  We consider sparse inhomogeneous Erd\H{o}s-R\'enyi random graph ensembles
where edges are connected independently with probability $p_{ij}$. We assume
that $p_{ij}= \varepsilon_N f(w_i, w_j)$ where $(w_i)_{i\ge 1}$ is a sequence
of deterministic weights, $f$ is a bounded function and $N\varepsilon_N\to
\lambda\in (0,\infty)$. We characterise the limiting moments in terms of graph
homomorphisms and also classify the contributing partitions. We present an
analytic way to determine the Stieltjes transform of the limiting measure. The
convergence of the empirical distribution function follows from the theory of
local weak convergence in many examples but we do not rely on this theory and
exploit combinatorial and analytic techniques to derive some interesting
properties of the limit. We extend the methods of Khorunzhy et al. (2004) and
show that a fixed point equation determines the limiting measure. The limiting
measure crucially depends on $\lambda$ and it is known that in the homogeneous
case, if $\lambda\to\infty$, the measure converges weakly to the semicircular
law (Jung and Lee (2018)). We extend this result of interpolating between the
sparse and dense regimes to the inhomogeneous setting and show that as
$\lambda\to \infty$, the measure converges weakly to a measure which is known
as the operator-valued semicircular law.
