  We study the structure-stability relationship of the Lennard-Jones (LJ)
clusters from a point of view of vibrations. By assuming the size up to
$N=1610$, we demonstrate that the $N$-dependence of the maximum vibrational
frequency reflects the geometry of the core (the interior of cluster) that will
determine the overall geometry of the cluster. This allows us to identify the
formation of non-icosahedral structures for $N\le 150$, the vacancy formation
at the core for $N\ge 752$, and the transition from icosahedral to decahedral
structures at $N = 1034$. We apply the maximum frequency analysis to classify
metastable clusters for $19\le N \le 39$, where transformation pathways between
different structures are visualized, and the energy barrier height is estimated
simultaneously.
