Pyrazolate thiocarbonylrhodium complexes. X-ray structure of [Rh(μ-3,5-Me2Pz)(CS)(PPh3)]2

The preparation and properties of complexes of general formulae [Rh(CS)-(HL)(PR₃)₂]ClO₄ (HL = pyrazole (HPz), 3-methylpyrazole (H3-MePz), 3,5-dimethylpyrazole (H3,5-Me₂Pz), PR₃ = triphenylphosphine, tricyclohexylphosphine) and [(PR₃)₂(CS)Rh(μ-Pz)AuPPh₃]ClO₄ are reported. Complexes of the first set react with potassium hydroxide to give [Rh(μ-L)(CS)(PPh₃)₂ or RhPz(CS)(PR₃)₂ complexes. The structure of the complex [Rh(3,5-Me₂Pz)(CS)(PPh₃)]₂ has been determined by X-ray diffraction methods. The crystals are monoclinic, space group P2₁/c, with Z = 4 in a unit cell of dimensions a = 12.700(11), b = 17.217(16), c = 23.041(18) Å, β = 116.55(8)°. The structure has been solved by Patterson and Fourier methods and refined by full-matrix least-squares to R = 0.059 for 1978 independent reflections. The structure consists of dimeric complexes, in which each rhodium atom is in a square-planar environment being bonded to a carbon atom of a thiocarbonyl ligand, a phosphorus atom of a triphenylphosphine molecule and to two nitrogen atoms of pyrazolate ligands bridging the metal atoms. The dihedral angle of 71.1° between such two square planes leads to a bent configuration with an intramolecular rhodium-rhodium distance of 3.220 Å. The thiocarbonyl and triphenylphosphine ligands are in a trans disposition.     

Adjacency matrices of probe interval graphs

In this paper we obtain several characterizations of the adjacency matrix of a probe interval graph. In course of this study we describe an easy method of obtaining interval representation of an interval bigraph from its adjacency matrix. Finally, we note that if we add a loop at every probe vertex of a probe interval graph, then the Ferrers dimension of the corresponding symmetric bipartite graph is at most 3.     

Spanning 3-colourable subgraphs of small bandwidth in dense graphs

A conjecture by Bollobás and Komlós states the following: For every γ>0 and integers r2 and Δ, there exists β>0 with the following property. If G is a sufficiently large graph with n vertices and minimum degree at least and H is an r-chromatic graph with n vertices, bandwidth at most βn and maximum degree at most Δ, then G contains a copy of H.This conjecture generalises several results concerning sufficient degree conditions for the containment of spanning subgraphs. We prove the conjecture for the case r=3.     

Schrödinger operators on periodic discrete graphs

We consider Schrödinger operators with periodic potentials on periodic discrete graphs. The spectrum of the Schrödinger operator consists of an absolutely continuous part (a union of a finite number of non-degenerated bands) plus a finite number of flat bands, i.e., eigenvalues of infinite multiplicity. We obtain estimates of the Lebesgue measure of the spectrum in terms of geometric parameters of the graph and show that they become identities for some class of graphs. Moreover, we obtain stability estimates and show the existence and positions of large number of flat bands for specific graphs. The proof is based on the Floquet theory and the precise representation of fiber Schrödinger operators, constructed in the paper.     

布尔矩阵平方根问题及其与图着色问题的关系

讨论了布尔矩阵平方根问题及其与图着色问题的关系.首先得到有平方根的布尔矩阵具有的一些性质;然后给出布尔矩阵存在平方根的一个充要条件;最后证明布尔矩阵的平方根问题可以转化为简单图的着色问题.     

图P^3n的奇优美标号算法

本文讨论了图P^3n的奇优美性,给出了图只奇优美标号算法．     

L（1, 2）-edge-labelings for lattices

For a graph G and two positive integers j and k, an m-L(j, k)-edge-labeling of G is an assignment on the edges to the set {0, 1, 2,..., m}, such that adjacent edges which receive labels differ at least by j, and edges which are distance two apart receive labels differ at least by kThe λ j,k-number of G is the minimum m such that an m-L(j, k)-edge-labeling is admitted by GIn this article, the L(1, 2)-edge-labeling for the hexagonal lattice, the square lattice and the triangular lattice are studied, and the bounds for λ j,k-numbers of these graphs are obtained.