  In this work we perform a detailed statistical analysis of topological and
spectral properties of random geometric graphs (RGGs); a graph model used to
study the structure and dynamics of complex systems embedded in a two
dimensional space. RGGs, $G(n,\ell)$, consist of $n$ vertices uniformly and
independently distributed on the unit square, where two vertices are connected
by an edge if their Euclidian distance is less or equal than the connection
radius $\ell \in [0,\sqrt{2}]$. To evaluate the topological properties of RGGs
we chose two well-known topological indices, the Randi\'c index $R(G)$ and the
harmonic index $H(G)$. While we characterize the spectral and eigenvector
properties of the corresponding randomly-weighted adjacency matrices by the use
of random matrix theory measures: the ratio between consecutive eigenvalue
spacings, the inverse participation ratios and the information or Shannon
entropies $S(G)$. First, we review the scaling properties of the averaged
measures, topological and spectral, on RGGs. Then we show that: (i) the
averaged--scaled indices, $\left\langle R(G) \right\rangle$ and $\left\langle
H(G) \right\rangle$, are highly correlated with the average number of
non-isolated vertices $\left\langle V_\times(G) \right\rangle$; and (ii)
surprisingly, the averaged--scaled Shannon entropy $\left\langle S(G)
\right\rangle$ is also highly correlated with $\left\langle V_\times(G)
\right\rangle$. Therefore, we suggest that very reliable predictions of
eigenvector properties of RGGs could be made by computing topological indices.
