Four Mathematical Results on a Theme by Paganini

In this thesis we explore four different problems, related through their use of graph theory. Firstly, we look at a pursuit game variant on graphs called Hunters and Rabbits and determine the parameters of this game for the hypercube, as well as a broader class of well behaved graphs. Secondly, we determine the maximum clique count across graphs of a fixed size under a maximum degree condition and a property of their clique complexes called shellability. Thirdly, we show that the 2-matching polynomial of a graph is always integral and identify a necessary and sufficient condition for a graph cover to be normal based only on its permutation representation. Lastly, we introduce two versions of the deck transformation monoid formed by taking the partial deck transformations of a space which respect a given immersion with and without a connected condition. We then explore the properties of these two variants and partial analogues to covering space theory