On graphs with given main eigenvalues

An eigenvalue of a graph is main if it has an eigenvector, the sum of whose entries is not equal to zero. Extending previous results of Hagos and Hou et al. we obtain two conditions for graphs with given main eigenvalues. All trees and connected unicyclic graphs with exactly two main eigenvalues were characterized by Hou et al. Extending their results, we determine all bicyclic connected graphs with exactly two main eigenvalues.