Fat Hoffman graphs with smallest eigenvalue at least −1 − τ

Abstract

In this paper, we show that all fat Hoffman graphs with smallest eigenvalue at least −1−τ , where τ is the golden ratio, can be described by a finite set of fat (−1 − τ )-irreducible Hoffman graphs. In the terminology of Woo and Neumaier, we mean that every fat Hoffman graph with smallest eigenvalue at least −1−τ is an H-line graph, where H is the set of isomorphism classes of maximal fat (−1−τ )-irreducible Hoffman graphs. It turns out that there are 37 fat (−1−τ )-irreducible Hoffman graphs, up to isomorphism.

Edge-signed graphs with smallest eigenvalue greater than −2

“…A Hoffman graph H is defined as a graph (V, E) with a distinguished coclique V f (H) ⊂ V called fat vertices, the remaining vertices V s (H) = V \V f (H) are called slim vertices. For more background on Hoffman graphs see some of the authors' previous papers [14,16,17]. Let H be a Hoffman graph and suppose its adjacency matrix A has the following form…”

Fat Hoffman graphs with smallest eigenvalue greater than −3

“…By assumption, we have Proof. By [11,Lemma 3.11], the number of (−)-edges in T cannot be two. Therefore, the number of (−)-edges in T is one or three.…”

“…In 1995, R. Woo and A. Neumaier [14] formulated Hoffman's idea by introducing the notion of Hoffman graphs and generalizations of line graphs. Hoffman graphs were subsequently studied in [9,11,12,13,15]. In particular, H. J. Jang, J. Koolen, A. Munemasa, and T. Taniguchi [9] proposed a scheme to classify fat indecomposable Hoffman graphs with smallest eigenvalue at least −3.…”

Problems on Graphs with Fixed Smallest Eigenvalue

“…2 . In 2014, Munemasa, Sano and Taniguchi [76] found that there are exactly 18 maximal (−1 − τ )-irreducible Hoffman graphs, up to isomorphism, and they also gave a list of these 18 Hoffman graphs.…”