  It is a classic result in spectral theory that the limit distribution of the
spectral measure of random graphs G(n, p) converges to the semicircle law in
case np tends to infinity with n. The spectral measure for random graphs G(n,
c/n) however is less understood. In this work, we combine and extend two
combinatorial approaches by Bauer and Golinelli (2001) and Enriquez and Menard
(2016) and approximate the moments of the spectral measure by counting walks
that span trees.
