The adjacency matrix of a simple labeled graph is the matrix $A$ with $A_{[i, j]}$ or 0 according to whether the vertex $v_j$, is adjacent to the vertex $v_j$ or not. For simple graphs without self-loops, the adjacency matrix has $0 \mathrm{~s}$ on the diagonal. For undirected graphs, the adjacency matrix is symmetric. Further the Laplacian $\mathbf{L}$ is always diagonalizable as it is symmetric.
What happens if the adjacency matrix is not symmetric, i.e., if, for example, the graph is directed and weighted? I imagine that it may happen that the Laplacian matrix is not diagonalizable. Is there a class of graphs (directed, weighted), as general as possible, for which we know that the Laplacian matrix is diagonalizable?