In this paper, we show that all fat Hoffman graphs with smallest eigenvalue at least -1-\tau, where \tau is the golden ratio, can be described by a finite set of fat (-1-\tau)-irreducible Hoffman graphs. In the terminology of Woo and Neumaier, we mean that every fat Hoffman graph with smallest eigenvalue at least -1-\tau is an H-line graph, where H is the set of isomorphism classes of maximal fat (-1-\tau)-irreducible Hoffman graphs. It turns out that there are 37 fat (-1-\tau)-irreducible Hoffman graphs, up to isomorphism.
1 Introduction P. J. Cameron, J. M. Goethals, J. J. Seidel, and E. E. Shult [1] characterized graphs whose adjacency matrices have smallest eigenvalue at least -2 by using root systems. Their results revealed that graphs with smallest eigenvalue at least -2 are generalized line graphs, except a finite number of graphs represented by the root system E 8 . Another characterization for generalized line graphs were given by D. Cvetković, M. Doob, and S. Simić [3] by determinig minimal forbidden subgraphs (see also [4]). Note that graphs with smallest eigenvalue greater than -2 were studied by A. J. Hoffman [5].
Hoffman [6] also studied graphs whose adjacency matrices have smallest eigenvalue at least -1 -√ 2 by using a technique of adding cliques to graphs. R. Woo and A. Neumaier [12] formulated Hoffman’s idea by introducing the notion of Hoffman graphs. A Hoffman graph is a simple graph with a distinguished independent set of vertices, called fat vertices, which can be considered as cliques of size infinity in a sense (see Definition 2.1, and also [8,Corollary 2.15]). To deal with graphs with bounded smallest eigenvalue, Woo and Neumaier introduced a generalization of line graphs by considering decompositions of Hoffman graphs. They gave a characterization of graphs with smallest eigenvalue at least -1 -√ 2 in terms of Hoffman graphs by classifying fat indecomposable Hoffman graphs with smallest eigenvalue at least -1 -√ 2. This led them to prove a theorem which states that every graph with smallest eigenvalue at least -1 -√ 2 and sufficiently large minimum degree is a subgraph of a Hoffman graph admitting a decomposition into subgraphs isomorphic to only four Hoffman graphs. In the terminology of [12], this means that every graph with smallest eigenvalue at least -1-√ 2 and sufficiently large minimum degree is an H-line graph, where H is the set of four isomorphism classes of Hoffman graphs. For further studies on graphs with smallest eigenvalue at least -1 -√ 2, see the papers by T. Taniguchi [10,11] and by H. Yu [13].
Recently, H. J. Jang, J. Koolen, A. Munemasa, and T. Taniguchi [8] made the first step to classify the fat indecomposable Hoffman graphs with smallest eigenvalue -3. However, it seems that there are so many such Hoffman graphs. A key to solve this problem is the notion of special graphs introduced by Woo and Neumaier. A special graph is an edge-signed graph defined for each Hoffman graph. Although non-isomorphic Hoffman graphs may have isomorphic special graphs, it is not difficult to recover all the Hoffman graphs with a given special graph in some cases.
In this paper, we introduce irreducibility of Hoffman graphs and classify fat (-1τ )-irreducible Hoffman graphs, where τ :
is the golden ratio. This is a somewhat more restricted class of Hoffman graphs than those considered in [8], and there are only 37 such Hoffman graphs. As a consequence, every fat Hoffman graph with smallest eigenvalue at least -1-τ is a subgraph of a Hoffman graph admitting a decomposition into subgraphs isomorphic to only 18 Hoffman graphs. In the terminology of [12], this means that every fat Hoffman graph with smallest eigenvalue at least -1τ is an H-line graph, where H is the set of 18 isomorphism classes of maximal fat (-1τ )-irreducible Hoffman graphs.
2.1 Hoffman graphs and eigenvalues Definition 2.1. A Hoffman graph H is a pair (H, µ) of a graph H and a vertex labeling µ : V (H) → {slim, fat} satisfying the following conditions: (i) every vertex with label fat is adjacent to at least one vertex with label slim; (ii) the vertices with label fat are pairwise non-adjacent.
Let
, and E(H) := E(H). We call a vertex in V s (H) a slim vertex, and a vertex in V f (H) a fat vertex of H. We represent a Hoffman graph H also by the triple
For a vertex x of a Hoffman graph H, we define N f H (x) (resp. N s H (x)) to be the set of fat (resp. slim) neighbors of x in H. The set of all neighbors of x is denoted by
(see Figure 1).
In particular, an irreducible Hoffman graph is indecomposable.
Example 2.13. For a non-negative integer t, let K 1,t be the connected Hoffman graph having exactly one slim vertex and t fat vertices, i.e.,
Then K 1,t is irreducible and λ min (K
The Hoffman graph H IV is indecomposable but reducible. Indeed, it is clear that H IV is indecomposable. Let H ′ be the Hoffman graph obtained from H IV by adding a new fat vertex f 3 and two edges {v 1 , f 3 } and {v 2 , f 3 }. The Hoffman graph H ′ is the sum of two copies of H II , where the newly added fat vertex is shared by both copies, that is, H ′ is decomposable. Furthermore,
Proof. Let G be a slim graph with maximum degree k. We define a Hoffman graph H by adding a fat vertex for each edge e of G and joining it to the two end vertices of e. Note that G is the slim subgraph of H. For each slim vertex
Definition 2.17. Let H be a family of isomorphism classes of Hoffman graphs. An H-line graph is an induced Hoffman subgraph of a Hoffman grap
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