Imprimitive symmetric graphs with cyclic blocks

Transcription

1 Imprimitive symmetric graphs with cyclic blocks Sanming Zhou Department of Mathematics and Statistics University of Melbourne Joint work with Cai Heng Li and Cheryl E. Praeger December 17, 2008

2 Outline 1 Introduction Symmetric graphs Imprimitive symmetric graphs

3 Outline 1 Introduction Symmetric graphs Imprimitive symmetric graphs 2 A family of imprimitive symmetric graphs

4 Outline 1 Introduction Symmetric graphs Imprimitive symmetric graphs 2 A family of imprimitive symmetric graphs 3 Imprimitive symmetric graphs with cyclic blocks Cyclic blocks: the simplest case Constructions

5 Outline 1 Introduction Symmetric graphs Imprimitive symmetric graphs 2 A family of imprimitive symmetric graphs 3 Imprimitive symmetric graphs with cyclic blocks Cyclic blocks: the simplest case Constructions 4 Questions

6 Arc and s-arc An arc of a graph is an ordered pair of adjacent vertices.

7 Arc and s-arc An arc of a graph is an ordered pair of adjacent vertices. An s-arc is a sequence of s + 1 vertices such that any two consecutive terms are adjacent and any three consecutive terms are distinct.

8 Arc and s-arc An arc of a graph is an ordered pair of adjacent vertices. An s-arc is a sequence of s + 1 vertices such that any two consecutive terms are adjacent and any three consecutive terms are distinct. s-arc path of length s.

9 Arc and s-arc An arc of a graph is an ordered pair of adjacent vertices. An s-arc is a sequence of s + 1 vertices such that any two consecutive terms are adjacent and any three consecutive terms are distinct. s-arc path of length s. Arc(Γ) := set of arcs of a graph Γ.

10 Symmetric and highly arc-transitive graphs Let Γ admit a group G as a group of automorphisms.

11 Symmetric and highly arc-transitive graphs Let Γ admit a group G as a group of automorphisms. That is, G acts on V (Γ) and preserves the adjacency of Γ, e.g. G Aut(Γ).

12 Symmetric and highly arc-transitive graphs Let Γ admit a group G as a group of automorphisms. That is, G acts on V (Γ) and preserves the adjacency of Γ, e.g. G Aut(Γ). Γ is called G-symmetric (or G-arc transitive) if G is transitive on V (Γ) and transitive on Arc(Γ).

13 Symmetric and highly arc-transitive graphs Let Γ admit a group G as a group of automorphisms. That is, G acts on V (Γ) and preserves the adjacency of Γ, e.g. G Aut(Γ). Γ is called G-symmetric (or G-arc transitive) if G is transitive on V (Γ) and transitive on Arc(Γ). That is, for any α, β V (Γ) there exists g G permuting α to β, and for any (α, β), (γ, δ) Arc(Γ) there exists g G mapping (α, β) to (γ, δ).

14 Symmetric and highly arc-transitive graphs Let Γ admit a group G as a group of automorphisms. That is, G acts on V (Γ) and preserves the adjacency of Γ, e.g. G Aut(Γ). Γ is called G-symmetric (or G-arc transitive) if G is transitive on V (Γ) and transitive on Arc(Γ). That is, for any α, β V (Γ) there exists g G permuting α to β, and for any (α, β), (γ, δ) Arc(Γ) there exists g G mapping (α, β) to (γ, δ). Γ is called (G, s)-arc transitive if G is transitive on V (Γ) and the set of s-arcs.

15 An example: Tutte s 8-cage Tutte s 8-cage is 5-arc transitive. It is a cubic graph of girth 8 with minimum order (30 vertices).

16 Group-theoretic language: coset graphs Let G be a group and H < G be core-free.

17 Group-theoretic language: coset graphs Let G be a group and H < G be core-free. Let g G be a 2-element such that g N G (H) and g 2 H H g.

18 Group-theoretic language: coset graphs Let G be a group and H < G be core-free. Let g G be a 2-element such that g N G (H) and g 2 H H g. Cos(G, H, HgH) is defined to be the graph with vertex set [G : H] such that Hx Hy if and only if xy 1 HgH.

