  In this paper, we study five-dimensional Dirac fermions of which
extra-dimension is compactified on quantum graphs. We find that there is a
non-trivial correspondence between matrices specifying boundary conditions at
the vertex of the quantum graphs and zero-dimensional Hamiltonians in gapped
free-fermion systems. Based on the correspondence, we provide a complete
topological classification of the boundary conditions in terms of
non-interacting fermionic topological phases. The ten symmetry classes of
topological phases are fully identified in the language of five-dimensional
Dirac fermions, and topological numbers of the boundary conditions are given.
In analogy with the bulk-boundary correspondence in non-interacting fermionic
topological phases, the boundary condition topological numbers predict
four-dimensional massless fermions localized at the vertex of the quantum
graphs and thus govern the low energy physics in four dimensions.
