  We explore algebraic and spectral properties of weighted graphs containing
twin vertices that are useful in quantum state transfer. We extend the notion
of adjacency strong cospectrality to arbitrary Hermitian matrices, with focus
on the generalized adjacency matrix and the generalized normalized adjacency
matrix. We then determine necessary and sufficient conditions such that a pair
of twin vertices in a weighted graph exhibits strong cospectrality with respect
to the above-mentioned matrices. We also generalize known results about
equitable and almost equitable partitions, and use these to determine which
joins of the form $X\vee H$, where $X$ is either the complete or empty graph,
exhibit strong cospectrality.
