Solution
= Solution
Let $A$ be the <adjacency matrix of a graph> and $Av=\lambda v$ with $v\ne0$. Choose a <vertex> $i$ maximizing $|v_i|>0$. Then
$$
|\lambda||v_i|=\left|\sum_{j\sim i}v_j\right|\leq\deg(i)|v_i|\leq\Delta|v_i|.
$$
Dividing gives
$$
\boxed{|\lambda|\leq\Delta.}
$$
This argument works even with a complex <eigenvector>; for a finite undirected <graph> the matrix is real symmetric and all <eigenvalues> are real.