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In a certain class of graphs, a graph is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum. A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to its number of vertices plus one. Let G1,nc and G2,ncbe the classes of the connected graphs of order n whose complements are bicyclic with exactly two and three cycles, respectively. In this paper, we characterize the unique minimizing graph among all the graphs which belong to Gnc= G1,nc [ G2,nc, a class of the connected graphs of order n whose complements are bicyclic. [ABSTRACT FROM AUTHOR]