A *two-graph* is a finite set provided with a collection of triples,
called *coherent*, such that each 4-set contains an even number
of coherent triples. The two-graph is called *regular* when
each pair of points is contained in the same number of coherent triples.
Trivial two-graphs are obtained by taking no triples, or all triples.

Given a graph Γ on a set V of vertices, one obtains a two-graph
by taking all triples that contain an odd number of edges of Γ.
Every two-graph arises in this way.
(One can for example stipulate that a given vertex *p* must be
isolated, and then *qr* must be an edge precisely when *pqr*
is a coherent triple.)
A graph and its complement give rise to two-graphs with complementary
sets of coherent triples.

Two graphs Γ, Δ with the same vertex set V are called
*switching equivalent* when V is the disjoint union of subsets A and B,
and Γ and Δ induce the same subgraph on A and on B, but
vertices a ∈ A and b ∈ B are joined in Γ iff they are not joined
in Δ. Being switching equivalent is an equivalence relation,
and two graphs are switching equivalent precisely when they give rise
to the same two-graph.

The *Seidel adjacency matrix* of a graph Γ is the matrix
S = J−I−2A that has zero diagonal, −1 for adjacency, and 1 for nonadjacency.
If Γ and Δ are switching equivalent, their Seidel adjacency
matrices have the same spectrum. This common spectrum is called the
Seidel spectrum of the two-graph. The Seidel spectrum of the complementary
two-graph equals minus the Seidel spectrum of a given two-graph.

graph | parameters | spectrum | Seidel spectrum |
---|---|---|---|

Clebsch graph | (16,10,6,6) | 10^{1} 2^{5} (−2)^{10} |
(−5)^{6} 3^{10} |

Shrikhande graph, Hamming graph H(2,4) | (16,6,2,2) | 6^{1} 2^{6} (−2)^{9} |
(−5)^{6} 3^{10} |

K_{1}+T(6) | 8^{1} 2^{5} 0^{1} (−2)^{9}
| (−5)^{6} 3^{10} | |

complement of Clebsch graph | (16,5,0,2) | 5^{1} 1^{10} (−3)^{5} |
5^{6} (−3)^{10} |

complement of Shrikhande, complement of H(2,4) |
(16,9,4,6) | 9^{1} 1^{9} (−3)^{6} |
5^{6} (−3)^{10} |

cone over GQ(2,2) | (3 ± 2√6)^{1} 1^{9} (−3)^{5} |
5^{6} (−3)^{10} |

The automorphism group of the regular two-graph here is
2^{4}:Sym(6) of order 11520. The automorphism groups
of the graphs in its switching class are subgroups.