Problem: Prove that Laplacian matrix $L$ of an undirected, connected graph $G$ ($G$ finite with $n$ vertices) is positive definite. Using the following property to prove $$L = D-A,$$ where $D$ is the degree matrix $(D_{ii} = \sum_{j=1}^n A_{ij})$ and $A$ is the adjacency matrix.
My attempt: Let $u \in \mathbb{R}^n$, then $$u^TLu = u^TDu - u^TAu = \sum_{i=1}^n(D_{ii} - A_{ii})u_i^2 - \sum_{i\neq j} A_{ij}u_iu_j.$$ On the other hand, since $G$ is connected then $$D_{ii} - A_{ii} = D_{ii} \ge 1 .$$ In addition, for all $i,j$ we have $$A_{ij} \le 1.$$ Therefore, $$u^T Lu \ge \sum_{i=1}^n u_i^2 - \sum_{i\neq j}u_iu_j.$$ Now, I am stuck here and do not know how to continue. I hope anyone can help me with some hints.