  We investigate the rank of the adjacency matrix of large diluted random
graphs: for a sequence of graphs $(G_n)_{n\geq0}$ converging locally to a
Galton--Watson tree $T$ (GWT), we provide an explicit formula for the
asymptotic multiplicity of the eigenvalue 0 in terms of the degree generating
function $\phi_*$ of $T$. In the first part, we show that the adjacency
operator associated with $T$ is always self-adjoint; we analyze the associated
spectral measure at the root and characterize the distribution of its atomic
mass at 0. In the second part, we establish a sufficient condition on $\phi_*$
for the expectation of this atomic mass to be precisely the normalized limit of
the dimension of the kernel of the adjacency matrices of $(G_n)_{n\geq 0}$. Our
proofs borrow ideas from analysis of algorithms, functional analysis, random
matrix theory and statistical physics.
