I know that we can determine the number of connected components of a graph from the eigenvalues of its Laplacian matrix. My question is:
*Is there a way to understand the size of each connected component of a graph from its eigenvalues?*


For example: if a graph has 3 connected components two of which are maximal then can we determine this from the graph's spectrum?


Explanation of terminology:
 By *maximal connected component*, I mean a connected component whose number of nodes at least greater (not strictly) than the number of nodes in every other connected component in the graph. 



Size of connected components of a graph via its spectrum
This isn't possible; here is some intuition. If $G$ is a graph with $k$ components, then the multiplicity of 0 as an eigenvalue is $k$. This is because we can write $$L=\begin{pmatrix} L\_1 & & \\ & \ddots & \\ & & L\_k \end{pmatrix}$$ where $L$ is the Laplacian for $G$, and $L\_i$ is the Laplacian for the $i$-th component. Each $L\_i$, as the Laplacian of a connected graph, has eigenvalue 0 with multiplicity 1. Since the components can be chosen independently, it seems very unlikely that the sizes of these blocks $L\_i$ would be uniquely determined by the union of their spectra.


I believe the following provides a counterexample in the specific case you asked about (but be sure to check my work):


$G\_1$ is the disjoint union of a 3-cycle with an extra node attached to one of the corners, two 3-cycles which share an edge, and a single node.


$G\_2$ is the union of a complete graph on 4 nodes, a 3-path, and a 2-path.


I believe both of these have spectrum $\{4,4,4,3,2,1,0,0,0\}$. However $G\_1$ has two maximal components while $G\_2$ does not.


