Entanglement entropy on the Cayley tree

The properties of the entanglement entropy (EE) of a clean and disordered Cayley tree (CT) are studied. The EE shows a completely different behavior depending on the way the CT is partitioned into two regions and whether we consider the ground-state or highly excited many-particle wave function. For a clean CT the ground-state EE increases logarithmically as a function of the number of generation if a single branch is pruned off the tree, while it grows exponentially if the region around the root is trimmed. On the other hand, in both cases the highly excited states' EE grows exponentially. In the presence of disorder the exponential behavior is preserved only for the latter dissection. The implications of these results to general graphs and disordered systems are shortly discussed.