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...
2026-03-25 08:07:34.1774426054
When/How An Undirected Graph Can Be Recovered From Graph Laplacian
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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.