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18
GRAPHS WHOSE MINIMAL RANK IS TWO
, 2004
"... Let F be a field, G = (V, E) be an undirected graph on n vertices, and let S(F,G) be the set of all symmetric n × n matrices whose nonzero offdiagonal entries occur in exactly the positions corresponding to the edges of G. For example, if G is a path, S(F,G) consists of the symmetric irreducible t ..."
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Cited by 38 (5 self)
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Let F be a field, G = (V, E) be an undirected graph on n vertices, and let S(F,G) be the set of all symmetric n × n matrices whose nonzero offdiagonal entries occur in exactly the positions corresponding to the edges of G. For example, if G is a path, S(F,G) consists of the symmetric irreducible tridiagonal matrices. Let mr(F,G) be the minimum rank over all matrices in S(F,G). Then mr(F,G) = 1 if and only if G is the union of a clique with at least 2 vertices and an independent set. If F is an infinite field such that charF = 2, then mr(F,G) ≤ 2 if and only if the complement of G is the join of a clique and a graph that is the union of at most two cliques and any number of complete bipartite graphs. A similar result is obtained in the case that F is an infinite field with charF = 2. Furthermore, in each case, such graphs are characterized as those for which 6 specific graphs do not occur as induced subgraphs. The number of forbidden subgraphs is reduced to 4 if the graph is connected. Finally, similar criteria is obtained for the minimum rank of a Hermitian matrix to be less than or equal to two. The complement is the join of a clique and a graph that is the union of any number of cliques and any number of complete bipartite graphs. The number of forbidden subgraphs is now 5, or in the connected case, 3.
A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs
"... 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 ..."
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Cited by 34 (8 self)
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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 Borsuktype 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.)
The Colin de Verdière graph parameter
, 1997
"... 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 emb ..."
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Cited by 20 (3 self)
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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.
The Colin de Verdière number and sphere representations of a graph
, 1996
"... 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, t ..."
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Cited by 14 (8 self)
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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.
On the null space of a Colin de Verdière matrix
"... Let G = (V, E) be a 3connected 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 ..."
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Cited by 4 (0 self)
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Let G = (V, E) be a 3connected 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 S², gives a proper embedding of G in S². This applies to the matrices associated with the parameter (G) introduced by Y. Colin de Verdière.
Discrete and Continuous: Two sides of the same?
"... 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. ..."
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Cited by 2 (0 self)
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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.
Spectre D'Opérateurs Differentiels Sur Les Graphes
, 2006
"... Dans cet exposé de survol, nous commençons par présenter des ensembles naturels d’opérateurs de type Schrödinger associés à un graphe fini. Nous étudions ensuite les limites singulières (au sens de la Γ−convergence) de tels opérateurs et montrons qu’elles sont associées à des relations naturelles en ..."
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Cited by 1 (0 self)
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Dans cet exposé de survol, nous commençons par présenter des ensembles naturels d’opérateurs de type Schrödinger associés à un graphe fini. Nous étudions ensuite les limites singulières (au sens de la Γ−convergence) de tels opérateurs et montrons qu’elles sont associées à des relations naturelles entre graphes (mineurs, transformation étoiletriangle) ou à des limites d’un intérêt indépendant (processus de Markov (recuit simulé), réseaux électriques). Cela conduit à introduire la notion de stabilité structurelle pour une valeur propre multiple d’un opérateur d’une famille en utilisant la transversalité dans les espaces d’opérateurs symétriques. Les invariants numériques de graphes ainsi construits sont liés à des problèmes classiques de la combinatoire des graphes: planarité, genre, plongement nonnoué dans R³, largeur d’arbre.