19 Group-theoretic language: coset graphs Let G be a group and H < G be core-free. Let g G be a 2-element such that g N G (H) and g 2 H H g. Cos(G, H, HgH) is defined to be the graph with vertex set [G : H] such that Hx Hy if and only if xy 1 HgH. Cos(G, H, HgH) is G-symmetric, and any symmetric graph is of this form.

20 Group-theoretic language: coset graphs Let G be a group and H < G be core-free. Let g G be a 2-element such that g N G (H) and g 2 H H g. Cos(G, H, HgH) is defined to be the graph with vertex set [G : H] such that Hx Hy if and only if xy 1 HgH. Cos(G, H, HgH) is G-symmetric, and any symmetric graph is of this form. Cos(G, H, HgH) is connected if and only if H, g = G.

21 Imprimitive symmetric graphs Let Γ be a G-symmetric graph.

22 Imprimitive symmetric graphs Let Γ be a G-symmetric graph. A partition B of V (Γ) is G-invariant if for any B B and g G, B g := {α g : α B} B, e.g. trivial partitions {{α} : α V (Γ)} and {V (Γ)}.

23 Imprimitive symmetric graphs Let Γ be a G-symmetric graph. A partition B of V (Γ) is G-invariant if for any B B and g G, B g := {α g : α B} B, e.g. trivial partitions {{α} : α V (Γ)} and {V (Γ)}. Γ is imprimitive if V (Γ) admits a nontrivial G-invariant partition B.

24 Imprimitive symmetric graphs Let Γ be a G-symmetric graph. A partition B of V (Γ) is G-invariant if for any B B and g G, B g := {α g : α B} B, e.g. trivial partitions {{α} : α V (Γ)} and {V (Γ)}. Γ is imprimitive if V (Γ) admits a nontrivial G-invariant partition B. This occurs if and only if G α is not a maximal subgroup of G, where G α := {g G : α g = α}.

25 Imprimitive symmetric graphs Let Γ be a G-symmetric graph. A partition B of V (Γ) is G-invariant if for any B B and g G, B g := {α g : α B} B, e.g. trivial partitions {{α} : α V (Γ)} and {V (Γ)}. Γ is imprimitive if V (Γ) admits a nontrivial G-invariant partition B. This occurs if and only if G α is not a maximal subgroup of G, where G α := {g G : α g = α}. Equivlently, Cos(G, H, HgH) is imprimitive if and only if H is not a maximal subgroup of G.

26 Imprimitive symmetric graphs Let Γ be a G-symmetric graph. A partition B of V (Γ) is G-invariant if for any B B and g G, B g := {α g : α B} B, e.g. trivial partitions {{α} : α V (Γ)} and {V (Γ)}. Γ is imprimitive if V (Γ) admits a nontrivial G-invariant partition B. This occurs if and only if G α is not a maximal subgroup of G, where G α := {g G : α g = α}. Equivlently, Cos(G, H, HgH) is imprimitive if and only if H is not a maximal subgroup of G. Γ B := quotient graph w.r.t B.

27 Imprimitive symmetric graphs Let Γ be a G-symmetric graph. A partition B of V (Γ) is G-invariant if for any B B and g G, B g := {α g : α B} B, e.g. trivial partitions {{α} : α V (Γ)} and {V (Γ)}. Γ is imprimitive if V (Γ) admits a nontrivial G-invariant partition B. This occurs if and only if G α is not a maximal subgroup of G, where G α := {g G : α g = α}. Equivlently, Cos(G, H, HgH) is imprimitive if and only if H is not a maximal subgroup of G. Γ B := quotient graph w.r.t B. Γ B is G-symmetric.

28 An example: Dodecahedron The dodecahedron graph is A 5 -symmetric and the partition with each part containing antipodal vertices is A 5 -invariant. The quotient graph is isomorphic to Petersen graph.

29 Symmetric graphs with k = v 2 1 Let (Γ, B) be an imprimitive G-symmetric graph.

30 Symmetric graphs with k = v 2 1 Let (Γ, B) be an imprimitive G-symmetric graph. v := block size of B.

31 Symmetric graphs with k = v 2 1 Let (Γ, B) be an imprimitive G-symmetric graph. v := block size of B. k := no. vertices in B having a neighbour in C (where B, C are adjacent).

