  The spectral gap of the graph Laplacian with Dirichlet boundary conditions is
computed for the graphs of several communication networks at the IP-layer,
which are subgraphs of the much larger global IP-layer network. We show that
the Dirichlet spectral gap of these networks is substantially larger than the
standard spectral gap and is likely to remain non-zero in the infinite graph
limit. We first prove this result for finite regular trees, and show that the
Dirichlet spectral gap in the infinite tree limit converges to the spectral gap
of the infinite tree. We also perform Dirichlet spectral clustering on the
IP-layer networks and show that it often yields cuts near the network core that
create genuine single-component clusters. This is much better than traditional
spectral clustering where several disjoint fragments near the periphery are
liable to be misleadingly classified as a single cluster. Spectral clustering
is often used to identify bottlenecks or congestion; since congestion in these
networks is known to peak at the core, our results suggest that Dirichlet
spectral clustering may be better at finding bona-fide bottlenecks.
