Self-Complementary Metacirculants
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1 Self-Complementary Metacirculants Grant Rao (joint work with Cai Heng Li and Shu Jiao Song) The University of Western Australia July 7, 2014
2 All graphs are assumed to be finite simple. Γ is vertex-transitive if AutΓ is transitive on V.
3 All graphs are assumed to be finite simple. Γ is vertex-transitive if AutΓ is transitive on V. For a finite group R, let R # := R \ {1}, S R #.
4 All graphs are assumed to be finite simple. Γ is vertex-transitive if AutΓ is transitive on V. For a finite group R, let R # := R \ {1}, S R #. Γ = Cay(R, S) is a Cayley graph where V = R, E = {(a, b) ba 1 S}
5 All graphs are assumed to be finite simple. Γ is vertex-transitive if AutΓ is transitive on V. For a finite group R, let R # := R \ {1}, S R #. Γ = Cay(R, S) is a Cayley graph where V = R, E = {(a, b) ba 1 S} Γ undirected S = S 1 = {s 1 s S}.
6 All graphs are assumed to be finite simple. Γ is vertex-transitive if AutΓ is transitive on V. For a finite group R, let R # := R \ {1}, S R #. Γ = Cay(R, S) is a Cayley graph where V = R, E = {(a, b) ba 1 S} Γ undirected S = S 1 = {s 1 s S}. A circulant is a Cayley graph of a cyclic group.
7 All graphs are assumed to be finite simple. Γ is vertex-transitive if AutΓ is transitive on V. For a finite group R, let R # := R \ {1}, S R #. Γ = Cay(R, S) is a Cayley graph where V = R, E = {(a, b) ba 1 S} Γ undirected S = S 1 = {s 1 s S}. A circulant is a Cayley graph of a cyclic group. A group R is metacyclic if exists N R such that N, R/N are cyclic. Γ is a metacirculant if R < AutΓ is transitive and metacyclic.
8 What is a self-complementary vertex-transitive graph? Γ is called self-complementary (SC) if Γ = Γ.
9 What is a self-complementary vertex-transitive graph? Γ is called self-complementary (SC) if Γ = Γ. Self-complementary vertex-transitive (SCVT) graphs are the graphs that are both self-complementary and vertex-transitive.
10 What is a self-complementary vertex-transitive graph? Γ is called self-complementary (SC) if Γ = Γ
11 What is a self-complementary vertex-transitive graph? Γ is called self-complementary (SC) if Γ = Γ
12 What is a self-complementary vertex-transitive graph? Γ is called self-complementary (SC) if Γ = Γ σ = (1342) maps Γ to Γ
13 What is a self-complementary vertex-transitive graph? Γ is called self-complementary (SC) if Γ = Γ σ = (1342) maps Γ to Γ An isomorphism σ : Γ Γ is a complementing isomorphism.
14 What is a self-complementary vertex-transitive graph? An isomorphism σ : Γ Γ is a complementing isomorphism. o(σ) = 2 e ; σ does not fix any edge 4 o(σ); σ 2 AutΓ σ normalises AutΓ.
15 Observation Γ = Cay(R, S) is an SCVT-graph. R = Z 2 3, S = {(0, 1), (2, 0), (0, 2), (1, 0)}
16 Observation Γ = Cay(R, S) is an SCVT-graph. R = Z 2 3, S = {(0, 1), (2, 0), (0, 2), (1, 0)} Define σ = ( ) 1 1 GL(2, 3). 2 1 Then S σ = R # \ S and S S σ = R #.
17 Background (Sachs, 1962) SC-graphs properties of SC-graphs construction method for SC-graphs conditions for the order of SC-circulants
18 Background (Sachs, 1962) SC-graphs (Zelinka, 1979) The order of an SCVT-graph 1 (mod 4).
19 Background (Sachs, 1962) SC-graphs (Zelinka, 1979) The order of an SCVT-graph 1 (mod 4). (Rao, 1985) Regular and strongly regular SC-graphs.
20 Background (Sachs, 1962) SC-graphs (Zelinka, 1979) The order of an SCVT-graph 1 (mod 4). (Rao, 1985) Regular and strongly regular SC-graphs. (Suprunenko, 1985) Construct SC-graphs using group theoretical method.
