Halved cube graph

Halved cube graph
The halved cube graph ${\displaystyle {\tfrac {1}{2}}Q_{3}}$
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	${\displaystyle {\tfrac {1}{2}}Q_{n}}$

In graph theory, the halved cube graph or half cube graph of order 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.

The construction of the halved cube graph can be reformulated in terms of binary numbers. The vertices of a hypercube may be labeled by binary numbers in such a way that two vertices are adjacent exactly when they differ in a single bit. The demicube may be constructed from the hypercube as the convex hull of the subset of binary numbers with an even number of nonzero bits (the evil numbers), and its edges connect pairs of numbers whose Hamming distance is exactly two.

It is also possible to construct the halved cube graph from a lower-order hypercube graph, without taking a subset of the vertices:

where the superscript 2 denotes the square of the hypercube graph Q_n − 1, the graph formed by connecting pairs of vertices whose distance is at most two in the original graph. For instance, the halved cube graph of order four may be formed from an ordinary three-dimensional cube by keeping the cube edges and adding edges connecting pairs of vertices that are on opposite corners of the same square face.

The halved cube graph ${\displaystyle {\tfrac {1}{2}}Q_{3}}$ of order 3 is the complete graph K₄, the graph of the tetrahedron. The halved cube graph ${\displaystyle {\tfrac {1}{2}}Q_{4}}$ of order 4 is K_2,2,2,2, the graph of the four-dimensional regular polytope, the 16-cell. The halved cube graph ${\displaystyle {\tfrac {1}{2}}Q_{5}}$ of order five is sometimes known as the Clebsch graph, and is the complement of the folded cube graph of order five which is more commonly called the Clebsch graph. It exists in the 5-dimensional uniform 5-polytope, the 5-demicube.