Spectral characterizations of signed cycles

A signed graph is a pair like (G,σ), where G is the underlying graph and σ:E(G)→{−1,+1} is a sign function on the edges of G. In this paper we study the spectral determination problem for signed n-cycles (Cn,σ) with respect to the adjacency spectrum and the Laplacian spectrum. In particular, for the Laplacian spectrum, we prove that balanced odd cycles and unbalanced cycles, denoted, respectively, by C2n+1+ and Cn−, are uniquely determined by their Laplacian spectra (i.e., they are DLS). On the other hand, we determine all Laplacian cospectral mates of the balanced even cycles C2n+, so that we show that C2n+ is not DLS. The same problem is then considered for the adjacency spectrum, hence we prove that odd signed cycles, namely, C2n+1+ and C2n+1−, are uniquely determined by their (adjacency) spectrum (i.e., they are DS). Moreover, we find cospectral mates for the even signed cycles C2n+ and C2n−, and we show that, except the signed cycle C4−, even signed cycles are not DS and we provide almost all cospectral mates.