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Edge-regular graphs and regular cliques Gary Greaves Nanyang - - PowerPoint PPT Presentation

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Edge-regular graphs and regular cliques Gary Greaves Nanyang Technological University, Singapore 23rd May 2018 joint work with J. H. Koolen Gary Greaves Edge-regular graphs and regular cliques 1/13 Gary Greaves Edge-regular graphs and

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Edge-regular graphs and regular cliques

Gary Greaves

Nanyang Technological University, Singapore

23rd May 2018

joint work with J. H. Koolen

Gary Greaves — Edge-regular graphs and regular cliques 1/13

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Gary Greaves — Edge-regular graphs and regular cliques 2/13

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k

Gary Greaves — Edge-regular graphs and regular cliques 2/13

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k

Gary Greaves — Edge-regular graphs and regular cliques 2/13

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k

Gary Greaves — Edge-regular graphs and regular cliques 2/13

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k

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k

Gary Greaves — Edge-regular graphs and regular cliques 2/13

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k k = 6

Gary Greaves — Edge-regular graphs and regular cliques 2/13

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k k = 6 λ

Gary Greaves — Edge-regular graphs and regular cliques 2/13

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k k = 6 λ

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k k = 6 λ

Gary Greaves — Edge-regular graphs and regular cliques 2/13

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k k = 6 λ

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k k = 6 λ

Gary Greaves — Edge-regular graphs and regular cliques 2/13

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k k = 6 λ λ = 3

Gary Greaves — Edge-regular graphs and regular cliques 2/13

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k k = 6 λ λ = 3 edge-regular erg(10, 6, 3)

Gary Greaves — Edge-regular graphs and regular cliques 2/13

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6 3 edge-regular erg(10, 6, 3)

Gary Greaves — Edge-regular graphs and regular cliques 3/13

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6 3 edge-regular erg(10, 6, 3) clique

  • f order 4

Gary Greaves — Edge-regular graphs and regular cliques 3/13

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6 3 edge-regular erg(10, 6, 3) clique

  • f order 4

Gary Greaves — Edge-regular graphs and regular cliques 3/13

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6 3 edge-regular erg(10, 6, 3) clique

  • f order 4

Gary Greaves — Edge-regular graphs and regular cliques 3/13

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6 3 edge-regular erg(10, 6, 3) clique

  • f order 4

Gary Greaves — Edge-regular graphs and regular cliques 3/13

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6 3 edge-regular erg(10, 6, 3) clique

  • f order 4

Gary Greaves — Edge-regular graphs and regular cliques 3/13

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6 3 edge-regular erg(10, 6, 3) clique

  • f order 4

Gary Greaves — Edge-regular graphs and regular cliques 3/13

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6 3 edge-regular erg(10, 6, 3) clique

  • f order 4

Gary Greaves — Edge-regular graphs and regular cliques 3/13

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6 3 edge-regular erg(10, 6, 3) 2-regular clique

  • f order 4

Gary Greaves — Edge-regular graphs and regular cliques 3/13

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Theorem (Neumaier 1981)

Let Γ be edge-regular with a regular clique. Suppose Γ is vertex-transitive and edge-transitive. Then Γ is strongly regular.

Gary Greaves — Edge-regular graphs and regular cliques 4/13

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Theorem (Neumaier 1981)

Let Γ be edge-regular with a regular clique. Suppose Γ is vertex-transitive and edge-transitive. Then Γ is strongly regular.

Gary Greaves — Edge-regular graphs and regular cliques 4/13

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Theorem (Neumaier 1981)

Let Γ be edge-regular with a regular clique. Suppose Γ is vertex-transitive and edge-transitive. Then Γ is strongly regular. 6 3

Gary Greaves — Edge-regular graphs and regular cliques 4/13

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Theorem (Neumaier 1981)

Let Γ be edge-regular with a regular clique. Suppose Γ is vertex-transitive and edge-transitive. Then Γ is strongly regular. 6 3 µ

Gary Greaves — Edge-regular graphs and regular cliques 4/13

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Theorem (Neumaier 1981)

Let Γ be edge-regular with a regular clique. Suppose Γ is vertex-transitive and edge-transitive. Then Γ is strongly regular. 6 3 µ

