Colin De Research Articles

Commentary: Peer Review and Incomplete Reference Lists

The well-known shortcomings and poor reliability of peer review (Colin de Gluocester, 2013; Ho et al., 2013; Newton, 2010; Onitilo et al., 2013; Smith, 2010) as well as the proverbial power and alm...

Geometry of nanostructures and eigenvectors of matrices

The visualization of graphs describing molecular structures or other atomic arrangements is necessary in theoretical studying or examining nanostructures of several atoms. In the present paper, we review first the previous results obtained by drawing graphs with the help of various matrices as the adjacency matrix, the Laplacian matrix and the Colin de Verdiere matrix. We explain why they are applicable on if the atoms are on spherical surfaces. We have found recently a matrix which could generate the Descartes coordinates for fullerenes, nanotubes and nanotori and also for nanotube junctions and coils as well. The construction of this matrix however is rather complicated in most of the cases. It needs energy minimization. Here will be shown that the spherical structures cut out from simple cubic, bcc, fcc and diamond lattices can be generated properly with the help of the matrix .

Colin de Verdiere parameters of chordal graphs

The Colin de Verdi`ere parameters, μ and ν, are defined to be the maximum nullity of certain real symmetric matrices associated with a given graph. In this work, both of these parametersare calculated for all chordal graphs. For ν the calculation is based solely on maximal cliques, while for μ the calculation depends on split subgraphs. For the case of μ our work extends some recent work on computing μ for split graphs.

Periodic elliptic operators with asymptotically preassigned spectrum

Abstract. We deal with operators in R n of the formA = −1b(x)X nk=1 ∂∂x k a(x)∂∂x k !where a(x),b(x) are positive, bounded and periodic functions. We denote by L per the set of such operators.The main result of this work is as follows: for an arbitrary L >0 and for arbitrary pairwise disjoint intervals(α j ,β j ) ⊂ [0,L], j = 1,...,m (m ∈ N) we construct the family of operators {A e ∈ L per } e such that thespectrum of A e has exactly m gaps in [0,L] when eis small enough, and these gaps tend to the intervals(α j ,β j ) as e→ 0. The idea how to construct the family {A e } e is based on methods of the homogenizationtheory.Keywords: periodic elliptic operators, spectrum, gaps, homogenization. IntroductionOur research is inspired by the following well-known resultof Y. Colin de Verdie`re [4]: forarbitrary numbers 0 = λ 1 <λ 2 <··· <λ m (m ∈ N) and n ∈ N {1} there is a n-dimensionalcompact Riemannian manifold M such that the first m eigenvalues of the corresponding Laplace-Beltrami operator −∆

Optimizing Colin de Verdière matrices of K4,4

We study the problem of finding and classifying the optimizing Colin de Verdière matrices of complete bipartite graphs. The main result is a complete classification of the optimizing Colin de Verdière matrices of K4,4. We then show a family of optimizing Colin de Verdière matrices for K4,q,q⩾4.

Classical and quantum ergodicity on orbifolds

We extend to orbifolds classical results on quantum ergodicity due to Shnirelman, Colin de Verdi\`ere and Zelditch, proving that, for any positive, first-order self-adjoint elliptic pseudodifferential operator P on a compact orbifold X with positive principal symbol p, ergodicity of the Hamiltonian flow of p implies quantum ergodicity for the operator P. We also prove ergodicity of the geodesic flow on a compact Riemannian orbifold of negative sectional curvature.

Parameters Related to Tree‐Width, Zero Forcing, and Maximum Nullity of a Graph

AbstractTree‐width, and variants that restrict the allowable tree decompositions, play an important role in the study of graph algorithms and have application to computer science. The zero forcing number is used to study the maximum nullity/minimum rank of the family of symmetric matrices described by a graph. We establish relationships between these parameters, including several Colin de Verdière type parameters, and introduce numerous variations, including the minor monotone floors and ceilings of some of these parameters. This leads to new graph parameters and to new characterizations of existing graph parameters. In particular, tree‐width, largeur d'arborescence, path‐width, and proper path‐width are each characterized in terms of a minor monotone floor of a certain zero forcing parameter defined by a color change rule.

Isospectrality for quantum toric integrable systems

We settle affirmatively the isospectral problem for quantum toric integrable systems: the semiclassical joint spectrum of such a system, given by a sequence of commuting Toeplitz operators on a sequence of Hilbert spaces, determines the classical integrable system given by the symplectic manifold and Poisson commuting functions, up to symplectomorphisms. We also give a full description of the semiclassical spectral theory of quantum toric integrable systems. This type of problem belongs to the realm of classical questions in spectral theory going back to pioneer works of Colin de Verdiere, Guillemin, Sternberg and others in the 1970s and 1980s.

