If a <regular graph> has $c>1$ connected components, the <indicator vector> of each component is a $\Delta$-<eigenvector>. They are <linearly independent>, and the Laplacian argument shows that every such <eigenvector> is constant on each component. Therefore its <multiplicity> is exactly $c$, in particular greater than one.

Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 4 / 16H / iv / Solution

 Solution (source code)