21 Background (Sachs, 1962) SC-graphs (Zelinka, 1979) The order of an SCVT-graph 1 (mod 4). (Rao, 1985) Regular and strongly regular SC-graphs. (Suprunenko, 1985) Construct SC-graphs using group theoretical method. (Mathon, 1988) Strongly regular SC-graphs adjacency matrices small order graphs
22 Background (Sachs, 1962) SC-graphs (Zelinka, 1979) The order of an SCVT-graph 1 (mod 4). (Rao, 1985) Regular and strongly regular SC-graphs. (Suprunenko, 1985) Construct SC-graphs using group theoretical method. (Mathon, 1988) Strongly regular SC-graphs (Fronček, Rosa and Širáň, 1996) (Alspach, Morris and Vilfred, 1999) The orders of SC-circulants.
23 Background (Sachs, 1962) SC-graphs (Zelinka, 1979) The order of an SCVT-graph 1 (mod 4). (Rao, 1985) Regular and strongly regular SC-graphs. (Suprunenko, 1985) Construct SC-graphs using group theoretical method. (Mathon, 1988) Strongly regular SC-graphs (Fronček, Rosa and Širáň, 1996) (Alspach, Morris and Vilfred, 1999) The orders of SC-circulants. (Muzychuk, 1999) An SCVT-graph of order n exists n p 1 (mod 4) p.
24 Background (Sachs, 1962) SC-graphs (Zelinka, 1979) The order of an SCVT-graph 1 (mod 4). (Rao, 1985) Regular and strongly regular SC-graphs. (Suprunenko, 1985) Construct SC-graphs using group theoretical method. (Mathon, 1988) Strongly regular SC-graphs (Fronček, Rosa and Širáň, 1996) (Alspach, Morris and Vilfred, 1999) The orders of SC-circulants. (Muzychuk, 1999) An SCVT-graph of order n exists n p 1 (mod 4) p. (Li, Sun and Xu, 2014) SC-circulants of prime-power order.
25 Background (Sachs, 1962) SC-graphs (Zelinka, 1979) The order of an SCVT-graph 1 (mod 4). (Rao, 1985) Regular and strongly regular SC-graphs. (Suprunenko, 1985) Construct SC-graphs using group theoretical method. (Mathon, 1988) Strongly regular SC-graphs (Fronček, Rosa and Širáň, 1996) (Alspach, Morris and Vilfred, 1999) The orders of SC-circulants. (Muzychuk, 1999) An SCVT-graph of order n exists n p 1 (mod 4) p. (Li, Sun and Xu, 2014) SC-circulants of prime-power order. (Li and Rao, 2014) SCVT-graphs of order pq.
26 Why study SCVT-graphs? The diagonal Ramsey number Example R(n, n) = min{ V Γ : K n Γ or K n Γ }
27 Why study SCVT-graphs? The diagonal Ramsey number Example The Shannon capacity R(n, n) = min{ V Γ : K n Γ or K n Γ } Θ(Γ ) = sup k k α(γ k ) = lim k k α(γ k )
28 Why study SCVT-graphs? The diagonal Ramsey number Example The Shannon capacity R(n, n) = min{ V Γ : K n Γ or K n Γ } Θ(Γ ) = sup k k α(γ k ) = lim k k α(γ k ) (Lovász, 1979) Θ(an SCVT-graph of order n) = n.
29 Why study SCVT-graphs? The diagonal Ramsey number Example The Shannon capacity R(n, n) = min{ V Γ : K n Γ or K n Γ } Θ(Γ ) = sup k k α(γ k ) = lim k k α(γ k ) (Lovász, 1979) Θ(an SCVT-graph of order n) = n. Homogeneous factorisations, transitive orbital decompositions
30 Why study SCVT-graphs? The diagonal Ramsey number Example The Shannon capacity R(n, n) = min{ V Γ : K n Γ or K n Γ } Θ(Γ ) = sup k k α(γ k ) = lim k k α(γ k ) (Lovász, 1979) Θ(an SCVT-graph of order n) = n. Homogeneous factorisations, transitive orbital decompositions (Zhang, 1992) Algebraic characterisation of arc-transitive SC-graphs. (Peisert, 2000) Classification of arc-transitive SC-graphs.
31 Why study SCVT-graphs? The diagonal Ramsey number Example The Shannon capacity R(n, n) = min{ V Γ : K n Γ or K n Γ } Θ(Γ ) = sup k k α(γ k ) = lim k k α(γ k ) (Lovász, 1979) Θ(an SCVT-graph of order n) = n. Homogeneous factorisations, transitive orbital decompositions (Zhang, 1992) Algebraic characterisation of arc-transitive SC-graphs. (Peisert, 2000) Classification of arc-transitive SC-graphs. (Beezer, 2006) In 50,502,031,367,952 non-isomorphic graphs of order 13, only 2 of them are SCVT.
