Hall conditions for edge-weighted bipartite graphs

A weighted variant of Hall's condition for the existence of matchings is shown to be equivalent to the existence of a matching in a lexicographic product. This is used to introduce characterizations of those bipartite graphs whose edges may be replicated so as to yield semiregular multigraphs or, equivalently, semiregular edge-weightings. Such bipartite graphs will be called semiregularizable. Some infinite families of semiregularizable trees are described and all semiregularizable trees on at most 11 vertices are listed. Matrix analogues of some of the results are mentioned and are shown to imply some of the known characterizations of regularizable graphs.

Variation in offspring size towards the high elevational range edge of a montane annual

Understanding the ecological determinants of species’ distribution is a fundamental goal of ecology, and is increasingly important with changing limits to species’ range. Species often reach distributional limits on gradients of resource availability, but the extent to which offspring provisioning varies towards range limits is poorly understood. Selection is generally expected to favour higher provisioning of individual offspring in environments with short growing seasons and limited moisture, nutrients, or hosts for parasitism. However, individual provisioning may decline if parent size is limited by resources. This thesis focuses on three major questions: 1) does seed size vary over an elevational gradient? 2) does this variation respond adaptively towards the range limit? and 3) is potential elevational variation environmentally or genetically controlled? I tested variation in seed investment towards the upper elevational limit of the hemiparasitic annual herb Rhinanthus minor, sampled across an elevational range of 1,000m in the Rocky Mountains of Alberta, Canada. I also used a reciprocal transplant experiment to address the heritability of seed mass. Seed mass increased marginally towards higher elevations, while seed number and plant size declined. There was a strong elevational increase in seed mass scaled by overall plant size. Therefore, investment in individual seeds was higher towards the upper range edge, indicating potential adaptation of the reproductive strategy to allow for establishment in marginal environments. Genetic, environmental, and genotype-by-environment interactions were observed in transplanted populations, but the relative proportions of these effects on seed size were unclear.