Spectral characterizations of signed cycles

A signed graph is a pair like (G, σ), where Gis 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 C^+_{2n+1} and C^−_{n}, 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 C^+_{2n}, so that we show that C^+_{2n} is not DLS. The same problem is then considered for the adjacency spectrum, hence we prove that odd signed cycles, namely, C^+_{2n+1} and C^−_{2n+1} , are uniquely determined by their (adjacency) spectrum (i.e., they are DS). Moreover, we find cospectral mates for the even signed cycles C^+_{2n} and C^−_{2n}, and we show that, except the signed cycle C^−_{4}, even signed cycles are not DS and we provide almost all cospectral mates.