Fat Hoffman graphs with smallest eigenvalue greater than −3

Abstract

In this paper, we give a combinatorial characterization of the special graphs of fat Hoffman graphs containing K 1,2 with smallest eigenvalue greater than −3, where K 1,2 is the Hoffman graph having one slim vertex and two fat vertices.

Edge-signed graphs with smallest eigenvalue greater than −2

“…Recently, Jang, Koolen, Munemasa, and Taniguchi [14] proposed a programme to classify fat Hoffman graphs with smallest eigenvalue at least −3. The present work fills a part of this programme, and includes the results of [17].…”

“…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…”

“…where the fat vertices come last. It is shown in [17] that if λ 1 (B(H)) > −3 then every slim vertex is adjacent to at most two fat vertices and at most one slim vertex can be adjacent to more than one fat vertex. It therefore follows that if a fat Hoffman graph H has smallest eigenvalue λ 1 (H) > −3 then the diagonal of B(H)+I consists of at most one −1 entry and the remaining entries 0.…”

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.…”

“…When the fat indecomposable Hoffman graph H has a vertex with exactly two fat neighbours, necessary and sufficient structural conditions for λ(H) > −3 are known from [79,Theorem 4.3]. One consequence is the following result, which should be compared with Theorem 8.1.…”