Solution to a problem on skew spectral radii of oriented graphs

Let G be a simple graph, and let Gσ be an oriented graph of G with skew adjacency matrix S(Gσ). The skew spectral radius ρs(Gσ) of Gσ is defined as the spectral radius of S(Gσ). When G is an odd-cycle graph (no even cycle), Cavers et al. (2012) [4] showed that the skew spectral radius of Gσ is the same for every orientation σ of G. They proposed a problem: If G is a connected graph and ρs(Gσ) is the same for all orientations σ of G, must G be an odd-cycle graph? In this paper, we solve this problem and give a positive answer.