Geodesic
The Least Eigenvalue of Graphs whose Complements Are Unicyclic
Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 2, pp. 249-260

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A graph in a certain graph class is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum among all graphs in that class. Bell et al. have identified a subclass within the connected graphs of order n and size m in which minimizing graphs belong (the complements of such graphs are either disconnected or contain a clique of size 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 class 𝒰_n^c of graphs whose complements are unicyclic), and characterize the unique minimizing graph in 𝒰_n^c when n ≥ 20.
Keywords: unicyclic graph, complement, adjacency matrix, least eigen- value 
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Wang, Yi; Fan, Yi-Zheng; Li, Xiao-Xin; Zhang, Fei-Fei. The Least Eigenvalue of Graphs whose Complements Are Unicyclic. Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 2, pp. 249-260. http://geodesic.mathdoc.fr/item/DMGT_2015_35_2_a4/