On the eigenvalues of Cayley graphs on the symmetric group generated by a complete multipartite set of transpositions

Given a finite simple graph \cG with $n$ vertices, we can construct the Cayley graph on the symmetric group $S_n$ generated by the edges of \cG, interpreted as transpositions. We show that, if \cG is complete multipartite, the eigenvalues of the Laplacian of \Cay(\cG) have a simple expression in terms of the irreducible characters of transpositions, and of the Littlewood-Richardson coefficients. As a consequence we can prove that the Laplacians of \cG and of \Cay(\cG) have the same first nontrivial eigenvalue. This is equivalent to saying that Aldous's conjecture, asserting that the random walk and the interchange process have the same spectral gap, holds for complete multipartite graphs.Comment: 29 pages. Includes modification which appear on the published version in J. Algebraic Combi

Ordering of Energy Levels in Heisenberg Models and Applications

In a recent paper we conjectured that for ferromagnetic Heisenberg models the smallest eigenvalues in the invariant subspaces of fixed total spin are monotone decreasing as a function of the total spin and called this property ferromagnetic ordering of energy levels (FOEL). We have proved this conjecture for the Heisenberg model with arbitrary spins and coupling constants on a chain. In this paper we give a pedagogical introduction to this result and also discuss some extensions and implications. The latter include the property that the relaxation time of symmetric simple exclusion processes on a graph for which FOEL can be proved, equals the relaxation time of a random walk on the same graph. This equality of relaxation times is known as Aldous' Conjecture.Comment: 20 pages, contribution for the proceedings of QMATH9, Giens, September 200

Selectivity control in Pt-catalyzed cinnamaldehyde hydrogenation

Chemoselectivity is a cornerstone of catalysis, permitting the targeted modification of specific functional groups within complex starting materials. Here we elucidate key structural and electronic factors controlling the liquid phase hydrogenation of cinnamaldehyde and related benzylic aldehydes over Pt nanoparticles. Mechanistic insight from kinetic mapping reveals cinnamaldehyde hydrogenation is structure-insensitive over metallic platinum, proceeding with a common Turnover Frequency independent of precursor, particle size or support architecture. In contrast, selectivity to the desired cinnamyl alcohol product is highly structure sensitive, with large nanoparticles and high hydrogen pressures favoring C=O over C=C hydrogenation, attributed to molecular surface crowding and suppression of sterically-demanding adsorption modes. In situ vibrational spectroscopies highlight the role of support polarity in enhancing C=O hydrogenation (through cinnamaldehyde reorientation), a general phenomenon extending to alkyl-substituted benzaldehydes. Tuning nanoparticle size and support polarity affords a flexible means to control the chemoselective hydrogenation of aromatic aldehydes