An exponentiation graph (that's a name I just made up myself) shows what happens to the last digit of a number (or, more strictly, to the last digit in its decimal representation) as it's raised to various powers. There's one such graph for each digit 0 through 9 (all of the graphs shown here are for numbers written in decimal; graphs for other number systems are left as an exercise for the reader, or the future, or both).
Each graph begins with a node for 1 (because for any number n, the number n0 is the number 1, which has a decimal representation ending in the digit “1”). Then, for each node in the graph, we calculate what will be the final digit when we raise the number to the next power. If necessary we add a new node and an arc connecting to it. Eventually, we don't need a new node but just direct the new arc to an existing node.