## The Colin de Verdière graph parameter (1997)

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Citations: | 16 - 1 self |

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@MISC{Holst97thecolin,

author = {Hein van der Holst and László Lovász and Alexander Schrijver},

title = {The Colin de Verdière graph parameter},

year = {1997}

}

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### Abstract

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.

### Citations

1139 |
Geometric Algorithms and Combinatorial Optimization
- Grötschel, Lovász, et al.
- 1981
(Show Context)
Citation Context ...Robertson, Seymour, and Thomas [26]), we know that χ(G) ≤ µ(G) +1 holds if µ(G) ≤ 4. An even weaker conjecture would be that ϑ(G) ≤ µ(G) +1. Here ϑ is the graph invariant introduced in [18] (cf. also =-=[10]-=-). Since ϑ is defined in terms of vector labellings and positive semidefinite matrices, it is quite close in spirit to µ (cf. sections 3.1, 3.2). The following results show that with the help of µ(G),... |

337 |
Three-Dimensional Geometry and Topology
- Thurston
- 1997
(Show Context)
Citation Context ... our method is that we consider labellings where adjacent nodes are labelled by circles intersecting at a given angle. In dimension 2, such representations were studied 7by Andre’ev [1] and Thurston =-=[31]-=-, generalizing Koebe’s theorem. (We use their proof method.) Sphere labellings give rise to a number of interesting geometric questions, which we don’t survey in this paper, but refer to [16]. Finally... |

334 |
On the Shannon Capacity of a Graph
- Lovasz
- 1979
(Show Context)
Citation Context ... any K6-minor (Robertson, Seymour, and Thomas [26]), we know that χ(G) ≤ µ(G) +1 holds if µ(G) ≤ 4. An even weaker conjecture would be that ϑ(G) ≤ µ(G) +1. Here ϑ is the graph invariant introduced in =-=[18]-=- (cf. also [10]). Since ϑ is defined in terms of vector labellings and positive semidefinite matrices, it is quite close in spirit to µ (cf. sections 3.1, 3.2). The following results show that with th... |

97 | Kontaktprobleme der konformen Abbildung - Koebe - 1936 |

47 |
Sachs’ linkless embedding conjecture
- Robertson, Seymor, et al.
- 1995
(Show Context)
Citation Context ...f and only if G is linklessly embeddable. Here (i), (ii), and (iii) are due to Colin de Verdière [7]. In (iv), direction =⇒ is due to Robertson, Seymour, and Thomas [25] (based on the hard theorem of =-=[27]-=- that the Petersen family (Figure 2 in Section 4.3) is the collection of forbidden minors for linkless embeddability), and direction ⇐= to Lovász and Schrijver [20]. In fact, in 1.4 each =⇒ follows fr... |

39 |
de Verdière, On a new graph invariant and a criterion for planarity
- Colin
- 1993
(Show Context)
Citation Context ...i,j =0whenever i = j or Mi,j = 0. Therefore, 2.3 is equivalent to 1.1(M3). 2.2 Monotonicity and components We start with proving the very important fact that µ(G) isminor-monotone (Colin de Verdière =-=[8]-=-). The proof is surprisingly nontrivial! Theorem 2.4 If H is a minor of G, then µ(H) ≤ µ(G). Proof. Let M be amatrix satisfying (M1)–(M3) for the graph H, with corank µ(H); we construct a matrix M ′ t... |

38 | de Verdiére. Multiplicites des valeurs propres et transformations étoiletriangle des graphes - Bacher, Colin - 1995 |

38 |
Hadwiger's conjecture for K6-free graphs, Combinatorica 13
- Robertson, Seymour, et al.
- 1993
(Show Context)
Citation Context ... denotes the chromatic number of G). This inequality was conjectured by Colin de Verdière [7]. Since Hadwiger’s conjecture holds for graphs not containing any K6-minor (Robertson, Seymour, and Thomas =-=[26]-=-), we know that χ(G) ≤ µ(G) +1 holds if µ(G) ≤ 4. An even weaker conjecture would be that ϑ(G) ≤ µ(G) +1. Here ϑ is the graph invariant introduced in [18] (cf. also [10]). Since ϑ is defined in terms ... |

34 |
de Verdière. Sur un nouvel invariant des graphes et un critère de planarité
- Colin
- 1990
(Show Context)
Citation Context ...nkless embeddability of G is characterized by the inequality µ(G) ≤ 4. In this paper we give an overview of results on µ(G) and of techniques to handle it. 21 Introduction In 1990, Colin de Verdière =-=[7]-=- (cf. [8]) introduced an interesting new parameter µ(G) for any undirected graph G. The parameter was motivated by the study of the maximum multiplicity of the second eigenvalue of certain Schrödinger... |

28 | A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs - Lovász, Schrijver - 1998 |

