Half-transitive graph
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Graph families defined by their automorphisms | ||||
distance-transitive | distance-regular | strongly regular | ||
symmetric (arc-transitive) | t-transitive, t ≥ 2 | skew-symmetric | ||
(if connected) vertex- and edge-transitive |
edge-transitive and regular | edge-transitive | ||
vertex-transitive | regular | (if bipartite) biregular |
||
Cayley graph | zero-symmetric | asymmetric |
In the mathematical field of graph theory, a half-transitive graph is a graph that is both vertex-transitive and edge-transitive, but not symmetric.[1] In other words, a graph is half-transitive if its automorphism group acts transitively upon both its vertices and its edges, but not on ordered pairs of linked vertices.
Every connected symmetric graph must be vertex-transitive and edge-transitive, and the converse is true for graphs of odd degree,[2] so that half-transitive graphs of odd degree do not exist. However, there do exist half-transitive graphs of even degree.[3] The smallest half-transitive graph is the Holt graph, with degree 4 and 27 vertices.[4][5]
References
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- ↑ Bouwer, Z. "Vertex and Edge Transitive, But Not 1-Transitive Graphs." Canad. Math. Bull. 13, 231–237, 1970.
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