Consider a large graph, minimum 1 000 vertices but it can easily go up to 50 000 vertices depending the case. The graph is the result of social relationships (followers, following, friendship) so it can be oriented but for the ease of the solution let's say it's not, so the matrix is symmetric. Moreover the matrix of adjacency is highly sparse.
I need to calculate the eigenvalues, but the major problem as I see it is that computing eigenvalues for large adjacency matrices cost a lot on time complexity due to matrices operations. If I can't efficiently compute all the eigenvalues, even a method to find the largest eigenvalue would be nice.
I've searched through the literature in vain, is there an algorithm or other approach to solve this?