 Kernel of a graph Laplacian

The kernel of the Graph Laplacian consists exactly of vectors constant on each connected component of a graph. Indeed, $Lx=0$ implies $x^{T}Lx=0$, so the Graph Laplacian quadratic form forces equality at the endpoints of every edge; equality then propagates along graph paths. Conversely, these equalities make every entry of $Lx$ zero. Thus the dimension of the kernel, equivalently the multiplicity of the zero eigenvalue, counts the connected components of a graph.

For the disconnected graph with just two edges, $AB$ and $CD$, in order $A,B,C,D$,The two displayed vectors form a basis of the kernel. The other two eigenvectors $(1,-1,0,0{)}^T$ and $(0,0,1,-1{)}^T$ each have eigenvalue two, so the full list of eigenvalues, including multiplicities, is $0,0,2,2$. Separate connected components of a graph allow independent constant levels; a single global constant is only one of these possibilities.