Van der Holst and Pendavingh introduced a graph parameter σ, which coincides with the more famous Colin de Verdière graph parameter μ for small values. However, the definition of a is much more geometric/topological directly reflecting embeddability properties of the graph. They proved μ(G) ≤ σ(G) + 2 and conjectured σ(G) ≤ σ(G) for any graph G. We confirm this conjecture. As far as we know, this is the first topological upper bound on σ(G) which is, in general, tight.

Equality between μ and σ does not hold in general as van der Holst and Pendavingh showed that there is a graph G with μ(G) ≤ 18 and σ(G) ≥ 20. We show that the gap appears at much smaller values, namely, we exhibit a graph H for which μ(H) ≥ 7 and σ(H) ≥ 8. We also prove that, in general, the gap can be large: The incidence graphs H_q of finite projective planes of order q satisfy μ(H_q) ∈ O(q^3/2) and σ(H_q) ≥ q².

Van der Holst 和 Pendavingh 引入了一个图参数σ，这与更著名的 Colin de Verdière 图参数μ的小值相吻合。然而，a的定义更多的是几何/拓扑，直接反映了图的可嵌入性。他们证明了μ ( G ) ≤ σ ( G ) + 2 并且猜想了任意图G的σ ( G ) ≤ σ ( G ) 。我们证实了这个猜想。据我们所知，这是σ ( G ) 上的第一个拓扑上界，通常是紧的。

μ和σ之间的等式通常不成立，因为 van der Holst 和 Pendavingh 表明存在图G，其中μ ( G ) ≤ 18 且σ ( G ) ≥ 20。我们表明差距出现在更小的值上，即，我们展示了一个图H，其中μ ( H ) ≥ 7 和σ ( H ) ≥ 8。我们还证明，一般情况下，间隙可以很大：q阶有限投影平面的入射图H _q满足μ ( H _q ) ∈ O( q ^3/2 ) 和σ ( H _q ) ≥ q ²。