  The spectral theory of the Laplace differential operator for biregular
quantum graphs is developed. Trees are studied in detail. Generating functions
for closed non backtracking walks appear when resolvents for trees are related
to resolvents for biregular graphs they cover. The relationship between
resolvent traces for finite graphs and walk generating functions is especially
productive. A detailed description of the rational extension of nonbacktracking
walk generating functions is presented.
