On the second largest eigenvalue of networks

From predicting the epidemic threshold of a disease outbreak to anticipating the stability of a complex system, analysis of spectra of the adjacency matrices of the underlying networks play a pivotal role. Despite spectra of networks considered as fingerprints of the corresponding complex systems, most works and review articles have circumscribed around the largest eigenvalue (lambda(1)) only. The second largest eigenvalue of a network that admits many applications in diverse fields, including mathematics and computer science, has not been thoroughly contemplated. This article first reviews existing literature on lambda(2), predominantly confined to the random regular graphs, followed by the results for various popular model networks. We emphasize the aspect that lambda(2) shows an entirely different behavior than lambda(1).