The Colin de Verdière graph parameter

In 1990, Y. Colin de Verdi ere 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 Lov asz 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. Contents

[1]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[2]  P. Mani.  Automorphismen von polyedrischen Graphen , 1971 .

[3]  I. Bárány,et al.  On a common generalization of Borsuk's and Radon's theorem , 1979 .

[4]  W. Whiteley.  Infinitesimally rigid polyhedra. I. Statics of frameworks , 1984 .

[5]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[6]  Y. C. Verdière,et al.  Sur la multiplicité de la première valeur propre non nulle du Laplacien , 1986 .

[7]  Charles R. Johnson.  The Theory of Matrices. Second Edition (with Applications) (Peter Lancaster and Miron Tismenetsky) , 1987 .

[8]  Rajeev Motwani,et al.  Constructive results from graph minors: linkless embeddings , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[9]  J. Reiterman,et al.  Embeddings of graphs in euclidean spaces , 1989, Discret. Comput. Geom..

[10]  Vojtech Rödl,et al.  Geometrical embeddings of graphs , 1989, Discret. Math..

[11]  J. G. Pierce,et al.  Geometric Algorithms and Combinatorial Optimization , 2016 .

[12]  L. Lovász,et al.  Orthogonal representations and connectivity of graphs , 1989 .

[13]  Y. D. Verdière.  On a novel graph invariant and a planarity criterion , 1990 .

[14]  Yves Colin de Verdière,et al.  Sur un nouvel invariant des graphes et un critère de planarité , 1990, J. Comb. Theory, Ser. B.

[15]  Yves Colin de Verdière,et al.  On a new graph invariant and a criterion for planarity , 1991, Graph Structure Theory.

[16]  Robin Thomas,et al.  A survey of linkless embeddings , 1991, Graph Structure Theory.

[17]  Oded Schramm.  How to cage an egg , 1992 .

[18]  Robin Thomas,et al.  Hadwiger's conjecture forK6-free graphs , 1993, Comb..

[19]  Alexander Schrijver,et al.  On the invariance of Colin de Verdière's graph parameter under clique sums , 1995 .

[20]  Alexander Schrijver,et al.  On a Minor-Monotone Graph Invariant , 1995, J. Comb. Theory, Ser. B.

[21]  Robin Thomas,et al.  Sachs' Linkless Embedding Conjecture , 1995, J. Comb. Theory, Ser. B.

[22]  R. Bacher,et al.  Multiplicités des valeurs propres et transformations étoile-triangle des graphes , 1995 .

[23]  Hein Vanderholst,et al.  A Short Proof of the Planarity Characterization of Colin de Verdière , 1995, J. Comb. Theory, Ser. B.

[24]  Paul D. Seymour,et al.  Graph Minors: XV. Giant Steps , 1996, J. Comb. Theory, Ser. B.

[25]  H. van der Holst,et al.  Topological and Spectral Graph Characterizations , 1996 .

[26]  Santosh S. Vempala,et al.  The Colin de Verdière number and sphere representations of a graph , 1997, Comb..

[27]  W. Thurston,et al.  Three-Dimensional Geometry and Topology, Volume 1 , 1997, The Mathematical Gazette.

[28]  Alexander Schrijver,et al.  Minor-monotone graph invariants. , 1997 .

[29]  Alexander Schrijver,et al.  A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs , 1998 .

[30]  Alexander Schrijver,et al.  A correction: orthogonal representations and connectivity of graphs , 2000 .