When/How An Undirected Graph Can Be Recovered From Graph Laplacian Let $L$ be a $d\times d$ symmetric positive definite matrix.  (When) does $L$ define a unique undirected graph?
Sorry if this is a basic question, I am very new to the field...

When a graph is loopless and contains no multiedges, its laplacian is always unique. Notice that the laplacian is a difference between matrices $D$ and $M$, where $D$ is a diagonal matrix containing graph's degree sequence on the main diagonal and $M$ is the graph's adjacency matrix. $M$ is unique for any graph and for a simple graph it has zeros on the main diagonal, therefore for each simple graph its laplacian is also unique.
