Let $G = (V,E)$ be undirected, simple graph. What are some relations from which you can (lower/upper) bound $\alpha(G)$ and $\omega(G)$ by simply knowing one of them? For example, we know that $\alpha(G) + \omega(G) \leq 2|V|$, so we can upper bound one of them by knowing the other with the formulas $\alpha(G) \leq 2|V| - \omega(G)$ and $\omega(G) \leq 2|V| - \alpha(G)$.
Now the question is, are there any other known upper bounds beside this one and the one in this question here: Proof Regarding the Upper Bound of the Product $\alpha(G) \omega(G)$ in a Graph ?
Lower/Upper bounds can include some other quantities (for example, $\delta(G)$, $\Delta(G)$, edge ratio) as long as those quantities can be calculated from the graph G in polynomial time.
The relationship that $\alpha(G) + \omega(G) \le 2n$ is useless in an $n$-vertex graph, because we independently know that $\alpha(G) \le n$ and $\omega(G) \le n$. But there is a closely related better bound: $$\alpha(G) + \omega(G) \le n+1.$$ This holds because a maximum independent set and a maximum clique can only intersect in at most one vertex, so once we've found a $k$-vertex independent set, at most $n-k+1$ vertices (the $n-k$ other vertices, plus one vertex of the independent set) can be part of the clique.
Before involving any other properties of the graph, this relationship is best possible: any pair of positive integers $(\alpha, \omega)$ with $\alpha+\omega=n+1$ is equal to $(\alpha(G), \omega(G))$ for some $n$-vertex graph $G$. (For example, take a complete graph on $\omega$ vertices, and add $n-\omega$ isolated vertices.)
I somehow don't expect most other properties of $G$ to give us better relationships between $\alpha(G)$ and $\omega(G)$. Generally, both $\alpha(G)$ and $\omega(G)$ are very local properties of graphs: they only tell us something about what a few of the vertices of $G$ are doing, and the rest of $G$ can do whatever it wants. Therefore, any relationship between the two is weak. Once we learn something else about $G$, that tells us much more: for example, if we know the maximum degree $\Delta(G)$, we know that $\alpha(G) \ge \frac{n}{\Delta(G)+1}$ and $\omega(G) \le \Delta(G)+1$. At this point, additional information about $\alpha(G)$ is unlikely to add much about our knowledge of $\omega(G)$, or vice versa.
One other notable connection appears in perfect graphs. In these graphs, and more generally in every graph where the chromatic number $\chi(G)$ is equal to the clique number $\omega(G)$, we conclude that $\alpha(G) \cdot \omega(G) \ge n$. (The argument is that the largest color class, which contains at least $\frac{n}{\chi(G)} = \frac{n}{\omega(G)}$ vertices, is an independent set.) Actually, perfect graphs are characterized by the property that for every $n'$-vertex induced subgraph $G'$, we have $\alpha(G') \cdot \omega(G') \ge n'$ (Lovász).
It is not trivial, but we can test a graph $G$ for perfection in $O(n^9)$ time (Chudnovsky, Cornuéjols, Liu, Seymour, Vušković).