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In this note, we present a structural description of certain connected cographs having k ≥ 2 main signless Laplacian eigenvalues. This result allows us to characterize the cographs which are quasi-threshold graphs with two main Q-eigenvalues. In addition, we describe all the quasi-threshold graphs belonging to the subclass of generalized core-satellite graphs with k ≥ 2 main Q-eigenvalues.
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Jones et al. (2024) studied this question.
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