2,379 research outputs found
Generalized Calogero-Moser systems from rational Cherednik algebras
We consider ideals of polynomials vanishing on the W-orbits of the
intersections of mirrors of a finite reflection group W. We determine all such
ideals which are invariant under the action of the corresponding rational
Cherednik algebra hence form submodules in the polynomial module. We show that
a quantum integrable system can be defined for every such ideal for a real
reflection group W. This leads to known and new integrable systems of
Calogero-Moser type which we explicitly specify. In the case of classical
Coxeter groups we also obtain generalized Calogero-Moser systems with added
quadratic potential.Comment: 36 pages; the main change is an improvement of section 7 so that it
now deals with an arbitrary complex reflection group; Selecta Math, 201
Spanning trees and even integer eigenvalues of graphs
For a graph , let and be the Laplacian and signless
Laplacian matrices of , respectively, and be the number of
spanning trees of . We prove that if has an odd number of vertices and
is not divisible by , then (i) has no even integer
eigenvalue, (ii) has no integer eigenvalue , and
(iii) has at most one eigenvalue and such an
eigenvalue is simple. As a consequence, we extend previous results by Gutman
and Sciriha and by Bapat on the nullity of adjacency matrices of the line
graphs. We also show that if with odd, then the multiplicity
of any even integer eigenvalue of is at most . Among other things,
we prove that if or has an even integer eigenvalue of
multiplicity at least , then is divisible by . As a very
special case of this result, a conjecture by Zhou et al. [On the nullity of
connected graphs with least eigenvalue at least , Appl. Anal. Discrete
Math. 7 (2013), 250--261] on the nullity of adjacency matrices of the line
graphs of unicyclic graphs follows.Comment: Final version. To appear in Discrete Mat
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