Zero-symmetric graph

<p>In the <a href="/facts/Mathematics/pxTouaz4">mathematical</a> field of <a href="/facts/Graph_theory/VVPCd4TX">graph theory</a>, a zero-symmetric graph is a <a href="/facts/Connected_graph/FSNjsYhz">connected graph</a> in which each vertex has exactly three incident edges and, for each two vertices, there is a unique <a href="/facts/Graph_automorphism/If8FTdFE">symmetry</a> taking one vertex to the other. Such a graph is a <a href="/facts/Vertex-transitive_graph/042b7rD7">vertex-transitive graph</a> but cannot be an <a href="/facts/Edge-transitive_graph/fnK9oyEJ">edge-transitive graph</a>: the number of symmetries equals the number of vertices, too few to take every edge to every other edge.
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<p>The name for this class of graphs was coined by <a href="/facts/R._M._Foster/UoVSMb8c">R. M. Foster</a> in a 1966 letter to  <a href="/facts/Harold_Scott_MacDonald_Coxeter/D2l5jWGG">H. S. M. Coxeter</a>. In the context of <a href="/facts/Group_theory/NfowcI5H">group theory</a>, zero-symmetric graphs are also called graphical regular representations of their symmetry groups.
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Zero-symmetric graph open-in-new

Zero-symmetric graph