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
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