A note on the stability number of an orthogonality graph

We consider the orthogonality graph Omega(n) with 2^n vertices corresponding to the 0-1 n-vectors, two vertices adjacent if and only if the Hamming distance between them is n/2. We show that the stability number of Omega(16) is alpha(Omega(16))= 2304, thus proving a conjecture by Galliard. The main tool we employ is a recent semidefinite programming relaxation for minimal distance binary codes due to Schrijver. As well, we give a general condition for Delsarte bound on the (co)cliques in graphs of relations of association schemes to coincide with the ratio bound, and use it to show that for Omega(n) the latter two bounds are equal to 2^n/n.Comment: 10 pages, LaTeX, 1 figure, companion Matlab code. Misc. misprints fixed and references update

Extended F_4-buildings and the Baby Monster

The Baby Monster group B acts naturally on a geometry E(B) with diagram c.F_4(t) for t=4 and the action of B on E(B) is flag-transitive. It possesses the following properties: (a) any two elements of type 1 are incident to at most one common element of type 2, and (b) three elements of type 1 are pairwise incident to common elements of type 2 iff they are incident to a common element of type 5. It is shown that E(B) is the only (non-necessary flag-transitive) c.F_4(t)-geometry, satisfying t=4, (a) and (b), thus obtaining the first characterization of B in terms of an incidence geometry, similar in vein to one known for classical groups acting on buildings. Further, it is shown that E(B) contains subgeometries E(^2E_6(2)) and E(Fi22) with diagrams c.F_4(2) and c.F_4(1). The stabilizers of these subgeometries induce on them flag-transitive actions of ^2E_6(2):2 and Fi22:2, respectively. Three further examples for t=2 with flag-transitive automorphism groups are constructed. A complete list of possibilities for the isomorphism type of the subgraph induced by the common neighbours of a pair of vertices at distance 2 in an arbitrary c.F_4(t) satisfying (a) and (b) is obtained.Comment: to appear in Inventiones Mathematica

Sandpile groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields

A maximal minor $M$ of the Laplacian of an $n$-vertex Eulerian digraph $\Gamma$ gives rise to a finite group $\mathbb{Z}^{n-1}/\mathbb{Z}^{n-1}M$ known as the sandpile (or critical) group $S(\Gamma)$ of $\Gamma$. We determine $S(\Gamma)$ of the generalized de Bruijn graphs $\Gamma=\mathrm{DB}(n,d)$ with vertices $0,\dots,n-1$ and arcs $(i,di+k)$ for $0\leq i\leq n-1$ and $0\leq k\leq d-1$, and closely related generalized Kautz graphs, extending and completing earlier results for the classical de Bruijn and Kautz graphs. Moreover, for a prime $p$ and an $n$-cycle permutation matrix $X\in\mathrm{GL}_n(p)$ we show that $S(\mathrm{DB}(n,p))$ is isomorphic to the quotient by $\langle X\rangle$ of the centralizer of $X$ in $\mathrm{PGL}_n(p)$. This offers an explanation for the coincidence of numerical data in sequences A027362 and A003473 of the OEIS, and allows one to speculate upon a possibility to construct normal bases in the finite field $\mathbb{F}_{p^n}$ from spanning trees in $\mathrm{DB}(n,p)$.Comment: I+24 page

Waveguide propagation of light in polymer porous films filled with nematic liquid crystals

We theoretically analyze the waveguide regime of light propagation in a cylindrical pore of a polymer matrix filled with liquid crystals assuming that the effective radial optical anisotropy is biaxial. From numerical analysis of the dispersion relations, the waveguide modes are found to be sensitive to the field-induced changes of the anisotropy. The electro-optic properties of the polymer porous polyethylene terephthalate (PET) films filled with the nematic liquid crystal 5CB are studied experimentally and the experimental results are compared with the results of the theoretical investigation.Comment: 15 pages, 10 figures, revtex4-

A linear programming reformulation of the standard quadratic optimization problem

