The spectrum of a graph G is the collection of eigenvalues of its adjacency matrix A(G). The rank of G, denoted by r(G), is the number of non-zero eigenvalues in its spectrum, or equivalently, the rank of A(G). The nullity of a graph G is the multiplicity of the eigenvalue zero in its spectrum. It is known that the rank is equal to the difference from the order to the nullity of the graph. Hu et al. in [On the nullity of bicyclic graphs, Lin. Algebra Appl., 429 (2008), 1387-1391.] characterized bicyclic graphs of order n with nullity n - 4. That is, they characterized bicyclic graphs of rank 4. But they missed some cases. In this paper, we will complete their proof and characterize unicyclic and bicyclic graphs of rank 5, respectively.
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