Skew spectra of graphs without even cycles

The study of the invariance of the skew spectrum of an oriented graph under different orientations leads to the special family GG of graphs without even cycles. Basic graphical and spectral properties of GG are observed. As a result, the Hosoya index of a graph in GG can be computed via its skew spectrum. We also focus on friendship graphs, a subfamily of GG. In particular, the condition under which the skew spectrum of a friendship graph is actually i times the spectrum of some graph is determined, where i=−1.