The problem of identifying those simple, undirected graphs with n vertices and k edges that have the smallest minimum eigenvalue of the adjacency matrix is considered. Several general properties of the minimizing graphs are described. These strongly suggest bipartition, to the extent possible for the number of edges. In the bipartite case, the precise structure of the minimizing graphs is given for a number of infinite classes. (C) 2012 Elsevier B.V. All rights reserved.
Johnson, C. R., & Sawikowska, A. (2012). Minimizing the least eigenvalue of graphs with fixed order and size. Discrete Mathematics, 312(15), 2272-2285.