  We give a new formula for computing the isospectral reduction of a matrix
(and graph) down to a submatrix (or subgraph). Using this, we generalize the
notion of isospectral reductions. In addition, we give a procedure for
constructing a matrix whose isospectral reduction down to a submatrix is given.
We also prove that the isospectral reduction completely determines the
restriction of the quantum walk transition matrix to a subset. Using these, we
construct new families of simple graphs exhibiting perfect quantum state
transfer.