24 |
Orthogonal representations and connectivity of graphs. Linear Algebra and its
- Lovasz, Saks, et al.
- 1989
(Show Context)
Citation Context ...e graph. A Gram labelling with property (U2) is nondegenerate. This result links Colin de Verdière’s number with a considerable amount of work done on various geometric representations of graphs, cf. =-=[22, 23, 9, 18, 19]-=-. By Theorem 3.1, we have Theorem 3.3 For every graph G different from K2, ν(G) =n − µ(G) − 1. We add that if G has no edges, then ν(G) =ν(Kn) =n − 2. An optimal Gram labelling is to use 3 copies of e... |

23 | de Verdière, Sur la multiplicité delapremière valeur propre non nulle du laplacien - Colin - 1986 |

22 | The Theory of Matrices Second Edition with Applications - Lancaster, Tismenetsky - 1985 |

22 |
Automorphismen von polyedrischen graphen
- Mani
- 1971
(Show Context)
Citation Context ...ve equality here. Then ν(H) ≤ 3. Corollary 5.12 Let H be a 3-connected planar graph with an edge-transitive automorphism group, different from K4 and K2,2,2. Then ν(H) =3. Proof. By a theorem of Mani =-=[21]-=-, H can be represented as the skeleton of a convex polytope P in R3 so that the group of congruences preserving P acts edge-transitively. We can translate the origin to the center of gravity of P , an... |

17 |
Convex polyhedra in Lobachevsky spaces
- Andreev
- 1970
(Show Context)
Citation Context ...ne way to look at our method is that we consider labellings where adjacent nodes are labelled by circles intersecting at a given angle. In dimension 2, such representations were studied 7by Andre’ev =-=[1]-=- and Thurston [31], generalizing Koebe’s theorem. (We use their proof method.) Sphere labellings give rise to a number of interesting geometric questions, which we don’t survey in this paper, but refe... |

17 |
A short proof of the planarity characterization of Colin de Verdière
- Holst
- 1995
(Show Context)
Citation Context ...aracterization 1.4(iii) uses a result of Cheng [5] on the maximum multiplicity of the second eigenvalue of Schrödinger operators defined on the sphere. A short direct proof was given by van der Holst =-=[11]-=-, based on a lemma that has other applications and also has motivated other research (see Section 2.5). Kotlov, Lovász, and Vempala [16] studied graphs for which µ is close to the number of nodes n. T... |

16 |
Topological and Spectral Graph Characterizations
- Holst
- 1996
(Show Context)
Citation Context ...K4, (b) follows directly from Corollary 2.11. □ 2.5 The null space of M In this section we study the null space of a matrix M satisfying 1.1. The main result will be Theorem 2.16 due to van der Holst =-=[11, 12]-=- and its extensions, but some of the preliminary results leading up to it will also be useful. Forany vector x, let supp(x) denote the support of x (i.e., the set {i|xi = 0}). Furthermore, we denote ... |

16 | How to cage an egg - Schramm - 1992 |

15 |
Infinitesimally rigid polyhedra
- Whiteley
- 1984
(Show Context)
Citation Context ... structures. Recall the classical theorem of Cauchy: 5.2 No nontrivial stress can act along the edges of a convex 3-dimensional polytope. We need a generalization of Cauchy’s theorem, due to Whiteley =-=[32]-=-. Let P be aconvex polyhedron in R 3 , and let H be a(planar) graph embedded in the surface of P , with straight edges. A stress on H is called facial if there is a facet of P containing all edges wit... |

14 | On the invariance of Colin de Verdière’s graph parameter under clique sums - Holst, Lovász, et al. - 1995 |

14 | The Colin de Verdière number and sphere representations of a graph
- Kotlov, Lovász, et al.
- 1997
(Show Context)
Citation Context ... on the sphere. A short direct proof was given by van der Holst [11], based on a lemma that has other applications and also has motivated other research (see Section 2.5). Kotlov, Lovász, and Vempala =-=[16]-=- studied graphs for which µ is close to the number of nodes n. They characterized graphs with µ(G) ≥ n − 3. They also found that the value n − µ(G) isclosely related to the outerplanarity and planarit... |

10 |
On a minor-monotone graph invariant
- Holst, Laurent, et al.
- 1995
(Show Context)
Citation Context ...ive eigenvalue. One easily checks that M has the Strong Arnold Property 1.1(M3). □ We end this section with a lemma that gives very useful information about the components of an induced subgraph of G =-=[13]-=-. Lemma 2.8 Let G =(V,E) be aconnected graph and let M be amatrix satisfying 1.1. Let S ⊆ V and let C1,...,Cm be the components of G − S. Then there are three alternatives: (i) either there exists an ... |

10 |
A survey of linkless embeddings, in: Graph Structure Theory
- Robertson, Seymour, et al.
- 1993
(Show Context)
Citation Context ...nly if G is planar. (iv) µ(G) ≤ 4 if and only if G is linklessly embeddable. Here (i), (ii), and (iii) are due to Colin de Verdière [7]. In (iv), direction =⇒ is due to Robertson, Seymour, and Thomas =-=[25]-=- (based on the hard theorem of [27] that the Petersen family (Figure 2 in Section 4.3) is the collection of forbidden minors for linkless embeddability), and direction ⇐= to Lovász and Schrijver [20].... |

