Spectral characterizations of signed cycles

A signed graph is a pair like (G, sigma), where G is the underlying graph and a : 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 (C-n, sigma) 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. (C) 2018 Elsevier Inc. All rights reserved.