Why is a graph Laplacian matrix positive semidefinite?

Question

Why is a graph's Laplacian matrix positive semidefinite? Can anyone provide an intuitive explanation and a proof?

Answer 1

If $B$ is the incidence matrix of an orientation of $G$, then $L=BB^T$. So

$$ x^TLx = \|Bx\|^2 \ge 0$$

for all $x$.

The matrix has rows indexed by the vertices, columns by edges and the $ij$-entry is 1 if the $i$-th vertex is the head of the $j$-th edge, $-1$ if its the tail and 0 otherwise.