  The celebrated Mar\v{c}enko-Pastur law, that considers the asymptotic
spectral density of random covariance matrices, has found a great number of
applications in physics, biology, economics, engineering, among others. Here,
using techniques from statistical mechanics of spin glasses, we derive simple
formulas concerning the spectral density of generalized diluted Wishart
matrices. These are defined as $\bm{F}\equiv \frac{1}{2d}\left( \bm{X}\bm{Y}^T+
\bm{Y}\bm{X}^T\right)$, where $\bm{X}$ and $\bm{Y}$ are diluted $N\times P$
rectangular matrices, whose entries correspond to the links of doubly-weighted
random bipartite Poissonian graphs following the distribution
$P(x_{i}^\mu,y_{i}^{\mu})=\frac{d}{N}\varrho(x_{i}^{\mu},y_{i}^{\mu})+\left(1-\frac{d}{N}\right)\delta_{x_{i}^{\mu},0}\delta_{y_{i}^{\mu},0}$,
with the probability density $\varrho(x,y)$ controlling the correlation between
the matrices entries of $\bm{X}$ and $\bm{Y}$. Our results cover several
interesting cases by varying the parameters of the matrix ensemble, namely, the
dilution of the graph $d$, the rectangularity of the matrices $\alpha=N/P$, and
the degree of correlation of the matrix entries via the density $\varrho(x,y)$.
Finally, we compare our findings to numerical diagonalisation showing excellent
agreement.
