The -spectrum and spanning trees of tensor products of bipartite graphs

Recently, Knuth and Ciucu independently proved the surprising fact, conjectured by Stanley, that one connected component of the tensor product of a path with itself (the so-called ``Aztec diamond graph') has four times as many spanning trees as the other connected component, independent of the length of the path. We show here that much more is true: the connected components of the tensor product of any connected bipartite multigraphs all have essentially the same -spectrum. It follows at once that there is a simple formula relating their numbers of spanning trees.

     

Fabrication, spectroscopy and laser performance of Nd-doped lead-silicate glass fibers

The fabrication of a highly efficient Nd³⁺-doped single-mode fiber laser operating at 1.06 μm is described. The Nd₃₊ is introduced by doping Nd₂O₃ into a multicomponent (flint) lead-silicate glass host, Schott commercial optical glass F7. A fabrication technique for doping rare-earth evenly into commercial optical glasses is demostrated. Spectroscopic properties relevant to laser operation in the Nd³⁺-doped lead-silicate glass fibers were measured and the influence of Pb₂₊ ions on the spectral properties was analyzed. Owing to the long lifetime and large absorption and emission cross-sections of Nd₃₊ in this lead-silicate glass host, a high-performance Nd³⁺-doped lead-silicate fibre laser device operating at 1.06 μm has been successfully demonstrated.     

Carbon–hydrogen and carbon–phosphorus bond cleavage in a triruthenium cluster complex containing dppm: The crystal and molecular structures of Ru3(CO)6{μ 3-OPPh2 C2H(C6H4)PPhCH2PPh}(μ 3-OPPh2)Ph and Ru3(CO)6 {μ-OPPh2C2H(C6H4)PPhO}(μ-PPh2)(μ-PPh2O)

The crystal and molecular structures of Ru₃(CO)₆{μ ₃-OPPh₂C₂H(C₆H₄)PPhCH₂PPh}-(μ ₃-OPPh₂)Ph (1) and Ru₃(CO)₆{μ-OPPh₂C₂H(C₆H₄)PPhO}(μ-PPh₂)(μ-PPh₂O) (2) have been determined by single crystal X-ray diffraction. Both complexes contain oxygen atoms oxidatively inserted into phosphorus–ruthenium bonds, and unique σ/π multidentate ligands formed from C $---{\text{H}}$ H and C $--$ P bond cleavage in bis(diphenylphosphino)acetylene and bis(diphenylphosphino)methane. Complex 1 crystallized in the triclinic space group ${\bar 1}$ , with lattice parameters a = 11.642(4) Å, b = 15.018(5) Å, c =16.587(5) Å, α = 2.48(3)°, β = 76.47(2)°, γ = 70.35(3)°, V = 2651.1(15) ų, Z = 2. Complex 2 crystallized in the centered monoclinic space group, C2/c, with lattice parameters a = 34.467(4) Å, b = 14.274(2) Å, c = 23.258(3) Å, β = 5.29(1)°, V = 11394(3) ų, Z = 8.