Chromatic roots and minor-closed families of graphs

Given a minor-closed class of graphs $\mathcal{G}$, what is the infimum of the non-trivial roots of the chromatic polynomial of $G \in \mathcal{G}$? When $\mathcal{G}$ is the class of all graphs, the answer is known to be $32/27$. We answer this question exactly for three minor-closed classes of graphs. Furthermore, we conjecture precisely when the value is larger than $32/27$.Comment: 18 pages, 5 figure

A zero-free interval for chromatic polynomials of graphs with 3-leaf spanning trees

It is proved that if $G$ is a graph containing a spanning tree with at most three leaves, then the chromatic polynomial of $G$ has no roots in the interval $(1,t_1]$, where $t_1 \approx 1.2904$ is the smallest real root of the polynomial $(t-2)^6 +4(t-1)^2(t-2)^3 -(t-1)^4$. We also construct a family of graphs containing such spanning trees with chromatic roots converging to $t_1$ from above. We employ the Whitney $2$-switch operation to manage the analysis of an infinite class of chromatic polynomials.Comment: 16 pages, 5 figure

Testing stellar population synthesis models with Sloan Digital Sky Survey colors of M31's globular clusters

Accurate stellar population synthesis models are vital in understanding the properties and formation histories of galaxies. In order to calibrate and test the reliability of these models, they are often compared with observations of star clusters. However, relatively little work has compared these models in the ugriz filters, despite the recent widespread use of this filter set. In this paper, we compare the integrated colors of globular clusters in the Sloan Digital Sky Survey (SDSS) with those predicted from commonly used simple stellar population (SSP) models. The colors are based on SDSS observations of M31's clusters and provide the largest population of star clusters with accurate photometry available from the survey. As such, it is a unique sample with which to compare SSP models with SDSS observations. From this work, we identify a significant offset between the SSP models and the clusters' g-r colors, with the models predicting colors which are too red by g-r\sim0.1. This finding is consistent with previous observations of luminous red galaxies in the SDSS, which show a similar discrepancy. The identification of this offset in globular clusters suggests that it is very unlikely to be due to a minority population of young stars. The recently updated SSP model of Maraston & Stromback better represents the observed g-r colors. This model is based on the empirical MILES stellar library, rather than theoretical libraries, suggesting an explanation for the g-r discrepancy.Comment: 8 pages, 4 figures, accepted for publication in Ap

Spectroscopy of globular clusters in the low-luminosity spiral galaxy NGC 45

CONTEXT: Extragalactic globular clusters have been studied in elliptical galaxies and in a few luminous spiral galaxies, but little is known about globular clusters in low-luminosity spirals. AIMS: Past observations with the ACS have shown that NGC 45 hosts a large population of globular clusters (19), as well as several young star clusters. In this work we aim to confirm the bona fide globular cluster status for 8 of 19 globular cluster candidates and to derive metallicities, ages, and velocities. METHODS: VLT/FORS2 multislit spectroscopy in combination with the Lick/IDS system was used to derive velocities and to constrain metallicities and [alpha/Fe] element ratio of the globular clusters. RESULTS: We confirm the 8 globular clusters as bona fide globular clusters. Their velocities indicate halo or bulge-like kinematics, with little or no overall rotation. From absorption indices such as H_beta, H_gamma, and H_delta and the combined [MgFe]' index, we found that the globular clusters are metal-poor [Z/H]<=-0.33 dex and [alpha/Fe]<=0.0 element ratio. These results argue in favor of a population of globular clusters formed during the assembling of the galaxy.Comment: Accepted for publication in Astronomy and Astrophysics. 10 pages, 6 figures. Table 6 and Fig. 6 will only be published in the electronic edition of the A&A journa

A six-parameter space to describe galaxy diversification

Galaxy diversification proceeds by transforming events like accretion, interaction or mergers. These explain the formation and evolution of galaxies that can now be described with many observables. Multivariate analyses are the obvious tools to tackle the datasets and understand the differences between different kinds of objects. However, depending on the method used, redundancies, incompatibilities or subjective choices of the parameters can void the usefulness of such analyses. The behaviour of the available parameters should be analysed before an objective reduction of dimensionality and subsequent clustering analyses can be undertaken, especially in an evolutionary context. We study a sample of 424 early-type galaxies described by 25 parameters, ten of which are Lick indices, to identify the most structuring parameters and determine an evolutionary classification of these objects. Four independent statistical methods are used to investigate the discriminant properties of the observables and the partitioning of the 424 galaxies: Principal Component Analysis, K-means cluster analysis, Minimum Contradiction Analysis and Cladistics. (abridged)Comment: Accepted for publicationin A\&

New Light on the Formation and Evolution of M31 and its Globular Cluster System

We present spectroscopic ages, metallicities, and [alpha/Fe] ratios for 70 globular clusters in M31 that were derived from Lick line-index measurements. In addition to a population of old (>10 Gyr) globular clusters with a wide range of metallicities, from about -2.0 dex to solar values, we find evidence for a population of intermediate-age globular clusters with ages between ~5 and 8 Gyr and a mean metallicity around [Z/H]=-0.6. We also confirm the presence of young M31 globular clusters that were recently identified by Beasley et al. (2004), which have ages <1 Gyr and relatively high metallicities around -0.4 dex. The M31 globular cluster system has a clearly super-solar mean [alpha/Fe]=0.14\pm0.04 dex. Intermediate-age and young objects show roughly solar abundance ratios. We find evidence for an age-[alpha/Fe] relation in the sense that younger clusters have smaller mean [alpha/Fe] ratios. From a comparison of indices, mostly sensitive to carbon and/or nitrogen abundance, with SSP model predictions for nitrogen-enhanced stellar populations, we find a dichotomy in nitrogen enhancement between young and old M31 globular clusters. The indices of objects older than 5 Gyr are consistent with a factor of three or higher in nitrogen enhancement compared to their younger counterparts. Using kinematical data from Morrison et al. (2004) we find that the globular cluster sub-population with halo kinematics is old (>9 Gyr), has a bimodal metallicity distribution, and super-solar [alpha/Fe]. Disk globular clusters have a wider range of ages, are on average more metal-rich, and have a slightly smaller mean [alpha/Fe] ratio. (abridged)Comment: 32 pages, 12 figures, accepted for publication in A&

An updated survey of globular clusters in M 31. III. A spectroscopic metallicity scale for the Revised Bologna Catalog

We present a new homogeneous set of metallicity estimates based on Lick indices for 245 old globular clusters of the M31 galaxy comprised in the Revised Bologna Catalog. The metallicity distribution of the M31 globular clusters is briefly discussed and compared with that of the Milky Way. Simple parametric statistics suggests that the [Fe/H] distribution is likely not unimodal. The strong correlation between metallicity and kinematics found in previous studies is confirmed. The most metal-rich GCs tend to be packed at the center of the system and share the galactic rotation as traced by the HI disk. Although the velocity dispersion around the curve increases with decreasing metallicity, also clusters with [Fe/H]<-1.0 display a clear rotational pattern, at odds with their Milky Way counterparts.Comment: 25 pages with 17 figures and 2 Appendices - To appear on Astronomy & Astrophysic