Gary Greaves — Edge-regular graphs and regular cliques 4/13

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Theorem (Neumaier 1981)

Let Γ be edge-regular with a regular clique. Suppose Γ is vertex-transitive and edge-transitive. Then Γ is strongly regular. 6 3 µ

Gary Greaves — Edge-regular graphs and regular cliques 4/13

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Theorem (Neumaier 1981)

Let Γ be edge-regular with a regular clique. Suppose Γ is vertex-transitive and edge-transitive. Then Γ is strongly regular. 6 3 µ

Gary Greaves — Edge-regular graphs and regular cliques 4/13

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Theorem (Neumaier 1981)

Let Γ be edge-regular with a regular clique. Suppose Γ is vertex-transitive and edge-transitive. Then Γ is strongly regular. 6 3 µ

Gary Greaves — Edge-regular graphs and regular cliques 4/13

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Theorem (Neumaier 1981)

Let Γ be edge-regular with a regular clique. Suppose Γ is vertex-transitive and edge-transitive. Then Γ is strongly regular. 6 3 µ µ = 4

Gary Greaves — Edge-regular graphs and regular cliques 4/13

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Theorem (Neumaier 1981)

Let Γ be edge-regular with a regular clique. Suppose Γ is vertex-transitive and edge-transitive. Then Γ is strongly regular. 6 3 µ µ = 4 strongly regular srg(10, 6, 3, 4)

Gary Greaves — Edge-regular graphs and regular cliques 4/13

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Question (Neumaier 1981)

Is every edge-regular graph with a regular clique strongly regular?

Gary Greaves — Edge-regular graphs and regular cliques 5/13

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Question (Neumaier 1981)

Is every edge-regular graph with a regular clique strongly regular?

Answer (GG and Koolen 2018)

No.

Gary Greaves — Edge-regular graphs and regular cliques 5/13

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Question (Neumaier 1981)

Is every edge-regular graph with a regular clique strongly regular?

Answer (GG and Koolen 2018)

  • No. There exist infinitely many non-strongly-regular,

edge-regular vertex-transitive graphs with regular cliques.

Gary Greaves — Edge-regular graphs and regular cliques 5/13

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An example

Gary Greaves — Edge-regular graphs and regular cliques 6/13

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Cayley graphs

◮ Let G be an (additive) group and S ⊆ G a (symmetric)

generating subset, i.e., s ∈ S =

⇒ −s ∈ S and G = S.

◮ The Cayley graph Cay(G, S) has vertex set G and edge

set

{{g, g + s} : g ∈ G and s ∈ S} .

Example

Γ = Cay(Z5, S) Generating set S = {−1, 1} 1 2

−2 −1

Gary Greaves — Edge-regular graphs and regular cliques 7/13

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An example

◮ Γ = Cay(Z2

2 ⊕ Z7, S)

(01, 0) (10, 0) (11, 0) (01, ±1) (10, ±2) (11, ±3)

Generating set S

Gary Greaves — Edge-regular graphs and regular cliques 8/13

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An example

◮ Γ = Cay(Z2

2 ⊕ Z7, S)

◮ Γ is edge-regular (28, 9, 2):

(01, 0) (10, 0) (11, 0) (01, ±1) (10, ±2) (11, ±3)

Generating set S

Gary Greaves — Edge-regular graphs and regular cliques 8/13

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An example

◮ Γ = Cay(Z2

2 ⊕ Z7, S)

◮ Γ is edge-regular (28, 9, 2):

(01, 0) (10, 0) (11, 0) (01, ±1) (10, ±2) (11, ±3)

Generating set S

(00, 0)

Gary Greaves — Edge-regular graphs and regular cliques 8/13

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An example

◮ Γ = Cay(Z2

2 ⊕ Z7, S)

◮ Γ is edge-regular (28, 9, 2):

(01, 0) (10, 0) (11, 0) (01, ±1) (10, ±2) (11, ±3)

Generating set S

(00, 0) (01, 0) (10, 0) (11, 0) (01, 1) (01, −1) (10, 2) (10, −2) (11, 3) (11, −3)

