Regular 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 graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree of each vertex are equal to each other. A regular graph with vertices of degree k is called a k‑regular graph or regular graph of degree k. Also, from the Handshaking lemma, a regular graph of odd degree will contain even number of vertices.

Regular graphs of degree at most 2 are easy to classify: A 0-regular graph consists of disconnected vertices, a 1-regular graph consists of disconnected edges, and a 2-regular graph consists of disconnected cycles and infinite chains.

A 3-regular graph is known as a cubic graph.

A strongly regular graph is a regular graph where every adjacent pair of vertices has the same number l of neighbors in common, and every non-adjacent pair of vertices has the same number n of neighbors in common. The smallest graphs that are regular but not strongly regular are the cycle graph and the circulant graph on 6 vertices.

The complete graph ${\displaystyle K_{m}}$ is strongly regular for any ${\displaystyle m}$.

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Wikipedia