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For any undirected graph G, let μ(G) be the graph parameter introduced by Colin de Verdiere. In this paper we show that μ(G) ≤ 4 if and only if G is linklessly embeddable (in R). This forms a spectral characterization of linklessly embeddable graphs, and was conjectured by Robertson, Seymour, and Thomas. A key ingredient is a Borsuk-type theorem on the existence of a pair of antipodal linked (k− 1)spheres in certain mappings φ : S → R. This result might be of interest in its own right. We also derive that λ(G) ≤ 4 for each linklessly embeddable graph G = (V,E), where λ(G) is the graph paramer introduced by van der Holst, Laurent, and Schrijver. (It is the largest dimension of any subspace L of R such that for each nonzero x ∈ L, the positive support of x induces a nonempty connected subgraph of G.)

A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs