  We study perfect state transfer on quantum networks represented by weighted
graphs. Our focus is on graphs constructed from the join and related graph
operators. Some specific results we prove include: (1) The join of a weighted
two-vertex graph with any regular graph has perfect state transfer. This
generalizes a result of Casaccino et al. [clms09] where the regular graph is a
complete graph or a complete graph with a missing link. In contrast, the
half-join of a weighted two-vertex graph with any weighted regular graph has no
perfect state transfer. This implies that adding weights in a complete
bipartite graph do not help in achieving perfect state transfer. (2) A Hamming
graph has perfect state transfer between each pair of its vertices. This is
obtained using a closure property on weighted Cartesian products of perfect
state transfer graphs. Moreover, on the hypercube, we show that perfect state
transfer occurs between uniform superpositions on pairs of arbitrary subcubes.
This generalizes results of Bernasconi et al. [bgs08] and Moore and Russell
[mr02]. Our techniques rely heavily on the spectral properties of graphs built
using the join and Cartesian product operators.
