We will be interested by the rank of the adjacency matrix of a random
diluted graph (i.e. when the degree is of a typical vertex is of order
1). This question has been previously considered by Costello, Tao and
Vu for dense and sparse graphs. We will exhibit a formula for the
asymptotic rank as the size of the graph grows to infinity.  This
problem is closely related to the size of the maximal matching of a
graph. Our work gives a new proof of a well-known result by Karp and
Sipser on maximal matching in Erdos-Renyi graphs.
This is a joint work with Marc Lelarge (ENS & INRIA).