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Van der Holst and Pendavingh introduced a graph parameter $\sigma$, whichcoincides with the more famous Colin de Verdi\`{e}re graph parameter $\mu$ forsmall values. However, the definition of $\sigma$ is much moregeometric/topological directly reflecting embeddability properties of thegraph. They proved $\mu(G) \leq \sigma(G) + 2$ and conjectured $\mu(G) \leq\sigma(G)$ for any graph $G$. We confirm this conjecture. As far as we know,this is the first topological upper bound on $\mu(G)$ which is, in general,tight.  Equality between $\mu$ and $\sigma$ does not hold in general as van der Holstand Pendavingh showed that there is a graph $G$ with $\mu(G) \leq 18$ and$\sigma(G)\geq 20$. We show that the gap appears on much smaller values,namely, we exhibit a graph $H$ for which $\mu(H)\leq 7$ and $\sigma(H)\geq 8$.We also prove that, in general, the gap can be large: The incidence graphs$H_q$ of finite projective planes of order $q$ satisfy $\mu(H_q) \inO(q^{3/2})$ and $\sigma(H_q) \geq q^2$.

Even maps, the Colin de~Verdi\`ere number and representations of graphs