32 Construction of Self-Complementary Cayley Graphs Observation Let R be a group and σ Aut(R). Then Cay(R, S) σ = Cay(R, S σ ).
33 Construction of Self-Complementary Cayley Graphs Observation Let R be a group and σ Aut(R). Then Cay(R, S) σ = Cay(R, S σ ). If there exists σ Aut(R) such that S σ = R # \ S, then Γ = Cay(R, S) = Cay(R, S σ ) = Cay(R, R # \ S) = Γ. Example
34 Properties of σ (i) σ does not fix any element of R # σ is fixed-point-free. (ii) σ 2 fixes S and R # \ S σ 2 AutΓ σ normalises AutΓ. (iii) o(σ) = 2 e, e 2.
35 Construction 1 1. σ 2 -orbits on R # : + 1, 1, + 2, 2,..., + r, r where ( + i ) σ = i for each i;
36 Construction 1 1. σ 2 -orbits on R # : + 1, 1, + 2, 2,..., + r, r where ( + i ) σ = i for each i; 2. S = r i=1 ε i i where ε i = + or
37 Construction 1 1. σ 2 -orbits on R # : + 1, 1, + 2, 2,..., + r, r where ( + i ) σ = i for each i; 2. S = r i=1 ε i i where ε i = + or
38 Construction 1 1. σ 2 -orbits on R # : + 1, 1, + 2, 2,..., + r, r where ( + i ) σ = i for each i; 2. S = r i=1 ε i i where ε i = + or
39 Construction 1 1. σ 2 -orbits on R # : + 1, 1, + 2, 2,..., + r, r where ( + i ) σ = i for each i; 2. S = r i=1 ε i i where ε i = + or Then Cay(R, S) is self-complementary.
40 Construction 1 1. σ 2 -orbits on R # : + 1, 1, + 2, 2,..., + r, r where ( + i ) σ = i for each i; 2. S = r i=1 ε i i where ε i = + or Construction 2 Let p 1 or 9 (mod 40), R = Z 2 p, and let σ Z(GL(2, p)) with 8 o(σ). Let H = σ, SL(2, 5), M = σ 2, SL(2, 5).
41 Construction 1 1. σ 2 -orbits on R # : + 1, 1, + 2, 2,..., + r, r where ( + i ) σ = i for each i; 2. S = r i=1 ε i i where ε i = + or. Construction 2 Let p 1 or 9 (mod 40), R = Z 2 p, and let σ Z(GL(2, p)) with 8 o(σ). Let H = σ, SL(2, 5), M = σ 2, SL(2, 5). 1. M-orbits on R # : + 1, 1, + 2, 2,..., + r, r where ( + i ) σ = i for each i; 2. S = r i=1 ε i i where ε i = + or.
42 Construction 1 1. σ 2 -orbits on R # : + 1, 1, + 2, 2,..., + r, r where ( + i ) σ = i for each i; 2. S = r i=1 ε i i where ε i = + or. Construction 2 Let p 1 or 9 (mod 40), R = Z 2 p, and let σ Z(GL(2, p)) with 8 o(σ). Let H = σ, SL(2, 5), M = σ 2, SL(2, 5). 1. M-orbits on R # : + 1, 1, + 2, 2,..., + r, r where ( + i ) σ = i for each i; 2. S = r i=1 ε i i where ε i = + or. Γ = Cay(R, S) is an SC-metacirculant with AutΓ insoluble.
43 Construction 1 1. σ 2 -orbits on R # : + 1, 1, + 2, 2,..., + r, r where ( + i ) σ = i for each i; 2. S = r i=1 ε i i where ε i = + or. Construction 2 Let p 1 or 9 (mod 40), R = Z 2 p, and let σ Z(GL(2, p)) with 8 o(σ). Let H = σ, SL(2, 5), M = σ 2, SL(2, 5). 1. M-orbits on R # : + 1, 1, + 2, 2,..., + r, r where ( + i ) σ = i for each i; 2. S = r i=1 ε i i where ε i = + or. Lemma 3 There exist SC-metacirculants of order p p2 t (p i distinct) with AutΓ Z 2 p 1...p t :(Z l SL(2, 5) t ).
44 Self-complementary metacirculants Theorem 4 (Li and Praeger, 2012) The automorphism group of a self-complementary circulant is soluble.
45 Self-complementary metacirculants Theorem 4 (Li and Praeger, 2012) The automorphism group of a self-complementary circulant is soluble. The following theorem extends the result. Theorem 5 (Li, Rao and Song, 2014) The automorphism group of a self-complementary metacirculant is either soluble, or contains composition factor A 5.