32 Symmetric graphs with k = v 2 1 Let (Γ, B) be an imprimitive G-symmetric graph. v := block size of B. k := no. vertices in B having a neighbour in C (where B, C are adjacent). Γ[B, C] := bipartite subgraph of Γ induced by B C with isolates deleted.

33 Symmetric graphs with k = v 2 1 Let (Γ, B) be an imprimitive G-symmetric graph. v := block size of B. k := no. vertices in B having a neighbour in C (where B, C are adjacent). Γ[B, C] := bipartite subgraph of Γ induced by B C with isolates deleted. Consider the case k = v 2 1.

34 Case k = v 2 1 k = v 2 1 B,C B C

35 An auxiliary graph when k = v 2 1 B, C := set of vertices of B not adjacent to any vertex of C.

36 An auxiliary graph when k = v 2 1 B, C := set of vertices of B not adjacent to any vertex of C. B, C = 2.

37 An auxiliary graph when k = v 2 1 B, C := set of vertices of B not adjacent to any vertex of C. B, C = 2. Γ B := multigraph with vertex set B and an edge joining the two vertices of B, C for each C adjacent to B in Γ B.

38 An auxiliary graph when k = v 2 1 B, C := set of vertices of B not adjacent to any vertex of C. B, C = 2. Γ B := multigraph with vertex set B and an edge joining the two vertices of B, C for each C adjacent to B in Γ B. Simple(Γ B ) := underlying simple graph of Γ B. Simple(Γ B ) is G B -vertex- and G B -edge-transitive, where G B := {g G : B g = B}.

39 Structure of Γ B Lemma [Iranmanesh, Praeger, Z] Suppose k = v 2 1. Then one of the following occurs: (a) Γ B is connected; (b) v is even and Simple(Γ B ) is a perfect matching between the vertices of B.

40 Known results when k = v 2 1 Γ B is (G, 2)-arc transitive (even if Γ is not (G, 2)-arc transitive) iff Γ B is simple and v = 3 or Γ B = (v/2) K 2 (Iranmanesh, Praeger, Z).

41 Known results when k = v 2 1 Γ B is (G, 2)-arc transitive (even if Γ is not (G, 2)-arc transitive) iff Γ B is simple and v = 3 or Γ B = (v/2) K 2 (Iranmanesh, Praeger, Z). We know when Γ B inherits (G, 2)-arc transitivity from Γ;

42 Known results when k = v 2 1 Γ B is (G, 2)-arc transitive (even if Γ is not (G, 2)-arc transitive) iff Γ B is simple and v = 3 or Γ B = (v/2) K 2 (Iranmanesh, Praeger, Z). We know when Γ B inherits (G, 2)-arc transitivity from Γ; and some information about Γ and Γ B in this case (Iranmanesh, Praeger, Z).

43 Known results when k = v 2 1 Γ B is (G, 2)-arc transitive (even if Γ is not (G, 2)-arc transitive) iff Γ B is simple and v = 3 or Γ B = (v/2) K 2 (Iranmanesh, Praeger, Z). We know when Γ B inherits (G, 2)-arc transitivity from Γ; and some information about Γ and Γ B in this case (Iranmanesh, Praeger, Z). But we know nothing about Γ and Γ B when Γ B is connected (except v = 3).

44 Known results when k = v 2 1 Γ B is (G, 2)-arc transitive (even if Γ is not (G, 2)-arc transitive) iff Γ B is simple and v = 3 or Γ B = (v/2) K 2 (Iranmanesh, Praeger, Z). We know when Γ B inherits (G, 2)-arc transitivity from Γ; and some information about Γ and Γ B in this case (Iranmanesh, Praeger, Z). But we know nothing about Γ and Γ B when Γ B is connected (except v = 3). Let us try the simplest case, namely Simple(Γ B ) has valency 2.

