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

Graphs whose minimal rank is two : the finite fields case

Let F be a finite 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 over F whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let mr(F,G) be the minimum rank of all matrices in S(F,G). If F is a finite field with pt elements, p ??= 2, it is shown that mr(F,G) = 2 if and only if the complement of G is the join of a complete graph with either the union of at most (pt+1)/2 nonempty complete bipartite graphs or the union of at most two nonempty complete graphs and of at most (pt - 1)/2 nonempty complete bipartite graphs. These graphs are also characterized as those for which 9 specific graphs do not occur as induced subgraphs. If F is a finite field with 2t elements, then mr(F,G) = 2 if and only if the complement of G is the join of a complete graph with either the union of at most 2t +1 nonempty complete graphs or the union of at most one nonempty complete graph and of at most 2t-1 nonempty complete bipartite graphs. A list of subgraphs that do not occur as induced subgraphs is provided for this case as well

Graphs whose minimal rank is two

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 off-diagonal entries occur in exactly the positions corresponding to the edges of G. For example, if G is a path, S(F,G) c onsists 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 char F ??= 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 char F = 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

One-dimensional conduction in Charge-Density Wave nanowires

We report a systematic study of the transport properties of coupled one-dimensional metallic chains as a function of the number of parallel chains. When the number of parallel chains is less than 2000, the transport properties show power-law behavior on temperature and voltage, characteristic for one-dimensional systems.Comment: 4 pages, 5 figures, submitted to Phys. Rev. Let

On the graph complement conjecture for minimum rank

AbstractThe 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

On the minimum rank of not necessarily symmetric matrices : a preliminary study

The minimum rank of a directed graph G is defined to be the smallest possible rank over all real matrices whose ijth entry is nonzero whenever (i, j) is an arc in G and is zero otherwise. The symmetric minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i _= j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. Maximum nullity is equal to the difference between the order of the graph and minimum rank in either case. Definitions of various graph parameters used to bound symmetric maximum nullity, including path cover number and zero forcing number, are extended to digraphs, and additional parameters related to minimum rank are introduced. It is shown that for directed trees, maximum nullity, path cover number, and zero forcing number are equal, providing a method to compute minimum rank for directed trees. It is shown that the minimum rank problem for any given digraph or zero-nonzero pattern may be converted into a symmetric minimum rank problem

Charge State Evolution in the Solar Wind. II. Plasma Charge State Composition in the Inner Corona and Accelerating Fast Solar Wind

In the present work, we calculate the evolution of the charge state distribution within the fast solar wind. We use the temperature, density, and velocity profiles predicted by Cranmer et al. to calculate the ionization history of the most important heavy elements in the solar corona and solar wind: C, N, O, Ne, Mg, Si, S, and Fe. The evolution of each charge state is calculated from the source region in the lower chromosphere to the final freeze-in point. We show that the solar wind velocity causes the plasma to experience significant departures from equilibrium at very low heights, well inside the field of view (within 0.6 R sun from the solar limb) of nearly all the available remote-sensing instrumentation, significantly affecting observed spectral line intensities. We also study the evolution of charge state ratios with distance from the source region, and the temperature they indicate if ionization equilibrium is assumed. We find that virtually every charge state from every element freezes in at a different height, so that the definition of freeze-in height is ambiguous. We also find that calculated freeze-in temperatures indicated by charge state ratios from in situ measurements have little relation to the local coronal temperature of the wind source region, and stop evolving much earlier than their correspondent charge state ratio. We discuss the implication of our results on plasma diagnostics of coronal holes from spectroscopic measurements as well as on theoretical solar wind models relying on coronal temperatures.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/98586/1/0004-637X_761_1_48.pd