图的Gram维数
The Gram dimension of a graph
摘要：
图的格兰姆维数$\gd(G)$是最小整数$k \ge 1$，使得对于图的节点的单位向量的每个分配，存在位于$\oR^k$中的单位向量的另一个分配，在图的边缘上具有相同的内积。满足$\gd(G) \le k$的图类对于固定的$k$是子闭的，因此它可以用一个有限的禁止子表来刻画.对于$k\le 3$，唯一被禁止的子进程是$K_{k+1}$。我们证明了一个图的Gram维数最多为4当且仅当它没有$K_5$和$K_{2,2,2}$作为子图。我们还展示了一些密切的联系$d$的概念-实现的图形。特别地，我们的结果包含了Belk和Connelly \cite{Belk,BC}的3-可实现图的特征.
英文摘要：
The Gram dimension $\gd(G)$ of a graph is the smallest integer $k \ge 1$ such that, for every assignment of unit vectors to the nodes of the graph, there exists another assignment of unit vectors lying in $\oR^k$, having the same inner products on the edges of the graph. The class of graphs satisfying $\gd(G) \le k$ is minor closed for fixed $k$, so it can characterized by a finite list of forbidden minors. For $k\le 3$, the only forbidden minor is $K_{k+1}$. We show that a graph has Gram dimension at most 4 if and only if it does not have $K_5$ and $K_{2,2,2}$ as minors. We also show some close connections to the notion of $d$-realizability of graphs. In particular, our result implies the characterization of 3-realizable graphs of Belk and Connelly \cite{Belk,BC}.
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