  This paper is an in-depth analysis of the generalized $\vartheta$-number of a
graph. The generalized $\vartheta$-number, $\vartheta_k(G)$, serves as a bound
for both the $k$-multichromatic number of a graph and the maximum $k$-colorable
subgraph problem. We present various properties of $\vartheta_k(G)$, such as
that the sequence $(\vartheta_k(G))_k$ is increasing and bounded from above by
the order of the graph $G$. We study $\vartheta_k(G)$ when $G$ is the strong,
disjunction or Cartesian product of two graphs. We provide closed form
expressions for the generalized $\vartheta$-number on several classes of graphs
including the Kneser graphs, cycle graphs, strongly regular graphs and
orthogonality graphs. Our paper provides bounds on the product and sum of the
$k$-multichromatic number of a graph and its complement graph, as well as lower
bounds for the $k$-multichromatic number on several graph classes including the
Hamming and Johnson graphs.
