The spectral geometry of biregular graphs: a quantum graph approach

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.