Perron-Wright asymptotics and critical component processes for regular inhomogeneous random graphs

We extend Wright's enumeration of connected labelled graphs with fixed surplus to irreducible critical finite type kernels and regular compact type kernels. Let $C_t(k,r)$ denote the total weight of connected labelled graphs with $k$ vertices and surplus $r$. Let $u>0$ satisfy $Ku=u$ and $\int u\,dμ