  We show that for any graph $G$, by considering "activation" through the
strong product with another graph $H$, the relation $\alpha(G) \leq
\vartheta(G)$ between the independence number and the Lov\'{a}sz number of $G$
can be made arbitrarily tight: Precisely, the inequality \[
  \alpha(G \times H) \leq \vartheta(G \times H) = \vartheta(G)\,\vartheta(H) \]
becomes asymptotically an equality for a suitable sequence of ancillary graphs
$H$.
  This motivates us to look for other products of graph parameters of $G$ and
$H$ on the right hand side of the above relation. For instance, a result of
Rosenfeld and Hales states that \[
  \alpha(G \times H) \leq \alpha^*(G)\,\alpha(H), \] with the fractional
packing number $\alpha^*(G)$, and for every $G$ there exists $H$ that makes the
above an equality; conversely, for every graph $H$ there is a $G$ that attains
equality.
  These findings constitute some sort of duality of graph parameters, mediated
through the independence number, under which $\alpha$ and $\alpha^*$ are dual
to each other, and the Lov\'{a}sz number $\vartheta$ is self-dual. We also show
duality of Schrijver's and Szegedy's variants $\vartheta^-$ and $\vartheta^+$
of the Lov\'{a}sz number, and explore analogous notions for the chromatic
number under strong and disjunctive graph products.
