On the nullity of complete bipartite signed graphs

Source abstract

In this paper, we investigate the nullity of complete bipartite signed graphs [Formula: see text]. We first give a simple proof of a known lower bound for the nullity of a signed complete bipartite graph in terms of its induced negative graph. We then identify infinite families of complete bipartite signed graphs that attain this lower bound; this provides a partial solution to an open problem raised in “Pirzada et al., On the eigenvalues of complete bipartite signed graphs, Ars Math. Contemp. 24 (2024), Paper no. 4.08, 17 pp.” Finally, we determine all switching equivalence classes of nonsingular complete bipartite signed graphs up to order 8, showing that there are exactly six such classes. We conclude the paper with several open problems.