Graphs G with nullity n(G)−g(G)−1

For a simple graph G, let n(G), η(G), r(G) and g(G) be respectively the order, the nullity, the rank and the girth of G. It was shown by Cheng and Liu (2007) that for every graph G, η(G)≤{n(G)−g(G)+2 if 4|g(G)n(G)−g(G) if 4∤g(G). Connected graphs G with η(G)=n(G)−g(G)+2 and n(G)−g(G) respectively have been characterized by Zhou et al. (2021). In this paper, we characterize connected graphs G with η(G)=n(G)−g(G)−1.