The Colin de Verdière number of a graph

In 1990, Y. Colin de Verdière introduced a new graph parameter μ(G), based on spectral properties of matrices associated with G. He showed that μ(G) is monotone under taking minors and that planarity of G is characterized by the inequality μ(G) ≤ 3. Small values of this parameter correspond to interesting graph properties, like μ ≤ 3 if and only if the graph is planar, and μ ≤ 4 if and only if the graph is linklessly embedable in 3-space. In these notes, which are an extended version of the survey paper [15], we give an overview of results on μ(G) and of techniques to handle it.