45 The case when Simple(Γ B ) has valency 2 D,B D C,B B B,C B,D C

46 The case when Simple(Γ B ) has valency 2 D,B D C,B B B,C B,D C

47 Local structures Theorem [Li, Praeger, Z] Suppose k = v 2 1, Γ B is connected and Simple(Γ B ) has valency 2. Then Simple(Γ B ) = C v, Γ B is of valency mv (where m is the multiplicity of each edge of Γ B ), and one of the following occurs. (a) v = 3 and Γ is of valency m; (b) v = 4, Γ[B, C] = K 2,2, and Γ is a connected graph of valency 4m; (c) v = 4, Γ[B, C] = 2 K 2, and Γ is of valency 2m.

48 Regular maps A map is a 2-cell embedding of a connected graph on a closed surface.

49 Regular maps A map is a 2-cell embedding of a connected graph on a closed surface. A map M is regular if its automorphism group Aut(M) is regular on incident vertex-edge-face triples.

50 v = 3 and m = (a) (b) 23 If v = 3 and m = 1, then Γ can be obtained from a regular map of valency 3 by truncation.

51 Incident vertex-face pairs Let M be a 4-valent regular map, and let G = Aut(M).

52 Incident vertex-face pairs Let M be a 4-valent regular map, and let G = Aut(M). For each edge {σ, σ } of M, let f, f denote the faces of M such that {σ, σ } is on the boundary of both f and f.

53 Incident vertex-face pairs Let M be a 4-valent regular map, and let G = Aut(M). For each edge {σ, σ } of M, let f, f denote the faces of M such that {σ, σ } is on the boundary of both f and f. Let O σ (f ) and O σ (f ) be the other two faces of M incident with σ and opposite to f and f respectively.

54 Incident vertex-face pairs Let M be a 4-valent regular map, and let G = Aut(M). For each edge {σ, σ } of M, let f, f denote the faces of M such that {σ, σ } is on the boundary of both f and f. Let O σ (f ) and O σ (f ) be the other two faces of M incident with σ and opposite to f and f respectively. Define O σ (f ) and O σ (f ) similarly.

55 Incident vertex-face pairs O σ (f ') f O σ' (f ') O σ (f) σ f ' σ' O σ' (f)

56 Flag graphs of 4-valent regular maps We construct four graphs Γ 1, Γ 2, Γ 3, Γ 4 as follows. They have vertices the incident vertex-face pairs of M such that for each edge {σ, σ } of M: Γ 1 : (σ, f ) (σ, f ), (σ, f ) (σ, f ) Γ 2 : (σ, f ) (σ, f ), (σ, f ) (σ, f ) Γ 3 : (σ, O σ (f )) (σ, O σ (f )), (σ, O σ (f )) (σ, O σ (f )) Γ 4 : (σ, O σ (f )) (σ, O σ (f )), (σ, O σ (f )) (σ, O σ (f ))

57 Flag graphs of 4-valent regular maps We construct four graphs Γ 1, Γ 2, Γ 3, Γ 4 as follows. They have vertices the incident vertex-face pairs of M such that for each edge {σ, σ } of M: Γ 1 : (σ, f ) (σ, f ), (σ, f ) (σ, f ) Γ 2 : (σ, f ) (σ, f ), (σ, f ) (σ, f ) Γ 3 : (σ, O σ (f )) (σ, O σ (f )), (σ, O σ (f )) (σ, O σ (f )) Γ 4 : (σ, O σ (f )) (σ, O σ (f )), (σ, O σ (f )) (σ, O σ (f )) These are examples of graphs in case (c) with m = 1.

58 Flag graphs of 4-valent regular maps We construct four graphs Γ 1, Γ 2, Γ 3, Γ 4 as follows. They have vertices the incident vertex-face pairs of M such that for each edge {σ, σ } of M: Γ 1 : (σ, f ) (σ, f ), (σ, f ) (σ, f ) Γ 2 : (σ, f ) (σ, f ), (σ, f ) (σ, f ) Γ 3 : (σ, O σ (f )) (σ, O σ (f )), (σ, O σ (f )) (σ, O σ (f )) Γ 4 : (σ, O σ (f )) (σ, O σ (f )), (σ, O σ (f )) (σ, O σ (f )) These are examples of graphs in case (c) with m = 1. No example is known in case (c) with m > 1.

