Star complementary characterization of oriented graphs whose skew spectral radius does not exceed 2

Abstract

We employ the method of star complements to classify all oriented graphs whose skew spectrum lies within the interval [–2, 2]. At the same time, we provide a structural characterisation of these graphs, showing that, with the sole exception of exactly one graph of order 14, every maximal oriented graph possessing this spectral property is determined by a fixed oriented cycle serving as a star complement for either –2 or 2. The exceptional oriented graph is uniquely determined by a fixed 7-vertex oriented path acting as the star complement. This work may be regarded as a counterpart to [13], where the corresponding oriented graphs were determined via associated signed graphs, without the present characterisation.