  Random unimodular lattice triangulations have been recently used as an
embedded random graph model, which exhibit a crossover behaviour between an
ordered, large-world and a disordered, small-world behaviour. Using the ergodic
Pachner flips that transform such triangulations into another and an energy
functional that corresponds to the degree distribution variance, Markov chain
Monte-Carlo simulations can be applied to study these graphs. Here, we consider
the spectra of the adja cency and the Laplacian matrix as well as the algebraic
connectivity and the spectral radius. Power law dependencies on the system size
can clearly be identified and compared to analytical solutions for periodic
ground states. For random triangulations we find a qualitative agreement of the
spectral properties with well-known random graph models. In the microcanonical
ensemble analytical approximations agree with numerical simulations. In the
canonical ensemble a crossover behaviour can be found for the algebraic
connectivity and the spectral radius, thus combining large-world and
small-world behavior in one model. The considered spectral properties can be
applied to transport problems on triangulation graphs and the crossover
behaviour allows a tuning of important transport quantities.
