I have come upon the statement that for a graph to be a good expander, the difference between the first and second eigenvalues of its adjacency matrix has to be "sufficiently large".
(Estrada E. 2006. Network robustness to targeted attacks. The interplay of expansibility and degree distribution. Eur. Phys. J. B 52, 563–574 DOI: 10.1140/epjb/e2006-00330-7)
However, i have not been able to find anything regarding this for non regular graphs.
Question 1: Does this hold true for non regular graphs?
Question 2: Does this hold true for directed graphs?