The uniqueness of covers of widely generalized line graphs

Abstract

As a natural generalization of line graphs, Hoffman line graphs were defined by Woo and Neumaier. Especially, Hoffman line graphs are closely related to the smallest eigenvalues of graphs, and the uniqueness of strict covers of a Hoffman line graph plays a key role in such a study. In this paper, we prove a theorem for the uniqueness of strict covers under a condition which can be checked in finite time. Our result gives a generalization and a short proof for the main part of Taniguchi's 2008 result on the uniqueness of strict covers.