Summary: We study the uniform random graph \(C_n\) with \(n\) vertices drawn from a subcritical class of connected graphs. Our main result is that the rescaled graph \(C_n/\sqrt{n}\) converges to the Brownian continuum random tree \(\mathcal{T}_e\) multiplied by a constant scaling factor that depends on the class under consideration. In addition, we provide subgaussian tail bounds for the diameter \(D(C_n)\) and height \(H(C_n^\bullet)\) of the rooted random graph \(C_n^\bullet\). We give analytic expressions for the scaling factor of several classes, including for example the prominent class of outerplanar graphs. Our methods also enable us to study first passage percolation on \(C_n\), where we show the convergence to \(\mathcal{T}_e\) under an appropriate rescaling.
For the entire collection see [Zbl 1333.05004].