  Mutualistic networks are used to study the structure and processes inherent
to mutualistic relationships. In this paper, we introduce a random matrix
ensemble (RME) representing the adjacency matrices of mutualistic networks
composed by two vertex sets of sizes $n$ and $m-n$. Our RME depends on three
parameters: the network size $n$, the size of the smaller set $m$, and the
connectivity between the two sets $\alpha$, where $\alpha$ is the ratio of
current adjacent pairs over the total number of possible adjacent pairs between
the sets. We focus on the the spectral, eigenvector and topological properties
of the RME by computing, respectively, the ratio of consecutive eigenvalue
spacings $r$, the Shannon entropy of the eigenvectors $S$, and the Randi\'c
index $R$. First, within a random matrix theory approach (i.e., a statistical
approach), we identify a parameter $\xi\equiv\xi(n,m,\alpha)$ that scales the
average normalized measures $\left< \overline{X} \right>$ (with $X$
representing $r$, $S$ and $R$). Specifically, we show that (i) $\xi\propto
\alpha n$ with a weak dependence on $m$, and (ii) for $\xi<1/10$ most vertices
in the mutualistic network are isolated, while for $\xi>10$ the network
acquires the properties of a complete network, i.e., the transition from
isolated vertices to a complete-like behavior occurs in the interval
$1/10<\xi<10$. Then, we demonstrate that our statistical approach predicts
reasonably well the properties of real-world mutualistic networks; that is, the
universal curves $\left< \overline{X} \right>$ vs. $\xi$ show good
correspondence with the properties of real-world networks.
