521,685 research outputs found
Triple crossing numbers of graphs
We introduce the triple crossing number, a variation of crossing number, of a
graph, which is the minimal number of crossing points in all drawings with only
triple crossings of the graph. It is defined to be zero for a planar graph, and
to be infinite unless a graph admits a drawing with only triple crossings. In
this paper, we determine the triple crossing numbers for all complete
multipartite graphs including all complete graphs.Comment: 34 pages, 53 figures: We reorganized the article and revised some
argument
Fermi level quantum numbers and secondary gap of conducting carbon nanotubes
For the single-wall carbon nanotubes conducting in the simplest tight binding
model, the complete set of line group symmetry based quantum numbers for the
bands crossing at Fermi level are given. Besides linear (k), helical (k'} and
angular momenta, emerging from roto-translational symmetries, the parities of U
axis and (in the zig-zag and armchair cases only) mirror planes appear in the
assignation. The helical and angular momentum quantum numbers of the crossing
bands never vanishes, what supports proposed chirality of currents. Except for
the armchair tubes, the crossing bands have the same quantum numbers and,
according to the non-crossing rule, a secondary gap arises, as it is shown by
the accurate tight-binding calculation. In the armchair case the different
vertical mirror parity of the crossing bands provides substantial conductivity,
though kF is slightly decreased.Comment: 6 pages, 2 figure
Production matrices for geometric graphs
We present production matrices for non-crossing geometric graphs on point sets in convex position, which allow us to derive formulas for the numbers of such graphs. Several known identities for Catalan numbers, Ballot numbers, and Fibonacci numbers arise in a natural way, and also new formulas are obtained, such as a formula for the number of non-crossing geometric graphs with root vertex of given degree. The characteristic polynomials of some of these production matrices are also presented. The proofs make use of generating trees and Riordan arrays.Postprint (updated version
Degenerate Crossing Numbers
Let G be a graph with n vertices and e≥4n edges, drawn in the plane in such a way that if two or more edges (arcs) share an interior point p, then they properly cross one another at p. It is shown that the number of crossing points, counted without multiplicity, is at least constant timese and that the order of magnitude of this bound cannot be improved. If, in addition, two edges are allowed to cross only at most once, then the number of crossing points must exceed constant times(e/n)
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