Edge-signed graphs with smallest eigenvalue greater than −2

Abstract

Dedicated to Alan J. Hoffman on the occasion of his ninetieth birthday.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.

On the smallest eigenvalues of the line graphs of some trees

“…. , d k )) has also been determined by Rojo and Jiménez [8] using a different method, but our result gives more concrete information about the smallest eigenvalue.…”

“…Indeed, as pointed out in [7,Remark 9], if H is the line graph of the Dynkin diagram E 6 , then choosing the vertex r appropriately, we can make χ H and χ H−v to have a common smallest zero. In this case, the smallest eigenvalue of G is a common zero of …”

A note on the least (normalized) laplacian eighva;ue of signed graphs

“…In [2], some parameters called frustration number and frustration index measuring how far a signed graph is to be balanced have been investigated in terms of the least eigenvalue of Laplacian of signed graph. More results on the spectra of signed graphs can be found in [1,2,3,6,11,12,16,13,14,5,7,8].…”

Graphs with least eigenvalue −2: Ten years on

“…Recently Greaves et al [53] proved Hoffman's conjecture (with an extension to signed graphs); another proof is given in [79]. This effectively deals with limit points greater than −2.…”

“…Those with λ(G σ ) > −2 were determined by the authors of [53], using representations in E 8 . They are listed in the Appendix to [53], and we can summarize the results as follows.…”