Notes on exceptional signed graphs

Abstract

A connected signed graph is called exceptional if it has a representation in the root system E 8 , but has not in any D k . In this study we obtain some properties of these signed graphs, mostly expressed in terms of those that are maximal with a fixed number of eigenvalues distinct from −2 . As an application, we characterize exceptional signed graphs with exactly 2 eigenvalues. In some particular cases, we prove the (non-)existence of such signed graphs.