8 |
Sinajová, Embeddings of graphs in Euclidean spaces
- Reiterman, Rödl, et al.
- 1989
(Show Context)
Citation Context ...e graph. A Gram labelling with property (U2) is nondegenerate. This result links Colin de Verdière’s number with a considerable amount of work done on various geometric representations of graphs, cf. =-=[22, 23, 9, 18, 19]-=-. By Theorem 3.1, we have Theorem 3.3 For every graph G different from K2, ν(G) =n − µ(G) − 1. We add that if G has no edges, then ν(G) =ν(Kn) =n − 2. An optimal Gram labelling is to use 3 copies of e... |

8 |
Minor-monotone graph invariants
- Schrijver
- 1997
(Show Context)
Citation Context ...duced and studied by van der Holst, Laurent and Schrijver [13]. For the interesting properties of these and other related graph invariants, we refer to [13] and to the forthcoming survey by Schrijver =-=[30]-=-. Avector x in the null space of M that violates the conclusion of Theorem 2.16 must have rather special properties: Theorem 2.17 Let G be agraphand let M be amatrix satisfying 1.1. Let x ∈ ker(M) be ... |

7 |
On spatial representations of graphs, in
- Böhme
- 1990
(Show Context)
Citation Context ... circuit on T .AsHis flatly embedded, for each edge t1t2 of H|T there is an open disk (“panel”) with boundary the triangle t1t2v3, insuch a way that the panels are pairwise disjoint (by Böhme’s lemma =-=[4]-=- (cf. [28, 27]). Since the union of H|({v3} ∪T ) with the panels is contractible, there is a curve C from v1 to v2 not intersecting any panel. This curve has the required properties, showing 4.13. We ... |

7 |
Constructive Results in Graph Minors: Linkless Embeddings
- Saran
- 1989
(Show Context)
Citation Context ...on T .AsHis flatly embedded, for each edge t1t2 of H|T there is an open disk (“panel”) with boundary the triangle t1t2v3, insuch a way that the panels are pairwise disjoint (by Böhme’s lemma [4] (cf. =-=[28, 27]-=-). Since the union of H|({v3} ∪T ) with the panels is contractible, there is a curve C from v1 to v2 not intersecting any panel. This curve has the required properties, showing 4.13. We now define φ o... |

6 |
On a common generalization of Borsuk’s and Radon’s theorem, Acta
- Bajmoczy, Barany
- 1979
(Show Context)
Citation Context ...s they intersect transversally. (In this paper, faces are relatively open.) Forany convex polytope P in R d , let ∂P denote its boundary. The following theorem extends a result of Bajmóczy and Bárány =-=[3]-=-. (The difference is that their theorem concludes that φ(F ) ∩ φ(F ′ )isnonempty. Their proof uses Borsuk’s theorem. We give an independent proof.) Theorem 4.6 Let P be afull-dimensional convex polyto... |

6 | Siňajová. Geometrical embeddings of graphs - Reiterman, Rödl, et al. - 1989 |

4 |
Eigenfunctions and nodal sets, Commentarii Mathematici Helvetici 51
- Cheng
- 1976
(Show Context)
Citation Context ...rove (iv), with the help of a certain Borsuk-type theorem on the existence of ‘antipodal links’. 6The proof by Colin de Verdière [7] of the planarity characterization 1.4(iii) uses a result of Cheng =-=[5]-=- on the maximum multiplicity of the second eigenvalue of Schrödinger operators defined on the sphere. A short direct proof was given by van der Holst [11], based on a lemma that has other applications... |

4 | Hadwiger's conjecture for K 6 -free graphs, Combinatorica 13 - Robertson, Seymour, et al. - 1993 |

4 |
Graph minors. XX. Wagner’s conjecture, preprint
- Robertson, Seymour
- 1988
(Show Context)
Citation Context ...3 is surprisingly nontrivial, and the Strong Arnold Property plays a crucial role. The minor-monotonicity of µ(G) isespecially interesting in the light of the Robertson-Seymour theory of graph minors =-=[24]-=-, which has as principal result that if C is a collection of graphs so that no graph in C is a minor of another graph in C, then C is finite. This can be equivalently formulated as follows. For any gr... |

2 |
Dot product representations of graphs (manuscript
- Fiduccia, Scheinerman, et al.
(Show Context)
Citation Context ...e graph. A Gram labelling with property (U2) is nondegenerate. This result links Colin de Verdière’s number with a considerable amount of work done on various geometric representations of graphs, cf. =-=[22, 23, 9, 18, 19]-=-. By Theorem 3.1, we have Theorem 3.3 For every graph G different from K2, ν(G) =n − µ(G) − 1. We add that if G has no edges, then ν(G) =ν(Kn) =n − 2. An optimal Gram labelling is to use 3 copies of e... |