  We present experimental results for the density of states (DOS) of a
superconducting microwave Dirac billiard which serves as an idealized model for
the electronic properties of graphene. The DOS exhibits two sharp peaks which
evolve into van Hove singularities with increasing system size. They divide the
band structure into regions governed by the \emph{relativistic} Dirac equation
and by the \emph{non-relativistic} Schr\"odinger equation, respectively. We
demonstrate that in the thermodynamic limit a topological transition appears as
a neck-disrupting Lifshitz transition in the number susceptibility and as an
excited state transition in the electronic excitations. Furthermore, we recover
the finite-size scaling typical for excited state quantum phase transitions
involving logarithmic divergences and identify a quasi-order parameter.
