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

Tetsuji Taniguchi

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 defined a new generalization of line graphs which depends on a family of isomorphism classes of graphs with a distinguished coclique. 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.


Keywords


Generalized line graph; Spectrum

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DOI: https://doi.org/10.26493/1855-3974.35.fe9

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications