  In order to obtain perfect state transfer between two sites in a network of
interacting qubits, their corresponding vertices in the underlying graph must
satisfy a combinatorial property called strong cospectrality. Here we determine
the structure of graphs containing pairs of vertices which are strongly
cospectral and satisfy a certain extremal property related to the spectrum of
the graph. If the graph satisfies this property globally and is regular, we
also show that the existence of a partition of the vertex set into pairs of
vertices at maximum distance admitting perfect state transfer forces the graph
to be distance-regular.