59 A group-theoretic construction Let p be a prime such that p 1 (mod 16), and let G = PSL(2, p).

60 A group-theoretic construction Let p be a prime such that p 1 (mod 16), and let G = PSL(2, p). Let H be a Sylow 2-subgroup of G.

61 A group-theoretic construction Let p be a prime such that p 1 (mod 16), and let G = PSL(2, p). Let H be a Sylow 2-subgroup of G. H = a : b = D 16, a 4, b = Z 2 2, N G ( a 4, b ) = S 4.

62 A group-theoretic construction Let p be a prime such that p 1 (mod 16), and let G = PSL(2, p). Let H be a Sylow 2-subgroup of G. H = a : b = D 16, a 4, b = Z 2 2, N G ( a 4, b ) = S 4. There exists an involution g N G ( a 4, b ) \ a 2, b such that g interchanges a 4 and b.

63 A group-theoretic construction Let p be a prime such that p 1 (mod 16), and let G = PSL(2, p). Let H be a Sylow 2-subgroup of G. H = a : b = D 16, a 4, b = Z 2 2, N G ( a 4, b ) = S 4. There exists an involution g N G ( a 4, b ) \ a 2, b such that g interchanges a 4 and b. Let L = a 4, ba = Z 2 2.

64 A group-theoretic construction Define Σ := Cos(G, H, HgH), Γ := Cos(G, L, LgL).

65 A group-theoretic construction Define Σ := Cos(G, H, HgH), Γ := Cos(G, L, LgL). Γ and Σ are G-symmetric, connected and of valency 4.

66 A group-theoretic construction Define Σ := Cos(G, H, HgH), Γ := Cos(G, L, LgL). Γ and Σ are G-symmetric, connected and of valency 4. Let B := [H : L] and B := {B x : x G}.

67 A group-theoretic construction Define Σ := Cos(G, H, HgH), Γ := Cos(G, L, LgL). Γ and Σ are G-symmetric, connected and of valency 4. Let B := [H : L] and B := {B x : x G}. B is a G-invariant partition of V (Γ) = [G : L] such that k = v 2 = 2, Γ B = C 4, Γ[B, C] = K 2,2 and Σ = Γ B.

68 A group-theoretic construction Define Σ := Cos(G, H, HgH), Γ := Cos(G, L, LgL). Γ and Σ are G-symmetric, connected and of valency 4. Let B := [H : L] and B := {B x : x G}. B is a G-invariant partition of V (Γ) = [G : L] such that k = v 2 = 2, Γ B = C 4, Γ[B, C] = K 2,2 and Σ = Γ B. These are examples of graphs in case (b) with m = 1.

69 A group-theoretic construction Define Σ := Cos(G, H, HgH), Γ := Cos(G, L, LgL). Γ and Σ are G-symmetric, connected and of valency 4. Let B := [H : L] and B := {B x : x G}. B is a G-invariant partition of V (Γ) = [G : L] such that k = v 2 = 2, Γ B = C 4, Γ[B, C] = K 2,2 and Σ = Γ B. These are examples of graphs in case (b) with m = 1. No example is known in case (b) with m > 1.

70 A group-theoretic construction Corollary [Li, Praeger, Z] There exists an infinite family of connected symmetric graphs Γ of valency 4 which have a quotient graph Γ B of valency 4 such that Γ is not a cover of Γ B. This is the first (infinite) family of graphs with these properties.

71 Questions In the case where k = v 2 1 and Γ B is connected, is v bounded by some function of the valency of Simple(Γ B )?

72 Questions In the case where k = v 2 1 and Γ B is connected, is v bounded by some function of the valency of Simple(Γ B )? Can Γ be determined for small values of m?

73 Questions In the case where k = v 2 1 and Γ B is connected, is v bounded by some function of the valency of Simple(Γ B )? Can Γ be determined for small values of m? Study the case when Simple(Γ B ) = C v and m is small (e.g. m = 3 in case (a), m = 2, 3 in case (b) and m = 2 in case (c)).