Characterization of tricyclic graphs with exactly two $Q$-main eigenvalues

Abstract

The signless Laplacian matrix of a graph G is defined to be the sum of its adjacency matrix and degree diagonal matrix, and its eigenvalues are called Q-eigenvalues of G. A Q-eigenvalue of a graph G is called a Q-main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. Chen and Huang [L. Chen, Q.X. Huang, Trees, unicyclic graphs and bicyclic graphs with exactly two Q-main eigenvalues, submitted for publication] characterized all trees, unicylic graphs and bicyclic graphs with exactly two main Q-eigenvalues, respectively. As a continuance of it, in this paper, all tricyclic graphs with exactly two Q-main eigenvalues are characterized.