Square graph

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:

It is the cycle graph on 4 vertices, denoted $C_{4}$ .
It is the complete bipartite graph $K_{2, 2}$
it is the 2-dimensional hypercube graph.

Arithmetic functions

Size measures

Function	Value	Explanation
size of vertex set	4	As cycle graph $C_{n}, n = 4$ : $n = 4$ As complete bipartite graph $K_{m, n}, m = n = 2$ : $m + n = 2 + 2 = 4$ As $n$ -dimensional hypercube, $n = 2$ : $2^{n} = 2^{2} = 4$
size of edge set	4	As cycle graph $C_{n}, n = 4$ : $n = 4$ As complete bipartite graph $K_{m, n}, m = n = 2$ : $m n = 2 (2) = 4$ As $n$ -dimensional hypercube, $n = 2$ : $n \cdot 2^{n - 1} = 2 \cdot 2^{2 - 1} = 4$

Numerical invariants associated with vertices

Since the graph is a vertex-transitive graph, any numerical invariant associated to a vertex must be equal on all vertices of the graph. Below are listed some of these invariants:

Function	Value	Explanation
degree of a vertex	2	As cycle graph $C_{n}, n = 4$ : 2 (independent of $n$ ) As complete bipartite graph $K_{m, n}, m = n = 2$ : Since $m, n$ are equal, the graph is vertex-transitive and $(m = n)$ -regular, so we get $m = n = 2$ As $n$ -dimensional hypercube, $n = 2$ : $n = 2$
eccentricity of a vertex	2	As cycle graph $C_{n}, n = 4$ : greatest integer of $n / 2$ equals greater integer of 4/2 equals 2 As complete bipartite graph $K_{m, n}, m = n = 2$ : 2 (independent of $m, n$ , though it uses that both numbers are greater than 1) As $n$ -dimensional hypercube, $n = 2$ : $n = 2$

Other numerical invariants

Function	Value	Explanation
clique number	2	As cycle graph $C_{n}, n = 4$ : 2 (independent of $n$ for $n \geq 4$ ) As $K_{m, n}, m = n = 2$ : 2 (independent of $m, n$ , follows from being bipartite) As $n$ -dimensional hypercube, $n = 2$ : 2 (independent of $n$ )
independence number	2	As cycle graph $C_{n}, n = 4$ : greatest integer of $n / 2$ equals greatest integer of 4/2 equals 2 As $K_{m, n}, m = n = 2$ : $max {m, n} = max {2, 2} = 2$ As $n$ -dimensional hypercube, $n = 2$ : $2^{n - 1} = 2^{2 - 1} = 2$
chromatic number	2	As cycle graph $C_{n}, n = 4$ : 2 (in general, it is 2 for even $n$ and 3 for odd $n$ As $K_{m, n}, m = n = 2$ : 2 (independent of $m, n$ , follows from being bipartite) As $n$ -dimensional hypercube, $n = 2$ : 2 (independent of $n$ , follows from being bipartite)
radius of a graph	2	Due to vertex-transitivity, the radius equals the eccentricity of any vertex, which has been computed above.
diameter of a graph	2	Due to vertex-transitivity, the radius equals the eccentricity of any vertex, which has been computed above.
odd girth	infinite
even girth	4	As cycle graph $C_{n}, n = 4$ : $n = 4$ (since $n$ even) As complete bipartite graph $K_{m, n}, m = n = 2$ : 4 (independent of $m, n$ as long as both are greater than 1) As $n$ -dimensional hypercube, $n = 2$ : 4 (independent of $n$ for $n \geq 2$ )
girth of a graph	4	As cycle graph $C_{n}, n = 4$ : $n = 4$ As complete bipartite graph $K_{m, n}, m = n = 2$ : 4 (independent of $m, n$ as long as both are greater than 1) As $n$ -dimensional hypercube, $n = 2$ : 4 (independent of $n$ for $n \geq 2$ )