Bipartite density of cubic graphs: the case of equality

Recently, Berman and Zhang (Discrete Math. 260 (2003) 27) obtained an upper bound for the bipartite density of cubic graphs in terms of the smallest eigenvalue of an adjacency matrix and attempted to characterize graphs for which the upper bound is attained. Here we do characterize graphs for which the upper bound is attained, and correct a few errors from Berman and Zhang's paper.