I have to prove that
$$L = \sum_{i,j \in E} (e_i-e_j)(e_i-e_j)^T.$$
where $e_i$ is the vector of length $n$ that is one in the $i^\mathrm{th}$ entry, and is zero in every other entry (i.e., $e_i$ is the $i^\mathrm{th}$ column of the $n \times n$ identity matrix).
My first question is, can we rewrite it following as,
$$L = \sum_{i,j \in E} (e_i-e_j)(e_i-e_j)^T = (e_i-e_j)^2$$
If so, can we than say $$L = \sum_{i,j \in E} (e_i-e_j)^2 = \hat{e}^TL\hat{e} $$
Where $\hat{e}$ represents the vector $e_i - e_j$. Not sure if I am expressing myself correctly here.
I believe I found the correct answer.
$$ \sum_{i,j \in E} x^TLx = \sum_{i,j \in E} x^T(e_i-e_j)(e_i-e_j)^Tx = x^T\bigg(\sum_{i,j \in E} (e_i - e_j)(e_i - e_j)\bigg)x = x^T \bigg(\sum_{i,j \in E} (e_i-e_j)^2\bigg)x = \sum_{i,j \in E} x^TLx \geq 0 $$