Suppose that $G$ and $H$ are both expander graphs on the same node set with a second largest eigenvalue of $\lambda_G$ resp. $\lambda_H$.
- What can be said about the expansion of graph $G \cup H$? In particular, is the spectral gap of $G \cup H$ at least as large as the minimum of the spectral gaps of $G$ and $H$?
- Does it make a difference whether $G$ and $H$ both have constant node degree?
This is certainly true for the edge expansion of $G \cup H$, since it can only increase by adding edges. I know that spectral expansion and edge expansion are related by the Cheeger inequality, but using this route we only get a bound on the spectral expansion of $G \cup H$ that is worse than $\lambda_G$ and $\lambda_H$.