For each $N, let G N$ be a simple random graph on the set of vertices $[N ] = {1, 2,. .. , N }$, which is invariant by relabeling of the vertices. The asymptotic behavior as N goes to infinity of certain correlation functions furnishes informations on the asymptotic spectral properties of the adjacency matrix $A N of G N$. Denote by $d N = N × P({i, j} ∈ G N)$ and assume $d N , N − d → N →∞$ . If the correlation functions are small enough, the standardized empirical eigenvalue distribution of A N converges in expectation to the semicircular law and the matrix satisfies asymptotic freeness properties in the sense of free probability theory. We provide such estimates for uniform d N-regular graphs $G N,d N ,$ under the additional assumption that $| N 2 − d N − η √ d N | → N →∞$ for some $η > 0$. Our method applies also for simple graphs whose edges are labelled by i.i.d. random variables.