Steinitz Representations of Polyhedra and the Colin de Verdière Number
نویسنده
چکیده
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.
منابع مشابه
The Colin de Verdière number and graphs of polytopes
The Colin de Verdière number μ(G) of a graph G is the maximum corank of a Colin de Verdière matrix for G (that is, of a Schrödinger operator on G with a single negative eigenvalue). In 2001, Lovász gave a construction that associated to every convex 3-polytope a Colin de Verdière matrix of corank 3 for its 1-skeleton. We generalize the Lovász construction to higher dimensions by interpreting it...
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We prove that a k-tree can be viewed as a subgraph of a special type of (k + 1)-tree that corresponds to a stacked polytope and that these “stacked” (k + 1)-trees admit representations by orthogonal spheres in Rk+1. As a result, we derive lower bounds for Colin de Verdière’s μ of complements of partial k-trees and prove that μ(G) + μ(G) ≥ |G| − 2 for all chordal G. Yves Colin de Verdière’s grap...
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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 ? 4, and if it is planar, then (G) n ? 5. We give a full characterization of maximal planar graphs with (G) = n ? 5. In the opposite di...
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We consider Schrödinger operators on threshold graphs and give an explicit construction of a Colin de Verdière matrix for each connected threshold graph G of n vertices. We conclude the Colin de Verdière graph parameter μ(G) satisfies μ(G) ≥ n− i− 1, where i is the number of isolates in the graph building sequence. The proof is algorithmic in nature, constructing a particular Colin de Verdiére ...
متن کاملColin de Verdière number and graphs of polytopes
To every convex d-polytope with the dual graph G a matrix is associated. The matrix is shown to be a discrete Schrödinger operator on G with the second least eigenvalue of multiplicity d. This implies that the Colin de Verdière parameter of G is greater or equal d. The construction generalizes the one given by Lovász in the case d = 3.
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عنوان ژورنال:
- J. Comb. Theory, Ser. B
دوره 82 شماره
صفحات -
تاریخ انتشار 2001