Variations on a Theme: Graph Homomorphisms

This thesis investigates three areas of the theory of graph homomorphisms: cores of graphs, the homomorphism order, and quantum homomorphisms. A core of a graph X is a vertex minimal subgraph to which X admits a homomorphism. Hahn and Tardif have shown that, for vertex transitive graphs, the size of the core must divide the size of the graph. This motivates the following question: when can the vertex set of a vertex transitive graph be partitioned into sets which each induce a copy of its core? We show that normal Cayley graphs and vertex transitive graphs with cores half their size always admit such partitions. We also show that the vertex sets of vertex transitive graphs with cores less than half their size do not, in general, have such partitions. Next we examine the restriction of the homomorphism order of graphs to line graphs. Our main focus is in comparing this restriction to the whole order. The primary tool we use in our investigation is that, as a consequence of Vizing's theorem, this partial order can be partitioned into intervals which can then be studied independently. We denote the line graph of X by L(X). We show that for all n ≥ 2, for any line graph Y strictly greater than the complete graph Kₙ, there exists a line graph X sitting strictly between Kₙ and Y. In contrast, we prove that there does not exist any connected line graph which sits strictly between L(Kₙ) and Kₙ, for n odd. We refer to this property as being ``n-maximal", and we show that any such line graph must be a core and the line graph of a regular graph of degree n. Finally, we introduce quantum homomorphisms as a generalization of, and framework for, quantum colorings. Using quantum homomorphisms, we are able to define several other quantum parameters in addition to the previously defined quantum chromatic number. We also define two other parameters, projective rank and projective packing number, which satisfy a reciprocal relationship similar to that of fractional chromatic number and independence number, and are closely related to quantum homomorphisms. Using the projective packing number, we show that there exists a quantum homomorphism from X to Y if and only if the quantum independence number of a certain product graph achieves |V(X)|. This parallels a well known classical result, and allows us to construct examples of graphs whose independence and quantum independence numbers differ. Most importantly, we show that if there exists a quantum homomorphism from a graph X to a graph Y, then ϑ̄(X) ≤ ϑ̄(Y), where ϑ̄ denotes the Lovász theta function of the complement. We prove similar monotonicity results for projective rank and the projective packing number of the complement, as well as for two variants of ϑ̄. These immediately imply that all of these parameters lie between the quantum clique and quantum chromatic numbers, in particular yielding a quantum analog of the well known ``sandwich theorem". We also briefly investigate the quantum homomorphism order of graphs

Parity Violation in Neutron Resonances of Palladium

Parity violation in p-wave neutron resonances of the palladium isotopes 104, 105, 106, and 108 has been measured by transmission of a longitudinally polarized neutron beam through a natural palladium target. The measurements were performed at the pulsed spallation neutron source of Los Alamos Neutron Science Center. The rms weak interaction matrix elements and the corresponding spreading widths were determined for 104 Pd, 105 Pd, and 106 P

Neutron Resonance Spectroscopy of 103Rh from 30 eV to 2 keV

Neutron resonances in 103Rh have been measured for neutron energies from 30 to 2000 eV using the time-of-flight method and the (n,γ) reaction. The rhodium resonance spectroscopy is essential for the analysis of parity violation measurements recently performed on neutron resonances in 103Rh. Neutron scattering and radiative widths were determined, and orbital angular momentum assignments made with a Bayesian analysis. The s-wave and p-wave strength functions and average level spacings were determined

Parity Violation in Neutron Resonances of 103Rh

Parity nonconservation (PNC) was studied in p-wave neutron resonances of 103Rh in the neutron energy range 30 to 490 eV. The helicity dependence of the neutron total cross section of rhodium was determined by capture measurements with the time-of-flight method at the Manuel Lujan Neutron Scattering Center at the Los Alamos National Laboratory. A total of 32 p-wave resonances were studied and statistically significant longitudinal asymmetries were observed for resonances at En=44.5, 110.8, 321.6, and 432.9 eV. A statistical analysis treating the PNC matrix elements as random variables yields a weak spreading widthΓw=(1.42-0.59+1.21)×10-7eV

Neutron Resonance Spectroscopy of 104Pd, 105Pd, and 110Pd

We have measured neutron resonances in the palladium isotopes 104, 105, and 110 for neutron energies from 1 to 2100 eV. Many new p-wave resonances have been observed. Their neutron widths and, in several cases, the radiative widths were measured. The average level spacings and the s-wave and p-wave neutron strength functions were determined. The time-of-flight method was used for both neutron total cross section measurements and total (n,γ) reaction yield measurements at the pulsed spallation neutron source of Los Alamos Neutron Science Center. Well established resonance spectroscopy for these isotopes is essential for the analysis of parity violation data that were recently measured in palladium

