  We show that the multiplicity of the second normalized adjacency matrix
eigenvalue of any connected graph of maximum degree $\Delta$ is bounded by $O(n
\Delta^{7/5}/\log^{1/5-o(1)}n)$ for any $\Delta$, and by
$O(n\log^{1/2}d/\log^{1/4-o(1)}n)$ for simple $d$-regular graphs when $d\ge
\log^{1/4}n$. In fact, the same bounds hold for the number of eigenvalues in
any interval of width $\lambda_2/\log_\Delta^{1-o(1)}n$ containing the second
eigenvalue $\lambda_2$. The main ingredient in the proof is a polynomial (in
$k$) lower bound on the typical support of a closed random walk of length $2k$
in any connected graph, which in turn relies on new lower bounds for the
entries of the Perron eigenvector of submatrices of the normalized adjacency
matrix.