Gary Greaves — Edge-regular graphs and regular cliques 8/13

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An example

◮ Γ = Cay(Z2

2 ⊕ Z7, S)

◮ Γ is edge-regular (28, 9, 2):

(01, 0) (10, 0) (11, 0) (01, ±1) (10, ±2) (11, ±3)

Generating set S

(00, 0) (01, 0) (10, 0) (11, 0) (01, 1) (01, −1) (10, 2) (10, −2) (11, 3) (11, −3)

Gary Greaves — Edge-regular graphs and regular cliques 8/13

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An example

◮ Γ = Cay(Z2

2 ⊕ Z7, S)

◮ Γ is edge-regular (28, 9, 2); ◮ Γ has a 1-regular 4-clique:

(01, 0) (10, 0) (11, 0) (01, ±1) (10, ±2) (11, ±3)

Generating set S

Gary Greaves — Edge-regular graphs and regular cliques 9/13

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An example

◮ Γ = Cay(Z2

2 ⊕ Z7, S)

◮ Γ is edge-regular (28, 9, 2); ◮ Γ has a 1-regular 4-clique:

(01, 0) (10, 0) (11, 0) (01, ±1) (10, ±2) (11, ±3)

Generating set S

(00, 0) (01, 0) (10, 0) (11, 0)

Gary Greaves — Edge-regular graphs and regular cliques 9/13

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An example

◮ Γ = Cay(Z2

2 ⊕ Z7, S)

◮ Γ is edge-regular (28, 9, 2); ◮ Γ has a 1-regular 4-clique:

(01, 0) (10, 0) (11, 0) (01, ±1) (10, ±2) (11, ±3)

Generating set S

(00, 0) (01, 0) (10, 0) (11, 0) (a, b)

b = 0

Gary Greaves — Edge-regular graphs and regular cliques 9/13

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An example

◮ Γ = Cay(Z2

2 ⊕ Z7, S)

◮ Γ is edge-regular (28, 9, 2); ◮ Γ has a 1-regular 4-clique:

(01, 0) (10, 0) (11, 0) (01, ±1) (10, ±2) (11, ±3)

Generating set S

(00, 0) (01, 0) (10, 0) (11, 0) (a, b)

b = 0

(∗, −b)

Gary Greaves — Edge-regular graphs and regular cliques 9/13

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An example

◮ Γ = Cay(Z2

2 ⊕ Z7, S)

◮ Γ is edge-regular (28, 9, 2); ◮ Γ has a 1-regular 4-clique; ◮ Γ is not strongly regular:

(01, 0) (10, 0) (11, 0) (01, ±1) (10, ±2) (11, ±3)

Generating set S

Gary Greaves — Edge-regular graphs and regular cliques 10/13

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An example

◮ Γ = Cay(Z2

2 ⊕ Z7, S)

◮ Γ is edge-regular (28, 9, 2); ◮ Γ has a 1-regular 4-clique; ◮ Γ is not strongly regular:

(01, 0) (10, 0) (11, 0) (01, ±1) (10, ±2) (11, ±3)

Generating set S

(00, 0) (01, 0) (10, 0) (11, 0) (01, 1) (01, −1) (10, 2) (10, −2) (11, 3) (11, −3)

Gary Greaves — Edge-regular graphs and regular cliques 10/13

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An example

◮ Γ = Cay(Z2

2 ⊕ Z7, S)

◮ Γ is edge-regular (28, 9, 2); ◮ Γ has a 1-regular 4-clique; ◮ Γ is not strongly regular:

(01, 0) (10, 0) (11, 0) (01, ±1) (10, ±2) (11, ±3)

Generating set S

(00, 0) (01, 0) (10, 0) (11, 0) (01, 1) (01, −1) (10, 2) (10, −2) (11, 3) (11, −3) (10, 1) (11, 1)

Gary Greaves — Edge-regular graphs and regular cliques 10/13

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An example

◮ Γ = Cay(Z2

2 ⊕ Z7, S)

◮ Γ is edge-regular (28, 9, 2); ◮ Γ has a 1-regular 4-clique; ◮ Γ is not strongly regular:

(01, 0) (10, 0) (11, 0) (01, ±1) (10, ±2) (11, ±3)

