We consider the enumeration of connected multitype (colored) graphs with prescribed vertex profile and edge matrix, in the sparse regime where the number of edges and the excess are proportional to the size of the graph. This extends the classical result of Bender, Canfield, and McKay for connected sparse graphs to the multitype setting. Our approach is probabilistic: we establish that a connected multitype graph with prescribed statistics can be identified, with high probability, as the giant component of an appropriately tuned supercritical inhomogeneous random graph. Combining this correspondence with large-deviation estimates for the connectivity probability of inhomogeneous random graphs, we derive the leading exponential asymptotics of the corresponding enumeration problem. We will outline the proof strategy and discuss the connectivity constraints that arise specifically in the multitype setting. This is a joint work with M. Veshaj (WIAS- Berlin).
Luisa Andreis is Associate Professor at the Department of Mathematics (Giuseppe Peano), University of Torino, in Italy. She received her PhD from the University of Padova, and held positions as a postdoc at the Weierstrass Institute in Berlin and as Assistant Professor at the University of Florence and Politecnico di Milano. Her research focuses on probability theory, with an emphasis on random graphs, large deviations, and interacting particle systems, and, most recently, on mathematical models for artificial intelligence.