32 research outputs found
The Strong Arnold Property for 4-connected flat graphs
We show that if is a 4-connected flat graph, then any real
symmetric matrix with exactly one negative eigenvalue and
satisfying, for any two distinct vertices and , if and
are adjacent, and if and are nonadjacent, has the Strong
Arnold Property: there is no nonzero real symmetric matrix with
and whenever and are equal or adjacent. (A graph
is {\em flat} if it can be embedded injectively in -dimensional Euclidean
space such that the image of any circuit is the boundary of some disk disjoint
from the image of the remainder of the graph.)
This applies to the Colin de Verdi\`ere graph parameter, and extends similar
results for 2-connected outerplanar graphs and 3-connected planar graphs
A short proof of the planarity characterization of Colin de Verdière
AbstractColin de Verdière introduced an interesting new invariant μ(G) for graphs G, based on algebraic and analytic properties of matrices associated with G. He showed that the invariant is monotone under taking miners and moreover, that μ(G) ≤ 3 if only if G is planar. In this paper we give a short proof of Colin de Verdière′s result that μ(G) ≤ 3 if G is planar
In search for a perfect shape of polyhedra: Buffon transformation
For an arbitrary polygon consider a new one by joining the centres of
consecutive edges. Iteration of this procedure leads to a shape which is affine
equivalent to a regular polygon. This regularisation effect is usually ascribed
to Count Buffon (1707-1788). We discuss a natural analogue of this procedure
for 3-dimensional polyhedra, which leads to a new notion of affine -regular
polyhedra. The main result is the proof of existence of star-shaped affine
-regular polyhedra with prescribed combinatorial structure, under partial
symmetry and simpliciality assumptions. The proof is based on deep results from
spectral graph theory due to Colin de Verdiere and Lovasz.Comment: Slightly revised version with added example of pentakis dodecahedro
A minor-monotone graph parameter based on oriented matroids
AbstractFor an undirected graph G = (V,E) let λ ′(G) be the largest d for which there exists an oriented matroid M on V of corank d such that for each nonzero vector (x+,x−) of M, x+ is nonempty and induces a connected subgraph of G.We show that λ′(G) is monotone under taking minors and clique sums. Moreover, we show that λ′(G) ⩽ 3 if and only if G has no K5- or V8-minor; that is, if and only if G arises from planar graphs by taking clique sums and subgraphs
Generalizations of the Strong Arnold Property and the minimum number of distinct eigenvalues of a graph
For a given graph G and an associated class of real symmetric matrices whose
off-diagonal entries are governed by the adjacencies in G, the collection of
all possible spectra for such matrices is considered. Building on the
pioneering work of Colin de Verdiere in connection with the Strong Arnold
Property, two extensions are devised that target a better understanding of all
possible spectra and their associated multiplicities. These new properties are
referred to as the Strong Spectral Property and the Strong Multiplicity
Property. Finally, these ideas are applied to the minimum number of distinct
eigenvalues associated with G, denoted by q(G). The graphs for which q(G) is at
least the number of vertices of G less one are characterized.Comment: 26 pages; corrected statement of Theorem 3.5 (a