Halved cube graph

Halved cube graph
Halved cube graph
	The halved cube graph ⁠1/2⁠Q3
Vertices	2n−1
Edges	n(n − 1)2n−3
Automorphisms	n! 2n−1, for n > 4; n! 2n, for n = 4; (2n−1)!, for n < 4
Properties	Symmetric; Distance regular
Notation	⁠1/2⁠Qn
	Table of graphs and parameters

In graph theory, the halved cube graph or half cube graph of dimension $n$ is the graph of the demihypercube, formed by connecting pairs of vertices at distance exactly two from each other in the hypercube graph. That is, it is the half-square of the hypercube. This connectivity pattern produces two isomorphic graphs, disconnected from each other, each of which is the halved cube graph.

Halved cube graph
The halved cube graph $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠1/2⁠Q3$
Vertices	$2 n -1$
Edges	$n (n - 1)2 n -3$
Automorphisms	$n! 2 n -1$ , for $n > 4$ $n! 2 n$ , for $n = 4$ $(2 n -1)!$ , for $n < 4$
Properties	Symmetric Distance regular
Notation	$⁠ 1 / 2 ⁠ Q n$
Table of graphs and parameters