  In this paper, we discuss various connections between the smallest eigenvalue
of the adjacency matrix of a graph and its structure. There are several
techniques for obtaining upper bounds on the smallest eigenvalue, and some of
them are based on Rayleigh quotients, Cauchy interlacing using induced
subgraphs, and Haemers interlacing with vertex partitions and quotient
matrices. In this paper, we are interested in obtaining lower bounds for the
smallest eigenvalue. Motivated by results on line graphs and generalized line
graphs, we show how graph decompositions can be used to obtain such lower
bounds.
