Molecular graph eigenvectors for molecular coordinates: System Demonstration

In [2] the eigenvectors of the adjacency matrix of a molecular graphs have been used to compute molecular coordinates. In [4] the same idea was used to solve a mathematical problem. Here we explain why the largest eigenvectors often give satisfactory results and also explain why it fails for some molecular graphs. We consider the problem of embedding a graph on n vertices in Euclidean space Ts k, for k< n. Typically k would be 3 or 2. By posing the problem as minimising the squared norm of the appropriately weighted distance between adjacent points subject to natural normalising conditions we arrive at a formulation of the problem for which the optimal solution can be simply computed in terms of the eigenvectors of the Laplacian matrix of the (weighted) graph. For the case where the weights are chosen to be unity the solution is independent of the uniform penalty given to non-adjacent vertices.