Cycle graph:C5

This article defines a particular undirected graph, i.e., the definition here determines the graph uniquely up to graph isomorphism.
View a complete list of particular undirected graphs

Definition

This undirected graph is defined in the following equivalent ways:

1. It is the cycle graph on 5 vertices, i.e., the graph $C_{5}$
2. It is the Paley graph $P_{5}$ corresponding to the field of 5 elements
3. It is the unique (up to graph isomorphism) self-complementary graph on a set of 5 vertices

Note that 5 is the only size for which the Paley graph coincides with the cycle graph. In general, the Paley graph $P_{q}$ can be expressed as an edge-disjoint union of $(q-1)/4$ cycle graphs. In our case, $(5-1)/4=1$, so the graphs coincide.

Arithmetic functions

Size measures

Function	Value	Explanation
size of vertex set	5	By either definition
size of edge set	5	As $C_{n},n=5$: $n=5$ As $P_{q},q=5$: $q(q-1)/4=5(4)/4=5$

Other numerical invariants

Function	Value	Explanation
clique number	2	As cycle graph $C_{n},n=5$: 2 (since $n\geq 4$)
independence number	2	As cycle graph $C_{n},n=5$: Greatest integer of $n/2$, which is greatest integer of $5/2$, which is 2
chromatic number	3	As cycle graph $C_{n},n=5$: 3 since $n$ is odd

Graph properties

Property	Satisfied?	Explanation
self-complementary graph	Yes	Paley graphs are self-complementary
strongly regular graph	Yes	Paley graphs are strongly regular
regular graph	Yes	Follows from being strongly regular. Also follows from the fact that cycle graphs are 2-regular.
conference graph	Yes	Paley graphs are conference graphs
symmetric graph	Yes
edge-transitive graph	Yes	Follows on account of being Paley and also on account of being cyclic
vertex-transitive graph	Yes