Semi-regular graphs of minimum independence number

There are many functions of the degree sequence of a graph which give lower bounds on the independence number of the graph. In particular, for every graph G, α(G) ≥ R(d(G)), where R is the residue of the degree sequence of G. We consider the precision of this estimate when it is applied to semi-regular degree sequences. We show that the residue nearly always gives the best possible estimate on independence number in the sense that, when d is semiregular and graphic, one can always construct a graph G realizing d with R(d) ≤ α(G) ≤ R(d) + 1. It is actually possible to determine explicitly, for any such d, which inequality is strict. We prove this fact directly for most semi-regular sequences, giving an outline of proof for the remainder.