Examples
Every directed path graph with an even number of vertices is skew-symmetric, via a symmetry that swaps the two ends of the path. However, path graphs with an odd number of vertices are not skew-symmetric, because the orientation-reversing symmetry of these graphs maps the center vertex of the path to itself, something that is not allowed for skew-symmetric graphs.
Similarly, a directed cycle graph is skew-symmetric if and only if it has an even number of vertices. In this case, the number of different mappings σ that realize the skew symmetry of the graph equals half the length of the cycle.
Read more about this topic: Skew-symmetric Graph
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