I am trying to form a symmetric Laplacian matrix from an ordinary directed graph but would I be able to form an isomorphism? My gut feeling says it is not injective. Are there any theorems on this.
I am seeing if there are 2 non isomorphic graphs with the same Laplacian matrix
No, at least not for undirected simple graphs. For the standard Laplacian the off-diagonal elements are identical to corresponding adjacency matrix elements multiplied with -1. The diagonal of the adjacency is zero.
Hence the Laplacian matrix uniquely specifies the adjacaency.
Graphs with the same same Laplacian matrix hence have the same adjacency matrix and are thus isomorphic.