Suppose that $G$ and $H$ are both expander graphs on the same node set $V$ with a second largest eigenvalue of $\lambda_G$ resp. $\lambda_H$. Let $G\cup H$ be the graph on $V$ with the smallest set of edges such that both, $G$ and $H$, are subgraphs of $G \cup H$.
- 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 $H$?
- Is the case where $G$ and $H$ both have constant node degree any different?
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 the one given by $\lambda_G$ and $\lambda_H$.