On the second largest eigenvalue of networks

Abstract

AbstractFrom 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 ($$\lambda _1$$ λ 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 $$\lambda _2$$ λ 2 , predominantly confined to the random regular graphs, followed by the results for various popular model networks. We emphasize the aspect that $$\lambda _2$$ λ 2 shows an entirely different behavior than $$\lambda _1$$ λ 1 .