An edge ordering of a graph G = (V,E) is an injection f : E → R, where R denotes the set of real numbers. A path in G for which the edge ordering f increases along its edge sequence is called an f-ascent; an f-ascent is maximal if it is not contained in a longer f-ascent. The depression of G is the smallest integer k such that any edge ordering f has a maximal f-ascent of length at most k. In this dissertation we discuss various results relating to the depression of a graph. We determine a formula for the depression of the class of trees known as double spiders. A k-kernel of a graph G is a set of vertices U ⊆ V (G) such that for any edge ordering f of G there exists a maximal f-ascent of length at most k which neither starts nor ends in U. We study the concept of k-kernels and discuss related depression results, including an improved upper bound for the depression of trees. We include a characterization of the class of graphs with depression three and without adjacent vertices of degree three or higher, and also construct a large class of graphs with depression three which contains graphs with adjacent vertices of high degree. Lastly, we apply the concept of ascents to edge colourings using possibly fewer than |E(G)| colours (integers). We consider the problem of determining the minimum number of colours for which there exists an edge colouring such that the length of a shortest maximal path of edges with increasing colors has a given length. / Graduate / 0405