Counting Independent Sets and Generalized Colorings in Graphs with Various Restrictions

An n-vertex, d-regular graph can have at most 2^(n/2+o(n)) independent sets. We address this upper bound when we impose the additional condition that the graph has independence number at most ?. In particular, we show that for a sequence of d_n-regular n-vertex graphs G_n with independence number at most ?_n, if d_n ? 8 and (?_n)/n ? c_? where 0