Main Q-eigenvalues and generalized Q-cospectrality of graphs |
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Authors: | Tianyi Bu Lizhu Sun Wenzhe Wang Jiang Zhou |
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Institution: | 1. College of Automation, Harbin Engineering University, Harbin, 150001, P. R. China 2. College of Science, Harbin Engineering University, Harbin, 150001, P. R. China 3. College of Computer Science and Technology, Harbin Engineering University, Harbin, 150001, P. R. China
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Abstract: | Let Q G denote the signless Laplacian matrix of a graph G. An eigenvalue μ of Q G is said to be a main Q-eigenvalue of G if μ has an eigenvector which is not orthogonal to an all-ones vector e. We give some basic properties of main Q-eigenvalues. For a graph G of order n, G is called Q-controllable if G has n distinct main Q-eigenvalues. We show that a graph H is generalized Q-cospectral with a Q-controllable G if and only if H is Q-controllable and there exists a unique rational orthogonal matrix R such that R e = e, Q H = R ? Q G R. |
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