For any undirected graph G, let µ(G) be the graph parameter introduced by Colin de Verdière. In this paper we show that (G) 4 if and only if G is linklessly embeddable (in R 3 ). 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 \Gamma 1)- spheres in certain mappings OE : S 2k ! R 2k\Gamma1 . This result might be of interest in its own right. We also derive that (G) 4 for each linklessly embeddable graph G = (V; E), where (G) is the graph paramer introduced by van der Holst, Laurent, and Schrijver. (It is the largest dimension of any subspace L of R V such that for each nonzero x 2 L, the positive support of x induces a nonempty connected subgraph of G.)

this paper, we are only concerned about the last question, which can be answered using semidefinite programming. For a survey of other aspects of such geometric representations, see [64].

Colin de Verdi`ere introduced an interesting linear algebraic invariant (G) of graphs. He proved that (G) 2 if and only if G is outerplanar, and (G) 3 if and only if G is planar. We prove that if the complement of a graph G on n nodes is outerplanar, then (G) n \Gamma 4, and if it is planar, then (G) n \Gamma 5. We give a full characterization of maximal planar graphs whose complements G have (G) = n \Gamma 5. In the opposite direction we show that if G does not have "twin" nodes, then (G) n \Gamma 3 implies that the complement of G is outerplanar, and (G) n \Gamma 4 implies that the complement of G is planar. Our main tools are a geometric formulation of the invariant, and constructing representations of graphs by spheres, related to the classical result of Koebe about representing planar graphs by touching disks. In particular we show that such sphere representations characterize outerplanar and planar graphs.

In 1990, Y. Colin de Verdière introduced a new graph parameter (G), based on spectral properties of matrices associated with G. He showed that (G) is monotone under taking minors and that planarity of G is characterized by the inequality (G) 3. Recently Lovasz and Schrijver showed that linkless embeddability of G is characterized by the inequality (G) 4. In this paper we give an overview of results on (G) and of techniques to handle it.

. Let G = (V; E) be a 3-connected planar graph, with V = f1; : : : ; ng. Let M = (m i;j ) be a symmetric n \Theta n matrix with exactly one negative eigenvalue (of multiplicity 1), such that for i; j with i 6= j, if i and j are adjacent then m i;j ! 0 and if i and j are nonadjacent then m i;j = 0, and such that M has rank n \Gamma 3. Then the null space ker M of M gives an embedding of G in S 2 as follows: Let a; b; c be a basis of ker M , and for i 2 V let OE(i) := (a i ; b i ; c i ) T ; then OE(i) 6= 0, and /(i) := OE(i)=kOE(i)k embeds V in S 2 such that connecting, for any two adjacent vertices i; j, the points /(i) and /(j) by a shortest geodesic on S 2 , gives a proper embedding of G in S 2 . This applies to the matrices associated with the parameter (G) introduced by Y. Colin de Verdi`ere. 1 Introduction and results Let G = (V; E) be an undirected graph, with vertex set f1; : : : ; ng. Let M(G) be the set of symmetric n \Theta n matrices M = (m i;j ) satisfying (i) M...

How deep is the dividing line between discrete and continuous mathematics? Basic structures and methods of both sides of our science are quite different. But on a deeper level, there is a more solid connection than meets the eye. 1

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In this lecture, I will introduce the Colin de Verdière number of a graph, and sketch the proof that it is three for planar graphs. Along the way, I will recall two important facts about planar graphs: 1. Three-connected planar graphs are the skeletons of three-dimensional convex polytopes. 2. Planar graphs are the graphs that do not have K5 or K3,3 minors.

Using multiplicities of eigenvalues of elliptic self-adjoint differential operators on graphs and transversality, we construct some new invariants of graphs which are related to tree-width.

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