@article{10335,
  abstract     = {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 Hq of finite projective planes of order q satisfy μ(Hq) ∈ O(q3/2) and σ(Hq) ≥ q2.},
  author       = {Kaluza, Vojtech and Tancer, Martin},
  issn         = {0209-9683},
  journal      = {Combinatorica},
  pages        = {1317--1345},
  publisher    = {Springer Nature},
  title        = {{Even maps, the Colin de Verdière number and representations of graphs}},
  doi          = {10.1007/s00493-021-4443-7},
  volume       = {42},
  year         = {2022},
}