3,803 research outputs found
Higher order matching polynomials and d-orthogonality
We show combinatorially that the higher-order matching polynomials of several
families of graphs are d-orthogonal polynomials. The matching polynomial of a
graph is a generating function for coverings of a graph by disjoint edges; the
higher-order matching polynomial corresponds to coverings by paths. Several
families of classical orthogonal polynomials -- the Chebyshev, Hermite, and
Laguerre polynomials -- can be interpreted as matching polynomials of paths,
cycles, complete graphs, and complete bipartite graphs. The notion of
d-orthogonality is a generalization of the usual idea of orthogonality for
polynomials and we use sign-reversing involutions to show that the higher-order
Chebyshev (first and second kinds), Hermite, and Laguerre polynomials are
d-orthogonal. We also investigate the moments and find generating functions of
those polynomials.Comment: 21 pages, many TikZ figures; v2: minor clarifications and addition
C*-algebras associated to coverings of k-graphs
A covering of k-graphs (in the sense of Pask-Quigg-Raeburn) induces an
embedding of universal C*-algebras. We show how to build a (k+1)-graph whose
universal algebra encodes this embedding. More generally we show how to realise
a direct limit of k-graph algebras under embeddings induced from coverings as
the universal algebra of a (k+1)-graph. Our main focus is on computing the
K-theory of the (k+1)-graph algebra from that of the component k-graph
algebras.
Examples of our construction include a realisation of the Kirchberg algebra
\mathcal{P}_n whose K-theory is opposite to that of \mathcal{O}_n, and a class
of AT-algebras that can naturally be regarded as higher-rank Bunce-Deddens
algebras.Comment: 44 pages, 2 figures, some diagrams drawn using picTeX. v2. A number
of typos corrected, some references updated. The statements of Theorem 6.7(2)
and Corollary 6.8 slightly reworded for clarity. v3. Some references updated;
in particular, theorem numbering of references to Evans updated to match
published versio
Generalized two-dimensional Yang-Mills theory is a matrix string theory
We consider two-dimensional Yang-Mills theories on arbitrary Riemann
surfaces. We introduce a generalized Yang-Mills action, which coincides with
the ordinary one on flat surfaces but differs from it in its coupling to
two-dimensional gravity. The quantization of this theory in the unitary gauge
can be consistently performed taking into account all the topological sectors
arising from the gauge-fixing procedure. The resulting theory is naturally
interpreted as a Matrix String Theory, that is as a theory of covering maps
from a two-dimensional world-sheet to the target Riemann surface.Comment: LaTeX, 10 pages, uses espcrc2.sty. Presented by A. D'adda at the
Third Meeting on Constrained Dynamics and Quantum Gravity, Villasimius
(Sardinia, Italy) September 13-17, 1999; to appear in the proceeding
Path coverings with prescribed ends in faulty hypercubes
We discuss the existence of vertex disjoint path coverings with prescribed
ends for the -dimensional hypercube with or without deleted vertices.
Depending on the type of the set of deleted vertices and desired properties of
the path coverings we establish the minimal integer such that for every such path coverings exist. Using some of these results, for ,
we prove Locke's conjecture that a hypercube with deleted vertices of each
parity is Hamiltonian if Some of our lemmas substantially
generalize known results of I. Havel and T. Dvo\v{r}\'{a}k. At the end of the
paper we formulate some conjectures supported by our results.Comment: 26 page
- …