Normalized Signless 1-Laplacian Energy and Minimum Left-Inverse Norms on θ- and ∞-Graphs

In this paper, we investigate the minimum normalized signless 1-Laplacian energy of two classes of bicyclic graphs, namely θ-graphs and ∞-graphs, and its relationship with generalized inverses of the corresponding normalized unsigned incidence matrices. By analyzing the structural properties of these graphs, we determine their minimum energies in terms of the parity and lengths of their constituent paths and cycles. Furthermore, using properties of unsigned incidence matrices and constructing suitable left inverses, we establish that, for the non-bipartite graphs in these two classes, the minimum normalized signless 1-Laplacian energy equals the reciprocal of the minimum induced 1-norm among all left inverses of the corresponding normalized unsigned incidence matrix. We further investigate whether this reciprocal identity holds for arbitrary connected non-bipartite graphs. By constructing a counterexample based on the complete graph K5, we demonstrate that the identity does not hold in general. These results characterize the minimum normalized signless 1-Laplacian energy for the two classes of bicyclic graphs and clarify the limitations of extending the reciprocal identity to more general graphs.