Properties of the Graph Laplacian

Please give careful mathematical proofs of the following properties of the Laplacian matrix \(\mathbf{L}\) of a simple graph.

Part A

If \(\mathbf{1}\) is the vector containing all ones and \(\mathbf{0}\) is the zero vector, then

\[ \begin{aligned} \mathbf{L}\mathbf{1} = \mathbf{0}\;. \end{aligned} \]

Part B

The matrix \(\mathbf{L}\) is real and symmetric, but not invertible.