The number of maximum matchings in a tree

We determine upper and lower bounds for the number of maximum matchings (i.e., matchings of maximum cardinality) $m(T)$ of a tree $T$ of given order. While the trees that attain the lower bound are easily characterised, the trees with largest number of maximum matchings show a very subtle structure. We give a complete characterisation of these trees and derive that the number of maximum matchings in a tree of order $n$ is at most $O(1.391664^n)$ (the precise constant being an algebraic number of degree 14). As a corollary, we improve on a recent result by G\'orska and Skupie\'n on the number of maximal matchings (maximal with respect to set inclusion).Comment: 38 page

Enumeration of Independent Sets in Graphs

An independent set is one of the most natural structures in a graph to focus on, from both a pure and applied perspective. In the realm of graph theory, and any concept it can represent, an independent set is the mathematical way of capturing a set of objects, none of which are related to each other. As graph theory grows, many questions about independent sets are being asked and answered, many of which are concerned with the enumeration of independent sets in graphs. We provide a detailed introduction to general graph theory for those who are not familiar with the subject, and then develop the basic language and notation of independent set theory before cataloging some of the history and major results of the field. We focus particularly on the enumeration of independent sets in various classes of graphs, with the heaviest focus on those defined by maximum and minimum degree restrictions. We provide a brief, specific history of this topic, and present some original results in this area. We then speak about some questions which remain open, and end the work with a conjecture for which we provide strong, original evidence. In the appendices, we cover all other necessary prerequisites for those without a mathematical background

Silent Sentinels: Archaeology, Magic, and the Gendered Control of Domestic Boundaries in New England, 1620-1725

The following dissertation is an historical archaeological study of the material culture of gendered protective magic used by Anglo-Europeans in seventeenth-century New England as a tactic to construct boundaries that mitigated perceived personal, social, spiritual, and environmental dangers. Such boundary construction was paramount in the seventeenth-century battle between good and evil epitomized by the belief in and struggle against witchcraft. This dissertation sought to answer three interrelated research questions: 1) What constitutes protective magical material culture in seventeenth-century contexts and how is it recognizable in the archaeological record? 2) What signifies gender specific protective magical practices and what can these differences relate about gender roles, identity, and social relationships? and 3) In what way and to what degree is the recourse to traditional beliefs significant in coping or risk management contexts? Synthesizing data from historical and folkloristic sources, and reviewing all accessible archaeological site reports and inventories from State Historic Preservation offices and principal site investigators for domestic structures in New England ca. 1620-1725 provided data to catalog and develop a typology of potential magical items. Analyzing these data then allowed the assessment of domestic and gendered patterns of magical risk management strategies. Magical content was frequently embedded within or symbolically encoded in architectural or artifactual details, whose gendered association tended to correspond with gender role activities or responsibilities; however, the general omission of magical interpretations in historical archaeology limits the visibility of potentially magical objects in site reports and inventories, so it is likely a wider range of materials and contexts exist. The final result of this dissertation was the construction of a criterion model for the identification and interpretation of magic in historical archaeological contexts, which extends the notion of ritual from specialized places and materials, and communal behaviors to include quotidian objects and settings, and individual practices. Ultimately, the results of this dissertation extend the field of the archaeology of ritual and magic in particular, and the broader field of archaeology more generally by providing theoretical and methodological tools for understanding and recognizing how magical belief contributes to physical and metaphoric boundary construction and maintenance