Parity Violation in Neutron Resonances in 107,109Ag

Parity nonconservation (PNC) was studied in p-wave resonances in Ag by measuring the helicity dependence of the neutron total cross section. Transmission measurements on natural Ag were performed in the energy range 32 to 422 eV with the time-of-flight method at the Manuel Lujan Neutron Scattering Center at Los Alamos National Laboratory. A total of 15 p-wave neutron resonances were studied in 107Ag and ninep-wave resonances in 109Ag. Statistically significant asymmetries were observed for eight resonances in 107Ag and for four resonances in109Ag. An analysis treating the PNC matrix elements as random variables yields a weak spreading width of Γw=(2.67-1.21+2.65)×10-7 eV for107Ag and Γw=(1.30-0.74+2.49)×10-7 eV for 109Ag

Sources of Klebsiella and Raoultella species on dairy farms: Be careful where you walk

Klebsiella spp. are a common cause of mastitis, milk loss, and culling on dairy farms. Control of Klebsiella mastitis is largely based on prevention of exposure of the udder to the pathogen. To identify critical control points for mastitis prevention, potential Klebsiella sources and transmission cycles in the farm environment were investigated, including oro-fecal transmission, transmission via the indoor environment, and transmission via the outdoor environment. A total of 305 samples was collected from 3 dairy farms in upstate New York in the summer of 2007, and included soil, feed crops, feed, water, rumen content, feces, bedding, and manure from alleyways and holding pens. Klebsiella spp. were detected in 100% of rumen samples, 89% of water samples, and approximately 64% of soil, feces, bedding, alleyway, and holding pen samples. Detection of Klebsiella spp. in feed crops and feed was less common. Genotypic identification of species using rpoB sequence data showed that Klebsiella pneumoniae was the most common species in rumen content, feces, and alleyways, whereas Klebsiella oxytoca, Klebsiella variicola, and Raoultella planticola were the most frequent species among isolates from soil and feed crops. Random amplified polymorphic DNA-based strain typing showed heterogeneity of Klebsiella spp. in rumen content and feces, with a median of 4 strains per 5 isolates. Observational and bacteriological data support the existence of an oro-fecal transmission cycle, which is primarily maintained through direct contact with fecal contamination or through ingestion of contaminated drinking water. Fecal shedding of Klebsiella spp. contributes to pathogen loads in the environment, including bedding, alleyways, and holding pens. Hygiene of alleyways and holding pens is an important component of Klebsiella control on dairy farms

Neutron Resonance Spectroscopy of 117Sn from1 eV to 1.5 keV

Parity violation has been studied recently for neutron resonances in 117Sn. The neutron resonance spectroscopy is essential for the analysis of the parity violation data. We have measured neutron resonances in 117Sn for neutron energies from 1 to 1500 eV using the time-of-flight method and the (n,γ) reaction. The sample was enriched to 87.6% 117Sn. Neutron scattering and radiative widths were determined, and orbital angular momentum assignments were made with a Bayesian analysis. The s-wave and p-wave strength functions and average level spacings were determined

Parity Violation in 232Th Neutron Resonances Above 250 eV

The analysis of parity nonconservation (PNC) measurements performed on 232Th by the TRIPLE Collaboration has been extended to include the neutron energy range of 250 to 1900 eV. Below 250 eV all ten statistically significant parity violations have the same sign. However, at higher energies PNC effects of both signs were observed in the transmission of longitudinally polarized neutrons through a thick thorium target. Although the limited experimental energy resolution precluded analysis in terms of the longitudinal asymmetry, parity violations were observed and the cross section differences for positive and negative neutron helicities were obtained. For comparison, a similar analysis was performed on the data below 250 eV, for which longitudinal asymmetries were obtained previously. For energies below 250 eV, the p-wave neutron strength functions for the J=1/2 and J=3/2 states were extracted: S1/21=(1.68±0.61)×10-4 and S3/21=(0.75±0.18)×10-4. The data provide constraints on the properties of local doorway states proposed to explain the PNC sign effect in thorium

Parity Violation in Neutron Resonances of 117 Sn

Parity nonconservation (PNC) has been studied in neutron p-wave resonances of 117Sn. The longitudinal asymmetries were measured for 29 p-wave resonances in the neutron energy range 0.8 eV to 1100 eV. Statistically significant PNC effects were observed for four resonances. A statistical analysis determined the rms weak mixing matrix element and the weak spreading width. A weak spreading width of Γw=(0.28-0.15+0.56)×10-7 eV was obtained for117Sn