  The nodal and effectively relativistic dispersion featuring in a range of
novel materials including two- dimensional graphene and three-dimensional Dirac
and Weyl semimetals has attracted enormous interest during the past decade.
Here, by studying the structure and symmetry of the diagrammatic expansion, we
show that these nodal touching points are in fact perturbatively stable to all
orders with respect to generic two-body interactions. For effective low-energy
theories relevant for single and multilayer graphene, type-I and type-II Weyl
and Dirac semimetals as well as Weyl points with higher topological charge,
this stability is shown to be a direct consequence of a spatial symmetry that
anti-commutes with the effective Hamiltonian while leaving the interaction
invariant. A more refined argument is applied to the honeycomb lattice model of
graphene showing that its Dirac points are also perturbatively stable to all
orders. We also give examples of nodal Hamiltonians that acquire a gap from
interactions as a consequence of symmetries different from those of Weyl and
Dirac materials.
