Can the Laplacian matrix of an undirected weighted graph be decomposed in the form of the Hadamard product ？

Question

I want to rewrite an incidence matrix M of an undirected weighted graph G into

$ M = W \circ A$

where W is the weight matrix, and $\circ$ denotes a Hadamard product. Now can someone tell me the factor expansion of the following formulation?

$(W \circ A)^T(W \circ A)\\=(W^T \circ A^T)(W \circ A)\\=?$

I guess there is a relationship among the Laplacian matrix, adjacency matrix and degree matrix.