first (trivial) answer: the spectrum of a bipartite graph is symmetric wrt to 0; hence, +1 is an eigenvalue iff -1 is an eigenvalue.

second (trivial) answer: an individual edge has eigenvalue +1 (and hence also -1).

further (much less trivial) answers: take a look at [this article][1] by chan and godsil, where several conditions answering your questions are presented, e.g. at page 76. in a slightly different version you can find on the internet (google is your friend), the same authors show that -1 is an eigenvalue if the graph has a perfect 1-code. also, in [this monograph][2] you can find some relevant information, e.g. the observation that +1 is an eigenvalue of so-called collinearity graphs.


  [1]: http://books.google.de/books?id=-tIaXdII8egC&lpg=PA75&ots=xGMs5wwI4Q&dq=chan%2520godsil%2520symmetry%2520and%2520eigenvectors&lr&hl=de&pg=PA75#v=onepage&q=chan%2520godsil%2520symmetry%2520and%2520eigenvectors&f=false
  [2]: http://www.win.tue.nl/~aeb/2WF02/spectra.pdf