HOSIGRA

Definition 1 (as the projection of the hypercubes)

Projective cube of dimension k, denoted PC(k) is obtained from the hypercube of dimension k+1 by identifying antipodal pairs of vertices.

Then by considering edges corresponding to dimension k+1 (e_k+1) as the negative edge we get signed projective cube of dimension k and denoted it by SPC(k).

As graphs, and defined as projection of hypercube, signed projective cubes are named as folded cubes in most literatures, but we see this construction more of a projection than folding and compare it to the construction of projective plane of dimension k from k-dimensional sphere.

The logo of the website shows two copies of H(4), which together with a matching of corresponding vertices they form H(5). A pair of antipodal vertices are circled, and dotted black edge shows how an of the matching forms a new edge after the project.

Definition 2 (as augmentation of hypercubes)

Projective cube of dimension k, is obtained from the hypercube of dimension by identifying antipodal pairs of vertices.

The signed projective cube of dimension k is then obtained by assigning negative sign to all newly added edges.

This definition has granted then the name augmented cubes or augmented hypercube in some literature.

Definition 3 (as Cayley graphs)

Projective cube of dimension k, is the Cayley graph (ℤ₂^k, {e₁, e₂,..., e_k, J})

Here {e₁, e₂,..., e_k} is the standard baises. That is to say e_i is the binary vector whose i^th coordinate is 1 and all other coordinates are 0, the last element, J is the all-1 vector.

The signed projective cube of dimension k is then obtained by assigning negative sign to the edges that correspond to J.

The class of Cayley graphs on a binary group are referred to as cube-like graphs by some authors. Thus projective cubes are among cube-like graphs.

Definition 4 (as Cayley graphs, a general setting)

Let S_k be a subset of order k+1 of ℤ₂^k with the properties that i. sum of the all the elements is 0 ii. for any nonempty proper subset, sum of the all the elements is not nonzero.

Projective cube of dimension k, is the Cayley graph (ℤ₂^k,S_k)

The signed projective cube of dimension k is then obtained by assigning negative sign to the edges that correspond to a given element s of S.

Remark If we take S_k as a subset of order k+1 of ℤ₂^l, l>k, having same two properties, then the Cayley graph (ℤ₂^l,S_k) consistes of 2^l-k isomorphic copies of PC(k).

Definition 5 (as a poset or a partition graph)

Let S be a set of size k+1 with k blue elements s₁, s₂, ..., s_k and one red element s₀. The signed projective cube of dimension k is defined as follows:

• Vertices are partitions of S into two parts, we denote them in the form of {A, B} (where B=S-A),
• Vertex {A,B} is adjacent to vertex {A',B'} with a negative sign if A⊊A' and A'=A�∪{s₀},
• Vertex {A,B} is adjacent to vertex {A',B'} with a positive sign if A⊊A' and A'=A∪{s_i} for some i in {1,2,...,k}.

Definition 6 (built inductively)

An inductive definition of the projective cubes is perhaps the first advantage of considering them as signed graph.

Step 1: The signed projective cube SPC(1) is the digon, that is the multigraph on two vertices with two parallel edges connecting them, one of the negative sign and the other of the positive sign.

Step k+1: Given SPC(k) with the set J of negative edges, SPC(k+1), the signed projective cube of dimension k+1, is built from two disjoint copies of SPC(k) where vertices are labeled x₁, x₁, ..., x_2^k in one copy and x'₁, x'₁, ..., x'_2^k in the other copy, x_i being the copy of x'_i. The edges are defined as follows:

• x_ix'_i form negative edges of SPC(k+1),
• if x_ix_j is a positive edge of SPC(k), then x_ix'_i and x_jx'_j are both positive edges of PC(2k),
• if x_ix_j is a negative edge of SPC(k), then x_ix'_j and x_jx'_i are both positive edges of PC(2k).

Definition 7 (as a power graph)

Given a signed graph (G, 𝛴 ) its power graph, denoted POW(G, 𝛴 ) is one of the two isomorphic copies of the signed graph built as follows:

• Vertices are subsests of V(G)
• A is adjacent to B with a positive sign if the symmetric difference of A and B is a positive edge of (G, 𝛴 )
• A is adjacent to B with a negative sign if the symmetric difference of A and B is a negative edge of (G, 𝛴 ) (i.e. A ∆ B ∈ 𝛴 )

Then the subsets of even order and subset of odd order induce two isomorphic copies either of which is the power graph of (G, 𝛴 ).

The signed projective cube of dimension k is the power graph of unbalanced cycle of length k.