On the First Eigenvalue of Bipartite Graphs
  In this paper we study the maximum value of the largest eigenvalue for simple bipartite graphs, where the number of edges is given and the number of vertices on each side of the bipartition is given. We state a conjectured solution, which is an analog of the Brualdi- Hoffman conjecture for general graphs, and prove the conjecture in some special cases. 
Amitava Bhattacharya, Shmuel Friedland and Uri N. Peled