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

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.