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Transcript of On highly regular strongly regular graphs Types Category of Graphs A graph-homomorphismh :...

• On highly regular strongly regular graphs

Christian Pech

TU Dresden

MTAGT’14, Villanova in June 2014

Ch. Pech On highly regular strongly regular graphs June 2014 1 / 20

• k -Homogeneity

The local-global principle

Every isomorphism between two substructures of a structure can be extended to an automorphism of the structure.

The formal definition for graphs

Let Γ = (V ,E) be a graph. Gamma is called k-homogeneous if for all V1,V2 ⊆ V with |V1| = |V2| ≤ k and for each isomorphism ψ : Γ(V1) → Γ(V2) there exists an automorphism ϕ of Γ such that ϕ|V1 = ψ.

Ch. Pech On highly regular strongly regular graphs June 2014 2 / 20

• Types

Category of Graphs

A graph-homomorphism h : Γ1 → Γ2 is a function from V (Γ1) to V (Γ2) that maps edges to edges. an embedding is an injective homomorphism with the property that the preimage of edges are edges, too.

Regularity-Types

A regularity-type (or type, for short) is an embedding of finite graphs. Regularity-types are denoted like

T : Γ1 →֒ Γ2.

Order of a Type

T : Γ1 →֒ Γ2 has order (n,m) if |V (Γ1)| = n, and |V (Γ2)| = m

Ch. Pech On highly regular strongly regular graphs June 2014 3 / 20

• T-regularity

Given: A graph Γ, a type T : ∆1 →֒ ∆2

Counting T:

Let ι : ∆1 →֒ Γ. #(Γ,T, ι) we define to be the number of embeddings ι̂ : ∆2 →֒ Γ that make the following diagram commute:

∆1 Γ

∆2

ι

ι̂

T

Ch. Pech On highly regular strongly regular graphs June 2014 4 / 20

• T-regularity (cont.)

T-regularity

Γ is called T-regular if the number #(Γ,T, ι) does not depend on the embedding

ι : ∆1 →֒ Γ.

In this case this number is denoted by #(Γ,T)

Remark

A concept equivalent to T-regularity, but in the category of complete colored graphs, was introduced and studied by Evdokimov and Ponomarenko (2000) in relation with the t-vertex condition for association schemes.

Example

If T is given by

x x =

then Γ is T-regular if and only if it is regular.

Ch. Pech On highly regular strongly regular graphs June 2014 5 / 20

• (n,m)-regularity

Definition

A graph Γ is (n,m)-regular if for all 1 ≤ k ≤ n and k < l ≤ m, and for every type T of order (k , l) we have that Γ is T-regular.

(1,2)-regular is the same as regular,

(2,3)-regular is the same as strongly regular,

(k , k + 1)-regular is the same as k-isoregular,

(2, t)-regular is the same as fulfilling the t-vertex condition

Ch. Pech On highly regular strongly regular graphs June 2014 6 / 20

• Known examples

Hestenes, Higman (1971): Point graphs of generalized quadrangles fulfill the 4-vc,

A.V.Ivanov (1989): found a graph on 256 vertices with the 4-vc (not 2-homogeneous),

Brouwer, Ivanov, Klin (1989): generalization to an infinite series,

A.V.Ivanov (1994): another infinite series of graphs with the 4-vc,

Reichard (2000): both series fulfill the 5-vc, A.A.Ivanov, Faradžev, Klin (1984) constructed a srg on 280

vertices with Aut(J2) as automorphism group, Reichard (2000): this graph fulfills the 4-vc, Reichard (2003): point graphs of GQ(s, t) fulfill the 5-vc, Reichard (2003): point graphs of GQ(q,q2) fulfill the 6-vc, Klin, Meszka, Reichard, Rosa (2003): the smallest srgs with

4-vc have parameters (36,14,4,6), CP (2004): point graphs of partial quadrangles fulfill the 5-vc,

Reichard (2005): point graphs of GQ(q,q2) fulfill the 7-vc, CP (2007): point graphs of PQ(q − 1,q2,q2 − q) fulfill the 6-vc,

Klin, CP (2007): found two self-complementary graphs that fulfill the 4-vc.

Ch. Pech On highly regular strongly regular graphs June 2014 7 / 20

• Proving the (n,m)-regularity

Counting types in graphs is algorithmically hard. Luckily, in general, it is not necessary to count all types.

Example (Hestenes-Higman-Theorem)

In order to prove that a graph fulfills the 4-vertex condition for a graph, it is enough to prove that it is T-regular for the following types:

x x y x y

x y x y

Ch. Pech On highly regular strongly regular graphs June 2014 8 / 20

• Composing types

Given: T1 : ∆1 →֒ ∆2, T2 : ∆3 →֒ ∆4, e : ∆3 →֒ ∆2.

