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On Graphs with Exactly Three Q-Main Eigenvalues

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 identied. In this paper we continue these investigations by identifying the trees with exactly three Q-main eigenvalues, where one of them is zero.