Hi all, I would like to ask about the eigenvalues of adjacency matrices of a simple undirected unweighted graph. I am mainly interested in the largest and second largest eigenvalues of a graph. From what I have read, it seems that the first eigenvalue gives an estimate on the average degree of the vertices, while the difference between the largest and second largest eigenvalue can tell us something interesting but I am not really sure what. The main point of my question, I was wondering, suppose we are not given the original graph, but we are given the first and second eigenvalues of the adjacency matrix. Are there any interesting deductions or observations we can make about the graph? Thank you very much!