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.

Abstract In this paper we discuss spectral properties of operators associated with the least-squares finite-element approximation elliptic partial differential equations. The convergence discrete eigenvalues and eigenfunctions towards corresponding continuous eigenmodes is studied analyzed help appropriate $L^2$ error estimates. A priori a posteriori estimates are proved.