  In this text, we consider an N by N random matrix X such that all but o(N)
rows of X have W non identically zero entries, the other rows having lass than
$W$ entries (such as, for example, standard or cyclic band matrices). We always
suppose that 1 << W << N. We first prove that if the entries are independent,
centered, have variance one, satisfy a certain tail upper-bound condition and W
>> (\log N)^{6(1+\alpha)}, where \alpha is a positive parameter depending on
the distribution of the entries, then the largest eigenvalue of X/\sqrt{W}
converges to the upper bound of its limit spectral distribution, that is 2, as
for Wigner matrices. This extends some previous results by Khorunzhiy and Sodin
where less hypotheses were made on W, but more hypotheses were made about the
law of the entries and the structure of the matrix. Then, under the same
hypotheses, we prove a delocalization result for the eigenvectors of X,
precisely that most of them cannot be essentially localized on less than
W/\log(N) entries. This lower bound on the localization length has to be
compared to the recent result by Steinerberger, which states that either the
localization length in the edge is << W^{7/5} or there is strong interaction
between two eigenvectors in an interval of length W^{7/5}.
