  We give a structural classification of edge-signed graphs with smallest
eigenvalue greater than -2. We prove a conjecture of Hoffman about the smallest
eigenvalue of the line graph of a tree that was stated in the 1970s.
Furthermore, we prove a more general result extending Hoffman's original
statement to all edge-signed graphs with smallest eigenvalue greater than -2.
Our results give a classification of the special graphs of fat Hoffman graphs
with smallest eigenvalue greater than -3.
