩的n个顶点上的图集上的线性算子。