The least eigenvalue of graphs whose complements are unicyclic

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 $n\mathrm{/}2$. 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 ${\mathcal{U}}_n^c$ of graphs whose complements are unicyclic), and characterize the unique minimizing graph in ${\mathcal{U}}_n^c$ when $n \geq 20$.