Consider the following diagram:

∆4

∆2 ∆3

∆1

T1

T2

e

Ch. Pech On highly regular strongly regular graphs June 2014 9 / 20

• Composing types

Given: T1 : ∆1 →֒ ∆2, T2 : ∆3 →֒ ∆4, e : ∆3 →֒ ∆2.

Consider the following diagram:

Λ ∆4

∆2 ∆3

∆1

ι4

ι3ι2

ι1

T1

T2

e

Let Λ be a colimes.

Ch. Pech On highly regular strongly regular graphs June 2014 9 / 20

• Composing types

Given: T1 : ∆1 →֒ ∆2, T2 : ∆3 →֒ ∆4, e : ∆3 →֒ ∆2.

Consider the following diagram:

Λ ∆4

∆2 ∆3

∆1

ι4

ι3ι2

ι1

T1

T2

e

Let Λ be a colimes. Then ι1 is a type. It is denoted by T1 ⊕e T2.

Ch. Pech On highly regular strongly regular graphs June 2014 9 / 20

• Comparison of Types

Given: T1 : ∆1 →֒ ∆2, T2 : ∆1 →֒ ∆3.

Definition

We define T1 � T2 if there is an epimorphism τ : ∆2 ։ ∆3 that makes the following diagram commute:

∆2

∆1 ∆3

T1

T2

τ

If τ is not an isomorphism, then we write T1 ≺ T2.

Ch. Pech On highly regular strongly regular graphs June 2014 10 / 20

• Type-Counting Lemma

Given: T1 : ∆1 →֒ ∆2, T2 : ∆3 →֒ ∆4, e : ∆3 →֒ ∆2,

a graph Γ.

Lemma

If Γ is T1- and T2-regular, and if Γ is T-regular for all T1 ⊕e T2 ≺ T, then Γ is also T1 ⊕e T2-regular.

Ch. Pech On highly regular strongly regular graphs June 2014 11 / 20

• First consequence of the type-counting lemma

Definition

Let T : ∆ →֒ Θ be a regularity-type of order (m,n). Suppose ∆ = (B,D), Θ = (T ,E). Let S ⊆ T be the image of T. Then we define

T̂ := (T ,E ∪ (S

2

) ),

Proposition

Let Γ be an (n,m)-regular graph (for m > n). Then, Γ is (n,m + 1)-regular if and only if it is T-regular for all graph-types T of order (n,m + 1) for which T̂ is (n + 1)-connected.

Ch. Pech On highly regular strongly regular graphs June 2014 12 / 20

• Partial Linear spaces

Definition

An incidence structure is a triple (P,L,I) such that 1 P is a set of points, 2 L is a set of lines, 3 I ⊆ P × L.

Definition

A partial linear space of order (s, t) is an incidence structure (P,L,I) such that

1 each line is incident with s + 1 points, 2 each point is incident with t + 1 lines, 3 every pair of points is incident with at most one line.

Definition

The point graph of a partial linear space is the graph that has as vertices the points such that two points form an edge whenever they are collinear.

Ch. Pech On highly regular strongly regular graphs June 2014 13 / 20

• Partial Linear spaces

Definition

An incidence structure is a triple (P,L,I) such that 1 P is a set of points, 2 L is a set of lines, 3 I ⊆ P × L.

Definition

A partial linear space of order (s, t) is an incidence structure (P,L,I) such that

1 each line is incident with s + 1 points, 2 each point is incident with t + 1 lines, 3 every pair of points is incident with at most one line.

Definition

The point graph of a partial linear space is the graph that has as vertices the points such that two points form an edge whenever they are collinear.

Ch. Pech On highly regular strongly regular graphs June 2014 13 / 20

Definition (Cameron 1975)

A partial quadrangle of order (s, t , µ) is a partial linear space of order (s, t) such that

1 if three points are pairwise collinear, then they are on one line,

2 if two points are non-collinear, then exactly t points are collinear with both.

Remarks

A strongly regular graph is isomorphic to the point-graph of a PQ if and only if it does not contain a subgraph isomorphic to K4 − e (Cameron ’75). Thus, the original PQ can be recovered from its point graph, up to isomorphism. Without loss of generality, we may identify a partial quadrangle with its point graph.

Ch. Pech On highly regular strongly regular graphs June 2014 14 / 20

Definition

A generalized quadrangle of order (s, t) is a partial linear space of order (s, t) such that for every line l and every point P not on l there is a unique point Q on l that is collinear with P.

Remark Every generalized quadrangle is also a partial quadrangle. Thus we may also identify a generalized quadrangle with its point graph.

Proposition (Higman 1971)

A generalized quadrangle has order (s, s2) if and only if every triad (i.e. triple of pairwise non-collinear points) has the same number of centers. In a GQ(s, s2) every triad has s + 1 centers.

Corollary

The point-graph of a G