Even Regular Graphs

There were several pictures of Perkel graphs floating around Tumblr, which got me interested in drawing them. This sketch only draws even degree - the k slider sets how many edges from each vertex, so then each vertex is on the end of that many edges also. s sets how many spaces between the edges ends. So from point i it goes to i+s, i+2s, ...i+ks. The animate box lets you see how the sketch builds the graph. Perkel graphs are 6-regular with 57 vertices. How many edges for a k-regular graph on n vertices?