  In this paper, we analyze the limiting spectral distribution of the adjacency
matrix of a random graph ensemble, proposed by Chung and Lu, in which a given
expected degree sequence $\overline{w}_n^{^{T}} = (w^{(n)}_1,\ldots,w^{(n)}_n)$
is prescribed on the ensemble. Let $\mathbf{a}_{i,j} =1$ if there is an edge
between the nodes $\{i,j\}$ and zero otherwise, and consider the normalized
random adjacency matrix of the graph ensemble: $\mathbf{A}_n$ $=$ $
[\mathbf{a}_{i,j}/\sqrt{n}]_{i,j=1}^{n}$. The empirical spectral distribution
of $\mathbf{A}_n$ denoted by $\mathbf{F}_n(\mathord{\cdot})$ is the empirical
measure putting a mass $1/n$ at each of the $n$ real eigenvalues of the
symmetric matrix $\mathbf{A}_n$. Under some technical conditions on the
expected degree sequence, we show that with probability one,
$\mathbf{F}_n(\mathord{\cdot})$ converges weakly to a deterministic
distribution $F(\mathord{\cdot})$. Furthermore, we fully characterize this
distribution by providing explicit expressions for the moments of
$F(\mathord{\cdot})$. We apply our results to well-known degree distributions,
such as power-law and exponential. The asymptotic expressions of the spectral
moments in each case provide significant insights about the bulk behavior of
the eigenvalue spectrum.
