We analyze the largest eigenvalue and eigenvector for the adjacency
matrices of sparse random graph.
Let $\lambda_1$ be the largest eigenvalue of
an $n$-vertex graph, and $v_1$ be its corresponding
normalized eigenvector.
For graphs of average degree $d\log n$,
where $d$ is a large enough constant,
we show $\lambda_1 = d\log n + 1 \pm o(1)$
and $\langle \mathbf{1}, v_1 \rangle = \sqrt{n}
\left( 1- \Theta \left(\frac{1}{\log n} \right) \right)$.
It shows a limitation of the existing method of 
analyzing spectral algorithms for NP-hard problems.