An efficient heuristic approach to detecting graph isomorphism based on combinations of highly discriminating invariants

The search for an easily computable, finite, complete set of graph invariants remains a challenging research topic. All measures characterizing the topology of a graph that have been developed thus far exhibit some degree of degeneracy, i.e., an inability to distinguish between non-isomorphic graphs. In this paper, we show that certain graph invariants can be useful in substantially reducing the computational complexity of isomorphism testing. Our findings are underpinned by numerical results based on a large scale statistical analysis.     

Synthesis and Thermal Transformations of 1-Polyfluoro-Alkyl-2-Trialkylsilylethyl Esters of Phosphorus Acids

Abstract

We have studied the phosphorylation of 1-polyfluorólkyl-2-trialkylsilylethanols (I) with phosphorus acids chlorides.     

Multinuclear NMR Studies of the Substitlent Effects in N-Phenyl-P,P,P-Triarylphospha-λ5-Azenes,Triaryl-Phosphines and Triarylphosphine Oxides

Abstract

A series of N-phenyl-P,P,P-triarylphospha-λ⁵-azenes (1) as well as their ^l5N labeled analogs was synthesized. The ¹³C, ³¹P, and ¹⁵N NMR spectra of this series and those of two other series of related compounds, namely triarylphosphines (2) and triarylphosphine oxides (3), were measured and are reported. Many satisfactory correlations using the mono-substituent parameter (MSP) and the Taft dual-substituent parameter (DSP) treatments with the ¹³C substituent chemical shifts (SCS), ³¹P SCS, ¹⁵N SCS and the one bond P-N, P-C and C-N coupling constants were observed and will be discussed. Thus, for example, the ³¹P and ¹⁵N chemical shifts in 1 correlated with [sgrave]^?with negative slopes while the ³¹P chemical shifts in 3 correlated with those in 1 with a slope of 2.0. The ¹³C chemical shifts in 1 correlated excellently with the corresponding ones in 3 with slopes very close to unity. The substituent effects on the chemical shifts of the various nuclei were shown to be mainly due to changes in the charge distribution on those nuclei. In 1 the one bond P-N and P-C coupling constants correlated with [sgrave]_p and [sgrave]_R respectively. The one bond P-C coupling constants of 1 correlated quite well with those of 3 with a slope of 0.93 while the corresponding correlation of 1 with 2 was quite poor. Taft DSP treatment of ¹J_PCin 1 and 3 were quite similar, ρI and ρR were both negative and ρR was much larger than ρI. Series 2 showed behavior which was different from that shown by 1 and 3 but similar to that shown by other systems with a lone electron pair on the atom bound to the phenyl ring. The substituent effects on the one bond P-N, P-C and C-N coupling constants will be discussed in terms of bonding and hybridization changes between the directly bonded nuclei.     

Essential selfadjointness of singular magnetic Schrödinger operators on Riemannian manifolds

In this paper we extend the well-known Leinfelder–Simader theorem on the essential selfadjointness of singular Schrödinger operators to arbitrary complete Riemannian manifolds. This improves some earlier results of Shubin, Milatovic and others.