The minimum span of L(2,1)-labelings of certain generalized Petersen graphs

In the classical channel assignment problem, transmitters that are sufficiently close together are assigned transmission frequencies that differ by prescribed amounts, with the goal of minimizing the span of frequencies required. This problem can be modeled through the use of an L(2,1)-labeling, which is a function f from the vertex set of a graph G to the non-negative integers such that |f(x)-f(y)|? 2 if xand y are adjacent vertices and |f(x)-f(y)|?1 if xand y are at distance two. The goal is to determine the λ-number of G, which is defined as the minimum span over all L(2,1)-labelings of G, or equivalently, the smallest number k such that G has an L(2,1)-labeling using integers from {0,1,…,k}. Recent work has focused on determining the λ-number of generalized Petersen graphs (GPGs) of order n. This paper provides exact values for the λ-numbers of GPGs of orders 5, 7, and 8, closing all remaining open cases for orders at most 8. It is also shown that there are no GPGs of order 4, 5, 8, or 11 with λ-number exactly equal to the known lower bound of 5, however, a construction is provided to obtain examples of GPGs with λ-number 5 for all other orders. This paper also provides an upper bound for the number of distinct isomorphism classes for GPGs of any given order. Finally, the exact values for the λ-number of n-stars, a subclass of the GPGs inspired by the classical Petersen graph, are also determined. These generalized stars have a useful representation on Möebius strips, which is fundamental in verifying our results.     

The photochemically induced reaction of sulfur dioxide with alkanes and carbon monoxide

Abstract— The gas phase photochemical reactions of SO₂ induced by 3130 Å radiation have been studied in the presence of added alkanes or added CO. The quantum yields obtained in the reactions with the low molecular weight alkanes employed are lower than those obtained by previous workers. The quantum yields were found to be pressure dependent increasing slowly with increasing pressure. A stoichiometric ratio of one SO₂ removed per molecule of hydrocarbon consumed was observed only under experimental conditions of [SO₂] < [RH]. For reaction mixtures where [SO₂] < [RH] the ratio of [SO₂]/[RH] reacted always exceeded unity. The quantum yields decreased slightly with increasing temperature. In all the alkane reaction systems studied, the deposition of viscous, nonvolatile reaction products was observed. In the experiments with added CO, the quantum yields were computed with respect to the rate of CO₂ formation. At 25°C and equal pressures of SO₂ and CO, φco₂ was observed to be 0.005 and it decreased slightly with increasing temperature. The results obtained are interpreted in terms of the sulfoxidation of the alkanes and the oxidation of CO proceeding by way of a ³SO₂ reaction intermediate.     

Infrared and Raman spectral interpretations of the ferroelectric phases of NaH3(SeO3)2 and NaD3(SeO3)2

The infrared and Raman spectra of NaH₃(SeO₃)₂ and NaD₃(SeO₃)₂ have been recorded from 24 to 300°K. The interpretation, assignments, and analysis of the spectral studies are presented on the paraelectric α phase (proton disordered), ferroelectric β phase (proton ordered) and ferroelectric γ phase (proton ordered). A discussion of a newly proposed proton-triggered phase transition mechanism and a possible origin of the hydrogen-bonded OH stretching region of KH₂PO₄-type ferroelectrics is given.     

The extraordinary ability of guanidinate derivatives to stabilize higher oxidation numbers in dimetal units by modification of redox potentials: structures of Mo(2)(5+) and Mo(2)(6+) compounds

Full characterization of the first homologous series of dimolybdenum paddlewheel compounds having electronic configurations of the types sigma(2)pi(4)delta(x), x = 2, 1, 0, and Mo-Mo bond orders of 4, 3.5, and 3, respectively, has been accomplished with the guanidinate-type ligand hpp (hpp = the anion of 1,3,4,6,7,8-hexahydro-2H-pyrimido[1,2-a]pyrimidine). Essentially quantitative oxidation of Mo(2)(hpp)(4), 1, by CH(2)Cl(2) gives Mo(2)(hpp)(4)Cl, 2. The halide in 2 can be replaced by reaction with TlBF(4) to produce Mo(2)(hpp)(4)(BF(4)), 3. Further oxidation of 2 by AgBF(4) produces Mo(2)(hpp)(4)ClBF(4), 4. The change from bond order 4 (in 1) to 3.5 in Mo(2)(hpp)(4)Cl is accompanied by an increase in the Mo-Mo bond length of 0.061 to 2.1280(4) A. A further increase of 0.044 A in the Mo-Mo distance to 2.172(1) A is observed as the bond order decreases to 3 in 4. At the same time, the Mo-N distances decrease smoothly as the oxidation state of the Mo atoms increases. Electrochemical studies have shown two chemically reversible processes at very negative potentials, E(1)(1/2)= -0.444 V and E(2)(1/2)= -1.271 V versus Ag/AgCl. These correspond to the processes Mo(2)(6+/5+) and Mo(2)(5+/4+), respectively. The latter potential is displaced by over 1.5 V relative to those of the Mo(2)(formamidinate)(4) compounds and the first one has never been observed in such complexes. Thus, in surprising contrast to previously observed behavior of the dimolybdenum unit, when it is surrounded by the very basic guanidinate ligand hpp, there is an extraordinary stabilization of the higher oxidation numbers of the molybdenum atoms.     

Determination of phenylalanine isotope ratio enrichment by liquid chromatography/time- of-flight mass spectrometry

The application of time-of-flight mass spectrometry to isotope ratio measurements has been limited by the relatively low dynamic range of the time-to-digital converter detectors available on commercial LC/ToF-MS systems. Here we report the measurement of phenylalanine isotope ratio enrichment by using a new LC/ToF-MS system with wide dynamic range. Underivatized phenylalanine was injected onto a C18 column directly with 0.1% formic acid/acetonitrile as the mobile phase. The optimal instrument parameters for the time-of-flight mass spectrometer were determined by tuning the instrument with a phenylalanine standard. The accuracy of the isotope enrichment measurement was determined by the injection of standard solutions with known isotope ratios ranging from 0.02% to 9.2%. A plot of the results against the theoretical values gave a linear curve with R2 of 0.9999. The coefficient of variation for the isotope ratio measurement was below 2%. The method is simple, rapid, and accurate and presents an attractive alternative to traditional GC/MS applications.