Locally triangular graphs and normal quotients of n-cubes

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1 Locally triangular graphs and normal quotients of n-cubes Joanna B. Fawcett The University of Western Australia 15 February, 2016

2 The triangular graph T n for n 2: Vertices 2-subsets of {1,...,n}. Adjacency {i,j} {k,l} {i,j} {k,l} = 1.

3 The triangular graph T n for n 2: Vertices 2-subsets of {1,...,n}. Adjacency {i,j} {k,l} {i,j} {k,l} = 1. A graph Γ is locally T n if for every vertex u V Γ, the graph induced by the neighbourhood Γ(u) is isomorphic to T n.

4 The triangular graph T n for n 2: Vertices 2-subsets of {1,...,n}. Adjacency {i,j} {k,l} {i,j} {k,l} = 1. A graph Γ is locally T n if for every vertex u V Γ, the graph induced by the neighbourhood Γ(u) is isomorphic to T n. vs A rectagraph is a connected triangle-free graph in which any 2-arc lies in a unique quadrangle.

5 The triangular graph T n for n 2: Vertices 2-subsets of {1,...,n}. Adjacency {i,j} {k,l} {i,j} {k,l} = 1. A graph Γ is locally T n if for every vertex u V Γ, the graph induced by the neighbourhood Γ(u) is isomorphic to T n. vs A rectagraph is a connected triangle-free graph in which any 2-arc lies in a unique quadrangle. e.g., the n-cube Q n for n 1: Vertices F n 2. Adjacency x,y F n 2 differing in exactly one coordinate.

6 The triangular graph T n for n 2: Vertices 2-subsets of {1,...,n}. Adjacency {i,j} {k,l} {i,j} {k,l} = 1. A graph Γ is locally T n if for every vertex u V Γ, the graph induced by the neighbourhood Γ(u) is isomorphic to T n. e.g., the halved n-cube 1 2 Q n for n 2. vs A rectagraph is a connected triangle-free graph in which any 2-arc lies in a unique quadrangle. e.g., the n-cube Q n for n 1: Vertices F n 2. Adjacency x,y F n 2 differing in exactly one coordinate.

7 Lemma (Neumaier, 1985) Let Γ be a graph. Let n 2. The following are equivalent. (i) Γ is a connected locally T n graph. (ii) Γ is a halved graph of an n-valent bipartite rectagraph with c 3 = 3.

8 Lemma (Neumaier, 1985) Let Γ be a graph. Let n 2. The following are equivalent. (i) Γ is a connected locally T n graph. (ii) Γ is a halved graph of an n-valent bipartite rectagraph with c 3 = 3. Goal: refine this result using groups!

9 Let K Aut(Q n ). The normal quotient (Q n ) K has Vertices {x K : x F n 2 }. Adjacency x K y K (distinct) x x K,y y K such that x y in Q n.

10 Let K Aut(Q n ). The normal quotient (Q n ) K has Vertices {x K : x F n 2 }. Adjacency x K y K (distinct) x x K,y y K such that x y in Q n. Let K Aut(Q n ). The minimum distance of K is { min{dqn (x,x d K := k ) : x VQ n,k K \ {1}} if K 1, otherwise. (Matsumoto, 1991)

11 Let K Aut(Q n ). The normal quotient (Q n ) K has Vertices {x K : x F n 2 }. Adjacency x K y K (distinct) x x K,y y K such that x y in Q n. Let K Aut(Q n ). The minimum distance of K is { min{dqn (x,x d K := k ) : x VQ n,k K \ {1}} if K 1, otherwise. (Matsumoto, 1991) Generalises minimum distance for binary linear codes C F n 2 :

12 Let K Aut(Q n ). The normal quotient (Q n ) K has Vertices {x K : x F n 2 }. Adjacency x K y K (distinct) x x K,y y K such that x y in Q n. Let K Aut(Q n ). The minimum distance of K is { min{dqn (x,x d K := k ) : x VQ n,k K \ {1}} if K 1, otherwise. (Matsumoto, 1991) Generalises minimum distance for binary linear codes C F n 2 : c C = d Qn (x,x c ) = d Qn (x,x + c) = c.

13 In a graph Γ, for u,v V Γ such that d Γ (u,v) = i, define a i (u,v) := Γ i (u) Γ(v) c i (u,v) := Γ i 1 (u) Γ(v).

14 In a graph Γ, for u,v V Γ such that d Γ (u,v) = i, define a i (u,v) := Γ i (u) Γ(v) c i (u,v) := Γ i 1 (u) Γ(v). Write a i and c i when there is no dependence on the choice of u,v.

