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

For a $\Delta$-regular undirected graph, its Graph Laplacian is $L=\Delta I-A$, and for any complex vector $v$,If $Av=\Delta v$, the left side is zero, hence $v_i=v_j$ on every edge. In a connected graph all coordinates coincide. Thus the eigenspace is exactly the constant vectors, of dimension one. Since the adjacency matrix of a graph is symmetric and diagonalizable,