  A graph in a certain graph class is called minimizing if the least eigenvalue
of the adjacency matrix of the graph attains the minimum among all graphs in
that class. Bell {\it et al.} have characterized the minimizing graphs in the
class of connected graphs of order $n$ and size $m$, whose complements are
either disconnected or contain a clique of order at least $n/2$. In this paper
we discuss the minimizing graphs of a special class of graphs of order $n$
whose complements are connected and contains exactly one cycle (namely the the
class $\mathscr {U}^{c}_{n}$ of graphs whose complements are unicyclic), and
characterize the unique minimizing graph in $\mathscr {U}^{c}_{n}$ when $n \geq
20$.
