The mapping class group of a locally finite graph Maps$(X)$ is the set of proper homotopy equivalences of $X$ up to proper homotopy. It is meant to be the analogue of the mapping class group of an infinite-type surface one dimension lower, but it also generalizes Out$(F_n)$ to a much larger class of possibly infinitely generated groups, establishing a “Big Out$(F_n)$.” In this talk, I plan to define the mapping class group for a locally finite graph, discuss its topology, and give motivation. I will then discuss which locally finite graphs $X$ are such that Maps$(X)$ contains a dense conjugacy class. Along the way, we will discuss end spaces and their structures.