For a given undirected graph, it is known that the signless Laplacian $$Q=D+W$$ is positive semidefinite, where $D$ is the degree matrix and $W$ is the adjacency matrix. In particular, the smallest eigenvalue of $Q$ is zero if and only if there is a bipartite component in the graph.
My questions are:
- Is there is a similar operator that helps to identify if the graph is $k$-partite?
- Is it possible to recognize a $k$-partite graph through the spectrum of the adjacency matrix?
Answers with references are strongly appreciated.