Using a new local cluster potential, cluster structures around double shell closures have been investigated. The calculations of spectra including electromagnetic transitions are in reasonable agreement with experimental data. Besides the nuclei, 20 Ne, 44 Ti, 60 Zn, and 212 Po, which have been intensively analyzed by cluster models, we have also studied two other good candidates, 52 Ti and 136 Te, for α-cluster structure in neutron-rich regions. In 136 Te, the unexpectedly small B(E2; 0 + → 2 + ) would be due to the cluster characteristic. The new cluster potential without any readjustment of the parameter values has been also applied to the investigations of possible 6 He-cluster structure in nuclei, 22 Ne, 46 Ti, 54 Ti, 62 Zn, and 138 Te, which have the same doubly magic cores as the α-cluster nuclei above. Similarly good results can be obtained, compared with experimental data. It is an advantage that the new cluster potential suits the different types of cluster structures with a set of fixed parameters which do not need any readjustment for different cluster-core structures.