On Graphs with Exactly Three $Q$-main Eigenvalues

Mehrnoosh Javarsineh, Gholam Hossein Fath-Tabar

For a simple graph $G$, the $Q$-eigenvalues are the eigenvalues of the signless Laplacian matrix $Q$ of $G$. A $Q$-eigenvalue is said to be a $Q$-main eigenvalue if it admits a corresponding eigenvector non orthogonal to the all-one vector, or alternatively if the sum of its component entries is non-zero. In the literature the trees, unicyclic, bicyclic and tricyclic graphs with exactly two $Q$-main eigenvalues have been recently identified. In this paper we continue these investigations by identifying the trees with exactly three $Q$-main eigenvalues, where one of them is zero.