Spectral and edge conditions for negative Hamiltonian cycles in unbalanced signed graphs
Spectral conditions for Hamiltonian cycles are a fundamental subject in extremal spectral graph theory. For simple unsigned graphs, Fiedler and Nikiforov [Linear Algebra Appl., 432 (2010), 2170-2173] pioneered this direction, and Li and Ning [Linear Multilinear Algebra, 64 (2016), 2252-2269] later extended these results by introducing minimum degree parameters. In this paper, we study the spectral conditions for the existence of a negative Hamiltonian cycle in unbalanced signed graphs of order n > 8. We first determine the maximum index among all n-vertex unbalanced signed graphs containing no negative Hamiltonian cycle, and characterize the corresponding extremal graphs up to switching isomorphism. Furthermore, we establish an upper bound on the number of edges in unbalanced signed graphs of order n >= 7 with frustration index one, and classify all extremal graphs achieving this bound.