Does the Laplacian spectrum of a graph give information on the size of the graph?
For example, is it possible that I have two disconnected graphs $G$ and $H$ with the following features:
1) $G$ and $H$ have the same Laplacian spectrum,
2) $G$ and $H$ have the same size,
3) $G$ and $H$ have two components such that $A$ and $B$ are the components of $G$ and both have the same size, and $C$ and $D$ are components of $H$ with different sizes.