Van der Holst and Pendavingh introduced a graph parameter σ, which
coincides with the more famous Colin de Verdi\`{e}re graph parameter μ for
small values. However, the definition of σ is much more
geometric/topological directly reflecting embeddability properties of the
graph. They proved μ(G)≤σ(G)+2 and conjectured μ(G)≤σ(G) for any graph G. We confirm this conjecture. As far as we know,
this is the first topological upper bound on μ(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 on 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
Hq of finite projective planes of order q satisfy μ(Hq)∈O(q3/2) and σ(Hq)≥q2.Comment: 28 pages, 4 figures. In v2 we slightly changed one of the core
definitions (previously "extended representation" now "semivalid
representation"). We also use it to introduce a new graph parameter, denoted
eta, which did not appear in v1. It allows us to establish an extended
version of the main result showing that mu(G) is at most eta(G) which is at
most sigma(G) for every graph