Unicyclic and bicyclic graphs with exactly three Q-main eigenvalues

Finding all the graphs with a certain number of Q-main eigenvalues is an algebraic graph theory problem that scientists have sought to answer it for many years. The purpose of this research is finding relationships between the algebraic properties of a signless Laplacian matrix of a graph and the other properties of that graph. In order to achieve this, we choose to characterize all the unicyclic and bicyclic graphs with exactly three distinct Q-main eigenvalues, one of which is zero.