On graphs with the smallest eigenvalue at least −1 − √2, part I

Abstract

There are many results on graphs with the smallest eigenvalue at least −2. As a next step, A. J. Hoffman proposed to study graphs with the smallest eigenvalue at least −1 − √ 2. In order to deal with such graphs, R. Woo and A. Neumaier introduced the concept of a Hoffman graph, and defined a new generalization of line graphs which depends on a family of Hoffman graphs. They proved a theorem analogous to Hoffman's, using a particular family consisting of four isomorphism classes.In this paper, we deal with a generalization based on a family H smaller than the one which they dealt with, yet including generalized line graphs in the sense of Hoffman. The main result is that the cover of an H -line graph with at least 8 vertices is unique.

Fat Hoffman graphs with smallest eigenvalue greater than −3

“…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.…”

Graphs with least eigenvalue −2: Ten years on

“…When −3 ≤ r < −2 the picture is not yet complete, but for r ≥ −1 − √ 2 forbidden subgraphs ensure that the slim subgraph of H is an F-line graph for a specific family of size 4 [160]; for minimal forbidden subgraphs see [116,117]. Analogous results for r ≥ −1 − τ can be found in [78].…”

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

“…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].…”