Zero-divisor Labelings of Graphs

This paper introduces the notions of a zero-divisor labeling and the zero-divisor index of a graph using the zero-divisors of a commutative ring. Viewed in this way, the usual zero-divisor graph is a maximal graph with respect to a zero-divisor labeling. We also study optimal zero-divisor labelings of a finite graph.     

Natural Abundance 17O DNP NMR Provides Precise O−H Distances and Insights into the Brønsted Acidity of Heterogeneous Catalysts

Heterogeneous Brønsted acid catalysts are tremendously important in industry, particularly in catalytic cracking processes. Here we show that these Brønsted acid sites can be directly observed at natural abundance by ¹⁷O DNP surface-enhanced NMR spectroscopy (SENS). We additionally show that the O−H bond length in these catalysts can be measured with sub-picometer precision, to enable a direct structural gauge of the lability of protons in a given material, which is correlated with the pH of the zero point of charge of the material. Experiments performed on materials impregnated with pyridine also allow for the direct detection of intermolecular hydrogen bonding interactions through the lengthening of O−H bonds.