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(C.n) of the rooted random graph C.n. 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.