Imprimitive symmetric graphs with cyclic blocks

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

Outline 1 Introduction Symmetric graphs Imprimitive symmetric graphs

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

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

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

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

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.

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 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.

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

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).

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).

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).

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.

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).

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

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.

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.

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.

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.

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

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)}.

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.

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}.

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.

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.

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.

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.

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

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

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).

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.

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.

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

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

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.

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.

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}.

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.

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).

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;

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).

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).

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.

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

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

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.

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

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.

v = 3 and m = 1 1 13 12 13 14 41 12 14 41 31 43 4 42 21 43 42 31 34 24 21 34 24 3 32 (a) 23 2 32 (b) 23 If v = 3 and m = 1, then Γ can be obtained from a regular map of valency 3 by truncation.

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

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.

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.

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.

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

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 ))

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.

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.

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

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.

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.

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.

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.

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

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.

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}.

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.

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.

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.

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.

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 )?

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?

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)).