summary:In this paper, the upper and lower bounds for the quotient of spectral radius (Laplacian spectral radius, signless Laplacian spectral radius) and the clique number together with the corresponding extremal graphs in the class of connected graphs with $n$ vertices and clique number $\omega $ $(2\leq \omega \leq n)$ are determined. As a consequence of our results, two conjectures given in Aouchiche (2006) and Hansen (2010) are proved.
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