Edge-signed graphs with smallest eigenvalue greater than -2

Abstract

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.

25 pages

References in corpus (2)

• Fat Hoffman graphs with smallest eigenvalue greater than -3
• On the KNS Conjecture in type $E$

Cited by in corpus (6)

• On the smallest eigenvalues of the line graphs of some trees
• Recent progress on graphs with fixed smallest eigenvalue
• Forbidden induced subgraphs for graphs and signed graphs with eigenvalues bounded from below
• Baskets and fibred links realizing $A_n$
• Signed analogue of line graphs and their smallest eigenvalues
• Cohen-Macaulay and Gorenstein properties under the amalgamated construction