  We investigate the statistics of the largest eigenvalue, $\lambda_{\rm max}$,
in an ensemble of $N\times N$ large ($N\gg 1$) sparse adjacency matrices,
$A_N$. The most attention is paid to the distribution and typical fluctuations
of $\lambda_{\rm max}$ in the vicinity of the percolation threshold,
$p_c=\frac{1}{N}$. The overwhelming majority of subgraphs representing $A_N$
near $p_c$ are exponentially distributed linear subchains, for which the
statistics of the normalized largest eigenvalue can be analytically connected
with the Gumbel distribution. For the ensemble of {\rm all} subgraphs near
$p_c$ we suggest that under an appropriate modification of the normalization
constant the Gumbel distribution provides a reasonably good approximation.
Using numerical simulations we demonstrate that the proposed transformation of
$\lambda_{\rm max}$ is indeed Gumbel-distributed and the leading finite-size
corrections in the vicinity of $p_c$ scale with $N$ as $\sim \ln^{-2}N$. All
together, our results reveal a previously unknown universality in eigenvalue
statistics of sparse matrices close to the percolation threshold.