On Essential Self-Adjointness for Magnetic Schrödinger and Pauli Operators on the Unit Disc in $${\mathbb {R}^2}$$

We study the question of magnetic confinement of quantum particles on the unit disk $${\mathbb {D}}$$ in $${\mathbb {R}^2}$$ , i.e. we wish to achieve confinement solely by means of the growth of the magnetic field $${B(\vec x)}$$ near the boundary of the disk. In the spinless case, we show that $${B(\vec x)\ge \frac{\sqrt 3}{2}\cdot\frac{1}{(1-r)^2}-\frac{1}{\sqrt 3}\frac{1}{(1-r)^2\ln \frac{1}{1-r}}}$$ , for $${|\vec x|}$$ close to 1, insures the confinement provided we assume that the non-radially symmetric part of the magnetic field is not very singular near the boundary. Both constants $${\frac{\sqrt 3}{2}}$$ and $${-\frac{1}{\sqrt 3}}$$ are optimal. This answers, in this context, an open question from Colin de Verdière and Truc (Ann Inst Fourier 2011, Preprint, arXiv:0903.0803v3). We also derive growth conditions for radially symmetric magnetic fields which lead to confinement of spin 1/2 particles.

On the graph complement conjecture for minimum rank

The minimum rank of a graph has been an interesting and well studied parameter investigated by many researchers over the past decade or so. One of the many unresolved questions on this topic is the so-called graph complement conjecture, which grew out of a workshop in 2006. This conjecture asks for an upper bound on the sum of the minimum rank of a graph and the minimum rank of its complement, and may be classified as a Nordhaus–Gaddum type problem involving the graph parameter minimum rank. The conjectured bound is the order of the graph plus two. Other variants of the graph complement conjecture are introduced here for the minimum semidefinite rank and the Colin de Verdière type parameter ν. We show that if the ν-graph complement conjecture is true for two graphs then it is true for the join of these graphs. Related results for the graph complement conjecture (and the positive semidefinite version) for joins of graphs are also established. We also report on the use of recent results on partial k-trees to establish the graph complement conjecture for graphs of low minimum rank.

The Laplacian on planar graphs and graphs on surfaces

These are lecture notes for the Current Developments in Mathematics conference at Harvard, November, 2011. We discuss topological, probabilistic and combinatorial aspects of the Laplacian on a graph embedded on a surface. The three main goals are to discuss: (1) for circular planar networks, the characterization due to Colin de Verdi\`ere of Dirichlet-to-Neumann operator; (2) The connections with the random spanning tree model; and (3) the characteristic polynomial of the Laplacian on an annulus and torus.

Sunada's method and the covering spectrum

In 2004, Sormani and Wei introduced the covering spectrum: a geometric invariant that isolates part of the length spectrum of a Riemannian manifold. In their paper they observed that certain Sunada isospectral manifolds share the same covering spectrum, thus raising the question of whether the covering spectrum is a spectral invariant. In the present paper we describe a group theoretic condition under which Sunada’s method gives manifolds with identical covering spectra. When the group theoretic condition of our method is not met, we are able to construct Sunada isospectral manifolds with distinct covering spectra in dimension 3 and higher. Hence, the covering spectrum is not a spectral invariant. The main geometric ingredient of the proof has an interpretation as the minimum-marked-length-spectrum analogue of Colin de Verdiere’s classical result on constructing metrics where the first $k$ eigenvalues of the Laplace spectrum have been prescribed.

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 as minus the Hessian matrix of the volume of the polar dual. As a corollary, µ(G) ≥ d if G is the 1-skeleton of a convex d-polytope.Determination of the signature of the Hessian of the volume is based on the second Minkowski inequality for mixed volumes and on Bol’s condition for equality.

On the Colin de Verdière number of graphs

Let μ ( G ) and ω ( G ) be the Colin de Verdière and clique numbers of a graph G, respectively. It is well-known that μ ( G ) ⩾ ω ( G ) - 1 for all graphs. Our main results include μ ( G ) ⩽ ω ( G ) for all chordal graphs; μ ( G ) ⩽ tw ( G ) + 1 for all graphs (where tw is the tree-width), and a characterization of those split ( ⊆ chordal) graphs for which μ ( G ) = ω ( G ) . The bound μ ( G ) ⩽ tw ( G ) + 1 improves a result of Colin de Verdière by a factor of 2.

