  Twin vertices in simple unweighted graphs are vertices that have the same
neighbours and, in the case of weighted graphs with possible loops, the
corresponding incident edges have equal weights. In this paper, we explore the
role of twin vertices in quantum state transfer. In particular, we provide
characterizations of periodicity, perfect state transfer, and pretty good state
transfer between twin vertices in a weighted graph with respect to its
adjacency, Laplacian and signless Laplacian matrices. As an application, we
provide characterizations of all simple unweighted double cones on regular
graphs that exhibit periodicity, perfect state transfer, and pretty good state
transfer.
