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Using the Hartree-Fock plus random-phase-approximation (HF+RPA), we study theimpurity effect of $\Lambda$ hyperon on the collective vibrational excitationsof double-$\Lambda$ hypernuclei. To this end, we employ a Skyrme-type $\LambdaN$ and $\Lambda\Lambda$ interactions for the HF calculations, and the residualinteractions for RPA derived with the same interactions. We find that inclusionof two $\Lambda$ hyperons in $^{16}$O shifts the energy of the collectivestates towards higher energies. In particular, the energy of the giant monopoleresonance of $^{\,\,18}_{\Lambda\Lambda}$O, as well as that of$^{210}_{\Lambda\Lambda}$Pb, becomes larger. This implies that the effectiveincompressibility modulus increases due to the impurity effect of $\Lambda$particle, if the $\beta$-stability condition is not imposed.

Application of random phase approximation to vibrational excitations of double-$\Lambda$ hypernuclei