Generating set S

(00, 0) (01, 0) (10, 0) (11, 0) (01, 1) (01, −1) (10, 2) (10, −2) (11, 3) (11, −3) (10, 1) (11, 1)

Gary Greaves — Edge-regular graphs and regular cliques 10/13

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An example

◮ Γ = Cay(Z2

2 ⊕ Z7, S)

◮ Γ is edge-regular (28, 9, 2); ◮ Γ has a 1-regular 4-clique; ◮ Γ is not strongly regular:

(01, 0) (10, 0) (11, 0) (01, ±1) (10, ±2) (11, ±3)

Generating set S

(00, 0) (01, 0) (10, 0) (11, 0) (01, 1) (01, −1) (10, 2) (10, −2) (11, 3) (11, −3) (10, 1) (11, 1)

Gary Greaves — Edge-regular graphs and regular cliques 10/13

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General construction

◮ Generalise: Z2

2 ⊕ Z7

to Z(c+1)/2 ⊕ Z2

2 ⊕ Fq.

◮ Works for q ≡ 1 (mod 6) such that the 3rd cyclotomic

number c = c3

q(1, 2) is odd.

◮ Then there exists an erg(2(c + 1)q, 2c + q, 2c) having a

1-regular clique of order 2c + 2.

◮ Take p ≡ 1 (mod 3) a prime s.t. 2 ≡ x3 (mod p).

Then there exist a such that c3

pa(1, 2) is odd.

Gary Greaves — Edge-regular graphs and regular cliques 11/13

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Examples in the wild

Four erg(24, 8, 2) graphs with a 1-regular clique

Gary Greaves — Edge-regular graphs and regular cliques 12/13

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Open problems

◮ Find general construction that includes erg(24, 8, 2) ◮ Smallest non-strongly-regular, edge-regular graph with

regular clique (Neumaier graph)

◮ All known examples have 1-regular cliques

Gary Greaves — Edge-regular graphs and regular cliques 13/13

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Open Closed problems

◮ Find general construction that includes erg(24, 8, 2)

◮ GG and Koolen (2018+): New infinite construction

a-antipodal erg(v, k, λ) to erg(v(λ + 2)/a, k + λ + 1, λ).

◮ Smallest non-strongly-regular, edge-regular graph with

regular clique (Neumaier graph)

◮ Evans and Goryainov (2018+): Smallest is erg(16, 9, 4)

◮ All known examples have 1-regular cliques

◮ Evans and Goryainov (2018+): 2-regular cliques Gary Greaves — Edge-regular graphs and regular cliques 13/13

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Problems

◮ Find general construction that includes erg(24, 8, 2)

◮ GG and Koolen (2018+): New infinite construction

a-antipodal erg(v, k, λ) to erg(v(λ + 2)/a, k + λ + 1, λ).

◮ Smallest non-strongly-regular, edge-regular graph with

regular clique (Neumaier graph)

◮ Evans and Goryainov (2018+): Smallest is erg(16, 9, 4)

◮ All known examples have 1-regular cliques

◮ Evans and Goryainov (2018+): 2-regular cliques

◮ ∃ Neumaier graphs with 3-regular cliques?

Gary Greaves — Edge-regular graphs and regular cliques 13/13

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Problems

◮ Find general construction that includes erg(24, 8, 2)

◮ GG and Koolen (2018+): New infinite construction

a-antipodal erg(v, k, λ) to erg(v(λ + 2)/a, k + λ + 1, λ).

◮ Smallest non-strongly-regular, edge-regular graph with

regular clique (Neumaier graph)

◮ Evans and Goryainov (2018+): Smallest is erg(16, 9, 4)

◮ All known examples have 1-regular cliques

◮ Evans and Goryainov (2018+): 2-regular cliques

◮ ∃ Neumaier graphs with 3-regular cliques? ◮ ∃ Neumaier graphs with diameter 3?

Gary Greaves — Edge-regular graphs and regular cliques 13/13

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Thank you for your attention

Further reading:

  • G. R. W. Greaves and J. H. Koolen, Edge-regular graphs with regular cliques,

European J. Combin. 71 (2018), pp. 194–201.

Gary Greaves — Edge-regular graphs and regular cliques 14/13