On shortest disjoint paths in planar graphs

For a graph G and a collection of vertex pairs { ( s 1 , t 1 ) , … , ( s k , t k ) } , the k disjoint paths problem is to find k vertex-disjoint paths P 1 , … , P k , where P i is a path from s i to t i for each i = 1 , … , k . In the corresponding optimization problem, the shortest disjoint paths problem, the vertex-disjoint paths P i have to be chosen such that a given objective function is minimized. We consider two different objectives, namely minimizing the total path length (minimum sum, or short: Min-Sum), and minimizing the length of the longest path (Min-Max), for k = 2 , 3 . Min-Sum: We extend recent results by Colin de Verdière and Schrijver to prove that, for a planar graph and for terminals adjacent to at most two faces, the Min-Sum 2 Disjoint Paths Problem can be solved in polynomial time. We also prove that, for six terminals adjacent to one face in any order, the Min-Sum 3 Disjoint Paths Problem can be solved in polynomial time. Min-Max: The Min-Max 2 Disjoint Paths Problem is known to be NP-hard for general graphs. We present an algorithm that solves the problem for graphs with tree-width 2 in polynomial time. We thus close the gap between easy and hard instances, since the problem is weakly NP-hard for graphs with tree-width 3.

Expected values of parameters associated with the minimum rank of a graph

We investigate the expected value of various graph parameters associated with the minimum rank of a graph, including minimum rank/maximum nullity and related Colin de Verdière-type parameters. Let G ( v , p ) denote the usual Erdős-Rényi random graph on v vertices with edge probability p . We obtain bounds for the expected value of the random variables mr ( G ( v , p ) ) , M ( G ( v , p ) ) , ν ( G ( v , p ) ) and ξ ( G ( v , p ) ) , which yield bounds on the average values of these parameters over all labeled graphs of order v .

Tightening Nonsimple Paths and Cycles on Surfaces

We describe algorithms to compute the shortest path homotopic to a given path, or the shortest cycle freely homotopic to a given cycle, on an orientable combinatorial surface. Unlike earlier results, our algorithms do not require the input path or cycle to be simple. Given a surface with complexity $n$, genus $g\geq2$, and no boundary, we construct in $O(gn\log n)$ time a tight octagonal decomposition of the surface—a set of simple cycles, each as short as possible in its free homotopy class, that decompose the surface into a complex of octagons meeting four at a vertex. After the surface is preprocessed, we can compute the shortest path homotopic to a given path of complexity $k$ in $O(gnk)$ time, or the shortest cycle homotopic to a given cycle of complexity $k$ in $O(gnk\log(nk))$ time. A similar algorithm computes shortest homotopic curves on surfaces with boundary or with genus 1. We also prove that the recent algorithms of Colin de Verdiere and Lazarus for shortening embedded graphs and sets of cycles have running times polynomial in the complexity of the surface and the input curves, regardless of the surface geometry.

On the Strong Arnol'd Hypothesis and the connectivity of graphs

In the definition of the graph parameters µ(G) and �(G), introduced by Colin de Verdiere, and in the definition of the graph parameter �(G), introduced by Barioli, Fallat, and Hogben, a transversality condition is used, called the Strong Arnol'd Hypothesis. In this paper, we define the Strong Arnol'd Hypothesis for linear subspaces L ⊆ R n with respect to a graph G = (V,E), with V = {1,2,...,n}. We give a necessary and sufficient condition for a linear subspace L ⊆ R n with dimL ≤ 2 to satisfy the Strong Arnol'd Hypothesis with respect to a graph G, and we obtain a sufficient condition for a linear subspaceL ⊆ R n with dimL = 3 to satisfy the Strong Arnol'd Hypothesis with respect to a graph G. We apply these results to show that if G = (V,E) with V = {1,2,...,n} is a path, 2-connected outerplanar, or 3-connected planar, then each real symmetric n×n matrix M = (mi,j) with mi,j < 0 if ij ∈ E and mi,j = 0 if i 6 j and ij 6∈E (and no restriction on the diagonal), having exactly one negative eigenvalue, satisfies the Strong Arnol'd Hypothesis.

On a graph property generalizing planarity and flatness

We introduce a topological graph parameter σ(G), defined for any graph G. This parameter characterizes subgraphs of paths, outerplanar graphs, planar graphs, and graphs that have a flat embedding as those graphs G with σ(G)≤1,2,3, and 4, respectively. Among several other theorems, we show that if H is a minor of G, then σ(H)≤σ(G), that σ(K n )=n−1, and that if H is the suspension of G, then σ(H)=σ(G)+1. Furthermore, we show that µ(G)≤σ(G) + 2 for each graph G. Here µ(G) is the graph parameter introduced by Colin de Verdiere in [2].

On the Colin de Verdière numbers of Cartesian graph products

Let G be a connected graph with Colin de Verdière number μ ( G ) . We study the behaviour of μ with respect to the Cartesian product of graphs. We conjecture that if G = G 1 □ G 2 , with G 1 , G 2 connected, then μ ( G ) ⩾ μ ( G 1 ) + μ ( G 2 ) and prove that μ ( G ) ⩾ μ ( G 1 ) + h ( G 2 ) - 1 , where h is the Hadwiger number ( i.e. the order of the largest clique minor). In addition we provide an explicit construction of a Colin de Verdière matrix with corank μ ( G 1 ) + μ ( K n ) for the graph G = G 1 □ K n .