A Lyness equation for graphs

Abstract

The Lyness equation, x(n+1)=(x(n)+k)/x(n-1), can be though of as an equation defined on the 2-regular tree T2: we can think of every vertex of T2 as a variable so that if x and z are the vertices adjacent to y, then x,y,z satisfy xz=y+k. This makes sense for any 2-regular graph. We generalize this to 3-regular graphs by considering xyz=w+k and xy+xz+yz=w+k where x,y,z are the three neighbors of w. In the special case where an auxiliary condition x+y+z=f(w) also hold, a solutions is determined by (any) two values and, in some cases, an invariant can be found.