Some results on graphs with exactly two main eigenvalues

An eigenvalue of a graph G is called main if there is an associated eigenvector not orthogonal to j , the vector with each entry equal to 1. In this work, an error in a prior paper [Y. Hou and F. Tian, Unicyclic graphs with exactly two main eigenvalues, Appl. Math. Letters, 19 (2006), 1143–1147] is pointed out and the properties of the graphs with exactly two main eigenvalues and with pendent vertices are discussed. As an application, we obtain, together with known results, all connected bicyclic and tricyclic graphs with exactly two main eigenvalues.