  We consider spectral properties of sparse sample covariance matrices, which
includes biadjacency matrices of the bipartite Erd\H{o}s-R\'enyi graph model.
We prove a local law for the eigenvalue density up to the upper spectral edge.
Under a suitable condition on the sparsity, we also prove that the limiting
distribution of the rescaled, shifted extremal eigenvalues is given by the GOE
Tracy-Widom law with an explicit formula on the deterministic shift of the
spectral edge. For the biadjacency matrix of an Erd\H{o}s-R\'enyi graph with
two vertex sets of comparable sizes $M$ and $N$, this establishes Tracy-Widom
fluctuations of the second largest eigenvalue when the connection probability
$p$ is much larger than $N^{-2/3}$ with a deterministic shift of order
$(Np)^{-1}$.
