  The independence polynomial $I(G,x)$ of a finite graph $G$ is the generating
function for the sequence of the number of independent sets of each
cardinality. We investigate whether, given a fixed number of vertices and
edges, there exists optimally-least (optimally-greatest) graphs, that are least
(respectively, greatest) for all non-negative $x$. Moreover, we broaden our
scope to $k$-independence polynomials, which are generating functions for the
$k$-clique-free subsets of vertices. For $k \geq 3$, the results can be quite
different from the $k = 2$ (i.e. independence) case.
