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

Let $A$ be the adjacency matrix of a graph and $Av=\lambda v$ with $v\ne 0$. Choose a vertex $i$ maximizing $|v_i|>0$. ThenDividing givesThis argument works even with a complex eigenvector; for a finite undirected graph the matrix is real symmetric and all eigenvalues are real.