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asymmetric graph
In graph theory, a branch of mathematics, an undirected graph is called an asymmetric graph if it has no nontrivial symmetries.
Formally, an automorphism of a graph is a permutation ''p'' of its vertices with the property that any two vertices ''u'' and ''v'' are adjacent if and only if ''p''(''u'') and ''p''(''v'') are adjacent.
The identity mapping of a graph onto itself is always an automorphism, and is called the trivial automorphism of the graph. An asymmetric graph is a graph for which there are no other automorphisms.
==Examples==
The smallest asymmetric non-trivial graphs have 6 vertices.〔 The smallest asymmetric regular graphs have ten vertices; there exist ten-vertex asymmetric graphs that are 4-regular and 5-regular.〔.〕〔.〕 One of the two smallest asymmetric cubic graphs is the twelve-vertex Frucht graph discovered in 1939.〔.〕 According to a strengthened version of Frucht's theorem, there are infinitely many asymmetric cubic graphs.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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