  We consider random geometric graphs on the plane characterized by a
non-uniform density of vertices. In particular, we introduce a graph model
where $n$ vertices are independently distributed in the unit disc with
positions, in polar coordinates $(l,\theta)$, obeying the probability density
functions $\rho(l)$ and $\rho(\theta)$. Here we choose $\rho(l)$ as a normal
distribution with zero mean and variance $\sigma\in(0,\infty)$ and
$\rho(\theta)$ as an uniform distribution in the interval $\theta\in [0,2\pi)$.
Then, two vertices are connected by an edge if their Euclidian distance is less
or equal than the connection radius $\ell$. We characterize the topological
properties of this random graph model, which depends on the parameter set
$(n,\sigma,\ell)$, by the use of the average degree $\left\langle k
\right\rangle$ and the number of non-isolated vertices $V_\times$; while we
approach their spectral properties with two measures on the graph adjacency
matrix: the ratio of consecutive eigenvalue spacings $r$ and the Shannon
entropy $S$ of eigenvectors. First we propose a heuristic expression for
$\left\langle k(n,\sigma,\ell) \right\rangle$. Then, we look for the scaling
properties of the normalized average measure $\left\langle \overline{X}
\right\rangle$ (where $X$ stands for $V_\times$, $r$ and $S$) over graph
ensembles. We demonstrate that the scaling parameter of $\left\langle
\overline{V_\times} \right\rangle=\left\langle V_\times \right\rangle/n$ is
indeed $\left\langle k \right\rangle$; with $\left\langle \overline{V_\times}
\right\rangle \approx 1-\exp(-\left\langle k \right\rangle)$. Meanwhile, the
scaling parameter of both $\left\langle \overline{r} \right\rangle$ and
$\left\langle \overline{S} \right\rangle$ is proportional to $n^{-\gamma}
\left\langle k \right\rangle$ with $\gamma\approx 0.16$.
