Terwilliger graphs with μ ≤ 3

A Terwilliger graph is a noncomplete graph in which the intersection of the neighborhoods of any two vertices at distance 2 from each other is a μ-clique. We classify connected Terwilliger graphs with μ = 3 and describe the structure of Terwilliger graphs of diameter 2 with μ = 2.     

On the Nonexistence of Strongly Regular Graphs with Parameters (486, 165, 36, 66)

We prove that a strongly regular graph with parameters (486, 165, 36, 66) does not exist. Since the parameters indicated are parameters of a pseudogeometric graph for pG ₂(5, 32), we conclude that the partial geometries pG ₂(5, 32) and pG ₂(32, 5) do not exist. Finally, a neighborhood of an arbitrary vertex of a pseudogeometric graph for pG ₃(6, 80) is a pseudogeometric graph for pG ₂(5, 32) and, therefore, a pseudogeometric graph for the partial geometry pG ₃(6, 80) [i.e., a strongly regular graph with parameters (1127, 486, 165, 243)] does not exist.     

Anisotropy of permittivity of the hexagonal multiferroic HoMnO<Subscript>3</Subscript> in the spectral range 0.6–5.0 eV

The anisotropy of the optical properties of a single crystal of the hexagonal manganite HoMnO₃ has been investigated by spectroscopic ellipsometry in the spectral range 0.6–5.0 eV. It has been demonstrated that the optical absorption edge for the polarization E ⊥ c is determined by the intense narrow transition O(2p) → Mn(3d) centered at 1.5 eV, whereas this transition for the polarization E ‖ c is strongly suppressed and shifted toward higher energies by 0.2 eV. It has been revealed that, at the temperature T = 293 K, the spectra for both polarizations E ‖ c and E ⊥ c exhibit a broad absorption band centered at ∼2.4 eV, which was earlier observed in nonlinear spectra during optical second harmonic generation.     

On using the Lagrange coefficients for a posteriori error estimation

A posteriori error estimation of the objective functional is considered by means of a differential presentation of a finite-difference scheme and adjoint equations. The local approximation error is presented as a Taylor series remainder in the Lagrangian form. The field of the Lagrange coefficients is determined by a high-accuracy finite-difference template affecting the computation results. The feasibility of using the Lagrange coefficients for refining the solution and for estimating its uncertainty is considered.