The problem of minimizing a quadratic form over the standard simplex is known as the standard quadratic optimization problem (SQO). It is NP-hard, and contains the maximum stable set problem in graphs as a special case. In this note, we show that the SQO problem may be reformulated as an (exponentially sized) linear program (LP). This reformulation also suggests a hierarchy of polynomial-time solvable LP’s whose optimal values converge finitely to the optimal value of the SQO problem. The hierarchies of LP relaxations from the literature do not share this finite convergence property for SQO, and we review the relevant counterexamples.Accepted versio

Improved bounds for the crossing numbers of K_m,n and K_n

It has been long--conjectured that the crossing number cr(K_m,n) of the complete bipartite graph K_m,n equals the Zarankiewicz Number Z(m,n):= floor((m-1)/2) floor(m/2) floor((n-1)/2) floor(n/2). Another long--standing conjecture states that the crossing number cr(K_n) of the complete graph K_n equals Z(n):= floor(n/2) floor((n-1)/2) floor((n-2)/2) floor((n-3)/2)/4. In this paper we show the following improved bounds on the asymptotic ratios of these crossing numbers and their conjectured values: (i) for each fixed m >= 9, lim_{n->infty} cr(K_m,n)/Z(m,n) >= 0.83m/(m-1); (ii) lim_{n->infty} cr(K_n,n)/Z(n,n) >= 0.83; and (iii) lim_{n->infty} cr(K_n)/Z(n) >= 0.83. The previous best known lower bounds were 0.8m/(m-1), 0.8, and 0.8, respectively. These improved bounds are obtained as a consequence of the new bound cr(K_{7,n}) >= 2.1796n^2 - 4.5n. To obtain this improved lower bound for cr(K_{7,n}), we use some elementary topological facts on drawings of K_{2,7} to set up a quadratic program on 6! variables whose minimum p satisfies cr(K_{7,n}) >= (p/2)n^2 - 4.5n, and then use state--of--the--art quadratic optimization techniques combined with a bit of invariant theory of permutation groups to show that p >= 4.3593.Comment: LaTeX, 18 pages, 2 figure

Effects of polarization azimuth in dynamics of electrically assisted light-induced gliding of nematic liquid-crystal easy axis

We experimentally study the reorientation dynamics of the nematic liquid crystal easy axis at photoaligned azo-dye films under the combined action of in-plane electric field and reorienting UV light linearly polarized at varying polarization azimuth, $\phi_p$. In contrast to the case where the light polarization vector is parallel to the initial easy axis and $\phi_p=0$, at $\phi_p\ne 0$, the pronounced purely photoinduced reorientation is observed outside the interelectrode gaps. In the regions between electrodes with non-zero electric field, it is found that the dynamics of reorientation slows down with $\phi_p$ and the sense of easy axis rotation is independent of the sign of $\phi_p$.Comment: revtex-4.1, 4 pages, 3 figure

Bjorken Sum Rule and pQCD frontier on the move

The reasonableness of the use of perturbative QCD notions in the region close to the scale of hadronization, i.e., below \lesssim 1 \GeV is under study. First, the interplay between higher orders of pQCD expansion and higher twist contributions in the analysis of recent Jefferson Lab (JLab) data on the Generalized Bjorken Sum Rule function $\Gamma_1^{p-n} (Q^2)$ at $0.1<Q^2< 3 {\rm GeV}^2$ is studied. It is shown that the inclusion of the higher-order pQCD corrections could be absorbed, with good numerical accuracy, by change of the normalization of the higher-twist terms. Second, to avoid the issue of unphysical singularity (Landau pole at Q=\Lambda\sim 400 \MeV ), we deal with the ghost-free Analytic Perturbation Theory (APT) that recently proved to be an intriguing candidate for a quantitative description of light quarkonia spectra within the Bethe-Salpeter approach. The values of the twist coefficients $\mu_{2k}$ extracted from the mentioned data by using the APT approach provide a better convergence of the higher-twist series than with the common pQCD. As the main result, a good quantitative description of the JLab data down to $Q\simeq$ 350 MeV is achieved.Comment: 10 pages, 3 figures, minor change