46 Proof of theorem 5 Γ is a self-complementary metacirculant σ is a complementing isomorphism G = AutΓ X = G, σ = G.Z 2 R < G is transitive and metacyclic
47 Proof of theorem 5 Γ is a self-complementary metacirculant σ is a complementing isomorphism G = AutΓ X = G, σ = G.Z 2 R < G is transitive and metacyclic A block system B is a nontrivial X -invariant partition of V. (i) V has no block systems X is primitive. (ii) V has a block system X is imprimitive.
48 The primitive case Theorem 6 (Guralnick, Li, Praeger and Saxl, 2004) If X is primitive, then (i) X is affine, or (ii) X is of product action type with soc(x ) = PSL(2, q 2 ) l.
49 The primitive case Theorem 6 (Guralnick, Li, Praeger and Saxl, 2004) If X is primitive, then (i) X is affine, or (ii) X is of product action type with soc(x ) = PSL(2, q 2 ) l. 1. R is metacyclic X is affine of dimension 2.
50 The primitive case Theorem 6 (Guralnick, Li, Praeger and Saxl, 2004) If X is primitive, then (i) X is affine, or (ii) X is of product action type with soc(x ) = PSL(2, q 2 ) l. 1. R is metacyclic X is affine of dimension X is insoluble X = Z 2 p:(z l SL(2, 5)) construction
51 The imprimitive case Theorem 7 (Li and Praeger, 2003) If X is imprimitive, then: (i) [B] Γ is self-complementary, G B B Aut[B] Γ, and σ B is its complementing isomorphism; (ii) there is a self-complementary graph Σ with vertex set B such that G B AutΣ and each element of X B \ G B is its complementing isomorphism.
52 The imprimitive case Theorem 7 (Li and Praeger, 2003) If X is imprimitive, then: (i) [B] Γ is self-complementary, G B B Aut[B] Γ, and σ B is its complementing isomorphism; (ii) there is a self-complementary graph Σ with vertex set B such that G B AutΣ and each element of X B \ G B is its complementing isomorphism. 1. Let B be a minimal block system of V.
53 The imprimitive case Theorem 7 (Li and Praeger, 2003) If X is imprimitive, then: (i) [B] Γ is self-complementary, G B B Aut[B] Γ, and σ B is its complementing isomorphism; (ii) there is a self-complementary graph Σ with vertex set B such that G B AutΣ and each element of X B \ G B is its complementing isomorphism. 1. Let B be a minimal block system of V. 2. Then X = K.X B, and XB B primitive on B.
54 The imprimitive case Theorem 7 (Li and Praeger, 2003) If X is imprimitive, then: (i) [B] Γ is self-complementary, G B B Aut[B] Γ, and σ B is its complementing isomorphism; (ii) there is a self-complementary graph Σ with vertex set B such that G B AutΣ and each element of X B \ G B is its complementing isomorphism. 1. Let B be a minimal block system of V. 2. Then X = K.X B, and XB B primitive on B. 3. RB B < X B B If X B B insoluble, then X B B = Z2 p:(z l SL(2, 5)).
55 The imprimitive case Theorem 7 (Li and Praeger, 2003) If X is imprimitive, then: (i) [B] Γ is self-complementary, G B B Aut[B] Γ, and σ B is its complementing isomorphism; (ii) there is a self-complementary graph Σ with vertex set B such that G B AutΣ and each element of X B \ G B is its complementing isomorphism. 1. Let B be a minimal block system of V. 2. Then X = K.X B, and XB B primitive on B. 3. RB B < X B B If X B B insoluble, then X B B = Z2 p:(z l SL(2, 5)). 4. Consider K B X B B and K K B 1... K B 2.
56 The imprimitive case Theorem 7 (Li and Praeger, 2003) If X is imprimitive, then: (i) [B] Γ is self-complementary, G B B Aut[B] Γ, and σ B is its complementing isomorphism; (ii) there is a self-complementary graph Σ with vertex set B such that G B AutΣ and each element of X B \ G B is its complementing isomorphism. 1. Let B be a minimal block system of V. 2. Then X = K.X B, and XB B primitive on B. 3. RB B < X B B If X B B insoluble, then X B B = Z2 p:(z l SL(2, 5)). 4. Consider K B X B B and K K B 1... K B R B X B is transitive and metacyclic.
57 Conjecture 8 Self-complementary metacirculants are Cayley graphs? Example
58 Thank you!
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