the $\Lambda \Lambda$ sector of NF, and then the strength of the core part $(i=3)$ is adjusted so as to reproduce the experimental value of $B_{\Lambda \Lambda}(^{\ 6}_{\Lambda \Lambda}$He). The obtained values of parameters are given in Table IV. It is interesting that the resulting $\Lambda \Lambda$ interaction is almost equal to the interaction obtained by multiplying a factor 0.5 on the above ND-simulating interaction employed in Ref.[@Hiyama97].

Results and Discussions
=======================

Let us show the calculated results for a series of double-$\Lambda$ hypernuclei with $\alpha+x+ \Lambda +\Lambda$ structures ($x=0, n,p,d,t,^3{\rm He}, \alpha)$ studied in the microscopic four-body cluster model. In order to understand the role of two $\Lambda$ particles attached to the core nuclei, it is useful to compare the obtained level structures of the $\alpha+x+ \Lambda + \Lambda$ double-$\Lambda$ hypernuclei with those of the $\alpha+x$ nuclei and the $\alpha+x+ \Lambda$ single-$\Lambda$ hypernuclei. Then, we can see clearly how the ground and excited states of $\alpha+x$ nuclei are changed due to the participation of $\Lambda$ particles. It should be noted again here that, in the model description of $\alpha+x+ \Lambda +\Lambda$, the observed low-energy properties of the $\alpha+x$ nuclei and the existing $\Lambda$-binding energies of the $x+ \Lambda$ and 