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

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.