Scaling limits of random graphs from subcritical classes

We study the uniform random graph Cₙ with n vertices drawn from a subcritical class of connected graphs. Our main result is that the rescaled graph Cₙ/√n converges to the Brownian continuum random tree Tₑ multiplied by a constant scaling factor that depends on the class under consideration. In addition, we provide sub-Gaussian tail bounds for the diameter D(Cₙ) and height H(Cₙ•) of the rooted random graph Cₙ•. We give analytic expressions for the scaling factor in several cases, including for example the class of outerplanar graphs. Our methods also enable us to study first passage percolation on Cₙ, where we also show the convergence to Tₑ under an appropriate rescaling.