Cayley graph

Graph families defined by their automorphisms
distance-transitive	${\displaystyle {\boldsymbol {\rightarrow }}}$	distance-regular	${\displaystyle {\boldsymbol {\leftarrow }}}$	strongly regular
${\displaystyle {\boldsymbol {\downarrow }}}$
symmetric (arc-transitive)	${\displaystyle {\boldsymbol {\leftarrow }}}$	t-transitive, t ≥ 2		skew-symmetric
${\displaystyle {\boldsymbol {\downarrow }}}$
(if connected) vertex- and edge-transitive	${\displaystyle {\boldsymbol {\rightarrow }}}$	edge-transitive and regular	${\displaystyle {\boldsymbol {\rightarrow }}}$	edge-transitive
${\displaystyle {\boldsymbol {\downarrow }}}$		${\displaystyle {\boldsymbol {\downarrow }}}$		${\displaystyle {\boldsymbol {\downarrow }}}$
vertex-transitive	${\displaystyle {\boldsymbol {\rightarrow }}}$	regular	${\displaystyle {\boldsymbol {\rightarrow }}}$	(if bipartite) biregular
${\displaystyle {\boldsymbol {\uparrow }}}$
Cayley graph	${\displaystyle {\boldsymbol {\leftarrow }}}$	zero-symmetric		asymmetric

In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayley) and uses a specified, usually finite, set of generators for the group. It is a central tool in combinatorial and geometric group theory.

Suppose that ${\displaystyle G}$ is a group and ${\displaystyle S}$ is a generating set. The Cayley graph ${\displaystyle \Gamma =\Gamma (G,S)}$ is a colored directed graph constructed as follows:

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