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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 regular cliques 2/13

k

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

k

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

k

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

k

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

k

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

kk = 6

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

kk = 6

λ

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

kk = 6

λ

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

kk = 6

λ

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

kk = 6

λ

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

kk = 6

λ

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

kk = 6

λ

λ = 3

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

kk = 6

λ

λ = 3

edge-regularerg(10, 6, 3)

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

6

3

edge-regularerg(10, 6, 3)

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

6

3

edge-regularerg(10, 6, 3)

cliqueof order 4

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

6

3

edge-regularerg(10, 6, 3)

cliqueof order 4

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

6

3

edge-regularerg(10, 6, 3)

cliqueof order 4

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

6

3

edge-regularerg(10, 6, 3)

cliqueof order 4

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

6

3

edge-regularerg(10, 6, 3)

cliqueof order 4

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

6

3

edge-regularerg(10, 6, 3)

cliqueof order 4

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

6

3

edge-regularerg(10, 6, 3)

cliqueof order 4

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

6

3

edge-regularerg(10, 6, 3)

2-regular cliqueof order 4

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

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

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

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

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

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

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

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

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

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

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 regularsrg(10, 6, 3, 4)

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

Question (Neumaier 1981)Is every edge-regular graph with a regular clique stronglyregular?

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

Question (Neumaier 1981)Is every edge-regular graph with a regular clique stronglyregular?

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

Question (Neumaier 1981)Is every edge-regular graph with a regular clique stronglyregular?

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

An example

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

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

generating subset, i.e., s ∈ S =⇒ −s ∈ S and G = 〈S〉.

I The Cayley graph Cay(G, S) has vertex set G and edgeset

{{g, g + s} : g ∈ G and s ∈ S} .

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

0

1

2

−2 −1

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

An exampleI Γ = Cay(Z2

2 ⊕Z7, S)

I 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

An exampleI Γ = Cay(Z2

2 ⊕Z7, S)

I 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

An exampleI Γ = Cay(Z2

2 ⊕Z7, S)

I 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

An exampleI Γ = Cay(Z2

2 ⊕Z7, S)

I 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

An exampleI Γ = Cay(Z2

2 ⊕Z7, S)

I 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

An exampleI Γ = Cay(Z2

2 ⊕Z7, S)I Γ is edge-regular (28, 9, 2);

I 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

An exampleI Γ = Cay(Z2

2 ⊕Z7, S)I Γ is edge-regular (28, 9, 2);

I 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

An exampleI Γ = Cay(Z2

2 ⊕Z7, S)I Γ is edge-regular (28, 9, 2);

I 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 6= 0

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

An exampleI Γ = Cay(Z2

2 ⊕Z7, S)I Γ is edge-regular (28, 9, 2);

I 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 6= 0(∗,−b)

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

An example

I Γ = Cay(Z22 ⊕Z7, S)

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

I 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

An example

I Γ = Cay(Z22 ⊕Z7, S)

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

I 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

An example

I Γ = Cay(Z22 ⊕Z7, S)

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

I 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

An example

I Γ = Cay(Z22 ⊕Z7, S)

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

I 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

An example

I Γ = Cay(Z22 ⊕Z7, S)

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

I 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

General construction

I Generalise: Z22 ⊕Z7 to Z(c+1)/2 ⊕Z2

2 ⊕Fq.

I Works for q ≡ 1 (mod 6) such that the 3rd cyclotomicnumber c = c3

q(1, 2) is odd.

I Then there exists an erg(2(c + 1)q, 2c + q, 2c) having a1-regular clique of order 2c + 2.

I Take p ≡ 1 (mod 3) a prime s.t. 2 6≡ 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

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

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

I Smallest non-strongly-regular, edge-regular graph withregular clique (Neumaier graph)

I All known examples have 1-regular cliques

I ∃ Neumaier graphs with 3-regular cliques?

I ∃ Neumaier graphs with diameter > 3?

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

Open Closed problemsI Find general construction that includes erg(24, 8, 2)

I GG and Koolen (2018+): New infinite constructiona-antipodal erg(v, k, λ) to erg(v(λ + 2)/a, k + λ + 1, λ).

I Smallest non-strongly-regular, edge-regular graph withregular clique (Neumaier graph)

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

I All known examples have 1-regular cliquesI Evans and Goryainov (2018+): 2-regular cliques

I ∃ Neumaier graphs with 3-regular cliques?

I ∃ Neumaier graphs with diameter > 3?

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

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

I GG and Koolen (2018+): New infinite constructiona-antipodal erg(v, k, λ) to erg(v(λ + 2)/a, k + λ + 1, λ).

I Smallest non-strongly-regular, edge-regular graph withregular clique (Neumaier graph)

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

I All known examples have 1-regular cliquesI Evans and Goryainov (2018+): 2-regular cliques

I ∃ Neumaier graphs with 3-regular cliques?

I ∃ Neumaier graphs with diameter > 3?

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

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

I GG and Koolen (2018+): New infinite constructiona-antipodal erg(v, k, λ) to erg(v(λ + 2)/a, k + λ + 1, λ).

I Smallest non-strongly-regular, edge-regular graph withregular clique (Neumaier graph)

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

I All known examples have 1-regular cliquesI Evans and Goryainov (2018+): 2-regular cliques

I ∃ Neumaier graphs with 3-regular cliques?

I ∃ Neumaier graphs with diameter > 3?

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

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