Spectral Conditions for Balance in Signed Unicyclic Graphs.

Source abstract

A signed graph is balanced if every cycle has positive sign. For a connected signed unicyclic graph, balance is governed by the sign of the unique cycle. In this paper, we give equivalent adjacency-spectral balance tests in this setting. We prove that balance is equivalent to equality of the signed and unsigned adjacency characteristic polynomials, equality of their largest adjacency eigenvalues, and equality of the first cycle-length spectral moment. In the unbalanced case, the trace defect at the cycle length is exactly 4g, where g is the length of the unique cycle. We also obtain the separation formula \phi_\Sigma(x)=\phi_G(x)+4m_F(x) for a negative unique cycle, where F=G-V(C) and m_F(x) is the matching polynomial of F.