Largest eigenvalue of a graph

Question

I have $\lambda_1$ the largest eigenvalue of a graph, with $x = (x_v)_{v \in V(G)}$ the corresponding eigenvector.

$x_u$ is the entry of $x$ with maximum absolute value.

I don't understand why I then have:

$$\lambda_1 x_u = \sum_{v \in N(u)} x_v$$

$N(u)$ being the neighborhood of $u$.

Answer 1

Let $A$ be the adjacency matrix of $G$, that is $a_{uv} = 1 \iff v \in N(u)$ and $a_{uv} = 0$ otherwise. For any eigenvalue $\lambda$ and corresponding eigenvector $x$ we have $Ax = \lambda x$, that is for any $u \in V(G)$: \begin{align*} \lambda x_u &= (Ax)_u\\ &= \sum_{v\in V(G)} a_{uv} x_v\\ &= \sum_{v\in N(u)} x_v. \end{align*}