15 In a graph Γ, for u,v V Γ such that d Γ (u,v) = i, define a i (u,v) := Γ i (u) Γ(v) c i (u,v) := Γ i 1 (u) Γ(v). Write a i and c i when there is no dependence on the choice of u,v. Theorem (F., 2016) Let K Aut(Q n ). Let l 1. The following are equivalent. (i) (Q n ) K is n-valent with a i 1 = 0 and c i = i for 1 i l. (ii) d K 2l + 1.

16 In a graph Γ, for u,v V Γ such that d Γ (u,v) = i, define a i (u,v) := Γ i (u) Γ(v) c i (u,v) := Γ i 1 (u) Γ(v). Write a i and c i when there is no dependence on the choice of u,v. Theorem (F., 2016) Let K Aut(Q n ). Let l 1. The following are equivalent. (i) (Q n ) K is n-valent with a i 1 = 0 and c i = i for 1 i l. (ii) d K 2l + 1. In particular, the following are equivalent for a graph Π. (i) Π is an n-valent rectagraph with a 2 = 0 and c 3 = 3. (ii) Π (Q n ) K for some K Aut(Q n ) such that d K 7.

17 Theorem (F., 2016) Let Γ be a graph. Let n 2. The following are equivalent. (i) Γ is a connected locally T n graph. (ii) Γ is a halved graph of (Q n ) K for some K Aut(Q n ) such that K is even and d K 7.

18 Theorem (F., 2016) Let Γ be a graph. Let n 2. The following are equivalent. (i) Γ is a connected locally T n graph. (ii) Γ is a halved graph of (Q n ) K for some K Aut(Q n ) such that K is even and d K 7. K is even precisely when (Q n ) K is bipartite.

19 Theorem (F., 2016) Let Γ be a graph. Let n 2. The following are equivalent. (i) Γ is a connected locally T n graph. (ii) Γ is a halved graph of (Q n ) K for some K Aut(Q n ) such that K is even and d K 7. K is even precisely when (Q n ) K is bipartite. K acts semiregularly on F n 2 ; in particular K is a 2-group.

20 Theorem (F., 2016) Let Γ be a graph. Let n 2. The following are equivalent. (i) Γ is a connected locally T n graph. (ii) Γ is a halved graph of (Q n ) K for some K Aut(Q n ) such that K is even and d K 7. K is even precisely when (Q n ) K is bipartite. K acts semiregularly on F n 2 ; in particular K is a 2-group. K is unique up to conjugacy in Aut(Q n ).

21 Theorem (F., 2016) Let Γ be a graph. Let n 2. The following are equivalent. (i) Γ is a connected locally T n graph. (ii) Γ is a halved graph of (Q n ) K for some K Aut(Q n ) such that K is even and d K 7. K is even precisely when (Q n ) K is bipartite. K acts semiregularly on F n 2 ; in particular K is a 2-group. K is unique up to conjugacy in Aut(Q n ). Aut(Γ) = N En :S n (K)/K where E n = {c F n 2 : c 0 mod 2}.

22 If C F n 2 and d C 2, then (Q n ) C has isomorphic halved graphs.

23 If C F n 2 and d C 2, then (Q n ) C has isomorphic halved graphs. Does this generalise to K Aut(Q n ) with d K 2?

24 If C F n 2 and d C 2, then (Q n ) C has isomorphic halved graphs. Does this generalise to K Aut(Q n ) with d K 2? No: When n = 8, even K Aut(Q n ) with K Q 8 and d K = 4, but the halved graphs of (Q n ) K are regular with different valencies.

25 If C F n 2 and d C 2, then (Q n ) C has isomorphic halved graphs. Does this generalise to K Aut(Q n ) with d K 2? No: When n = 8, even K Aut(Q n ) with K Q 8 and d K = 4, but the halved graphs of (Q n ) K are regular with different valencies. Proposition Let K Aut(Q n ) be even where d K 2. If n is odd, then (Q n ) K has isomorphic halved graphs.

26 If C F n 2 and d C 2, then (Q n ) C has isomorphic halved graphs. Does this generalise to K Aut(Q n ) with d K 2? No: When n = 8, even K Aut(Q n ) with K Q 8 and d K = 4, but the halved graphs of (Q n ) K are regular with different valencies. Proposition Let K Aut(Q n ) be even where d K 2. If n is odd, then (Q n ) K has isomorphic halved graphs. What about n even? And if d K 7?

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