## Steinitz representations of polyhedra and the Colin de Verdière number (2000)

Venue: | J. COMB. THEORY, SER. B |

Citations: | 10 - 1 self |

### BibTeX

@ARTICLE{Lovász00steinitzrepresentations,

author = {László Lovász},

title = {Steinitz representations of polyhedra and the Colin de Verdière number},

journal = {J. COMB. THEORY, SER. B},

year = {2000},

volume = {82},

pages = {223--236}

}

### OpenURL

### Abstract

We show that the Steinitz representations of 3-connected planar graphs are correspond, in a well described way, to Colin de Verdière matrices of such graphs.

### Citations

337 |
Three-Dimensional Geometry and Topology
- Thurston
- 1997
(Show Context)
Citation Context ...2 using such a Steinitz representation, have any interesting special properties? Of particular interest is the Koebe--Andre'ev representation, in which ever edge touches the unit sphere (see Thurston =-=[10]-=-). In this case, the definition of the matrix M is simpler: -M ij is the ratio between the length of the edge of the polytope and the corresponding edge of the polar. Let us note that the Koebe--Andre... |

47 |
Sachs’ linkless embedding conjecture
- Robertson, Seymor, et al.
- 1995
(Show Context)
Citation Context ...d be quite interesting, because no e#cient way is known to construct a linkless embedding for a graph, even if we know (from, say, the excluded minor characterization of Robertson, Seymour and Thomas =-=[9]-=-) that G is linklessly embedable. Perhaps the methods of this paper can be extended to the four dimensional space and will help understand the structure of nullspace embeddings of linklessly embedable... |

39 | Realization Spaces of Polytopes
- Richter-Gebert
(Show Context)
Citation Context ...y one negative eigenvalue. Steinitz proved that any two 3-dimensional polytopes with isomorphic skeletons can be transformed into each other continuously through polytopes with the same skeleton (see =-=[8]-=-). Each vertex moves continuously, and hence so does their center of gravity. So we can translate each intermediate polytope so that the center of gravity of the vertices stays 0. Then the polar of ea... |

34 |
de Verdière. Sur un nouvel invariant des graphes et un critère de planarité
- Colin
- 1990
(Show Context)
Citation Context ...Abstract We show that the Steinitz representations of 3-connected planar graphs are correspond, in a well described way, to Colin de Verdiere matrices of such graphs. 1 Introduction Colin de Verdiere =-=[1]-=- introduced a spectral invariant (G) of a graph G. Roughly speaking, (G) is the multiplicity of the second largest eigenvalue of the adjacency matrix of G, maximized by weighting the edges and nodes (... |

28 | A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs
- Lovász, Schrijver
- 1998
(Show Context)
Citation Context ... related to the Colin de Verdiere number in another way too: it plays a key role in the study of graphs with (G) # |V (G)| - 4 (Kotlov, Lovasz and Vempala [5]). 4. Lex Schrijver and the author proved =-=[6]-=- that a graph has (G) # 4 if and only if it is linklessly embedable in R 3 . Let G be a linklessly embedable graph and M , a Colin de Verdiere matrix for M . We can carry out the construction leading ... |

14 | The Colin de Verdière number and sphere representations of a graph
- Kotlov, Lovász, et al.
- 1997
(Show Context)
Citation Context ...the Koebe--Andre'ev representation is closely related to the Colin de Verdiere number in another way too: it plays a key role in the study of graphs with (G) # |V (G)| - 4 (Kotlov, Lovasz and Vempala =-=[5]-=-). 4. Lex Schrijver and the author proved [6] that a graph has (G) # 4 if and only if it is linklessly embedable in R 3 . Let G be a linklessly embedable graph and M , a Colin de Verdiere matrix for M... |

11 |
The Colin de Verdière graph parameter, in: Graph Theory and Computational Biology
- Holst, Lovász, et al.
- 1996
(Show Context)
Citation Context ...f the rows and columns by positive real numbers (we call this briefly scaling). The Colin de Verdiere number of a graph has a number of surprising graph-theoretical properties; we refer to the survey =-=[4]-=- for most of these. Perhaps the most basic is that (G) is minor-monotone, i.e., if H is a minor of G, then (H) # (G). Colin de Verdiere characterized graphs with small . He proved that (G) # 1 if and ... |

10 |
On the null space of a Colin de Verdière matrix. Annales de l’ institut Fourier
- Lovász, Schrijver
- 1999
(Show Context)
Citation Context ...and planar if and only if (G) # 3. Suppose that G is a 3-connected planar graph, and consider its Colin de Verdiere matrix M . Then the appropriate eigenvalue of M has multiplicity 3. It was shown in =-=[7]-=- that one can use the corresponding eigensubspace to construct an embedding of G in the sphere with convex faces. 1 In this paper we go further by showing that after an appropriate rescaling of M , on... |

3 |
der Holst: A short proof of the planarity characterization of Colin de Verdiere
- van
- 1995
(Show Context)
Citation Context ...t of the theorem is relatively easy, using Kuratowski's Theorem and the minor-monotonicity of . The original proof of the "if " part was quite involved; an elementary proof was given by Van =-=der Holst [3]-=-. It will be important for us that for 3-connected planar graphs, Holst's proof does not use the Strong Arnold Hypothesis (M3); thus, it implies: Proposition 1 Let G be a 3-connected planar graph and ... |