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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 structurecan 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 iffor all V1,V2 ⊆ V with |V1| = |V2| ≤ k and for eachisomorphism ψ : Γ(V1) → Γ(V2) there exists an automorphismϕ 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 fromV (Γ1) to V (Γ2) that maps edges to edges.an embedding is an injective homomorphism with theproperty that the preimage of edges are edges, too.
Regularity-Types
A regularity-type (or type, for short) is an embedding offinite 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 embeddingsι̂ : ∆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 dependon 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 ofcomplete colored graphs, was introduced and studied byEvdokimov and Ponomarenko (2000) in relation with thet-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 generalizedquadrangles 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 infiniteseries,
A.V.Ivanov (1994): another infinite series of graphs with the4-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 thatfulfill 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 agraph, it is enough to prove that it is T-regular for the followingtypes:
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 thatmakes 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 allT1 ⊕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 wedefine
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-typesT 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 that1 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 hasas vertices the points such that two points form an edgewhenever 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 that1 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 hasas vertices the points such that two points form an edgewhenever they are collinear.
Ch. Pech On highly regular strongly regular graphs June 2014 13 / 20
Partial Quadrangles
Definition (Cameron 1975)
A partial quadrangle of order (s, t , µ) is a partial linear space oforder (s, t) such that
1 if three points are pairwise collinear, then they are on oneline,
2 if two points are non-collinear, then exactly t points arecollinear with both.
Remarks
A strongly regular graph is isomorphic to the point-graph ofa PQ if and only if it does not contain a subgraphisomorphic to K4 − e (Cameron ’75).Thus, the original PQ can be recovered from its pointgraph, up to isomorphism.Without loss of generality, we may identify a partialquadrangle with its point graph.
Ch. Pech On highly regular strongly regular graphs June 2014 14 / 20
Generalized quadrangles
Definition
A generalized quadrangle of order (s, t) is a partial linear spaceof order (s, t) such that for every line l and every point P not onl there is a unique point Q on l that is collinear with P.
RemarkEvery generalized quadrangle is also a partial quadrangle.Thus we may also identify a generalized quadrangle with itspoint graph.
Proposition (Higman 1971)
A generalized quadrangle has order (s, s2) if and only if everytriad (i.e. triple of pairwise non-collinear points) has the samenumber of centers.In a GQ(s, s2) every triad has s + 1 centers.
Corollary
The point-graph of a GQ(s, s2) is (3,4)-regular.
Ch. Pech On highly regular strongly regular graphs June 2014 15 / 20
Generalized quadrangles
Definition
A generalized quadrangle of order (s, t) is a partial linear spaceof order (s, t) such that for every line l and every point P not onl there is a unique point Q on l that is collinear with P.
RemarkEvery generalized quadrangle is also a partial quadrangle.Thus we may also identify a generalized quadrangle with itspoint graph.
Proposition (Higman 1971)
A generalized quadrangle has order (s, s2) if and only if everytriad (i.e. triple of pairwise non-collinear points) has the samenumber of centers.In a GQ(s, s2) every triad has s + 1 centers.
Corollary
The point-graph of a GQ(s, s2) is (3,4)-regular.
Ch. Pech On highly regular strongly regular graphs June 2014 15 / 20
PQ(s − 1, s2, s2 − s) and GQ(s, s2)
Step 1. Take a generalized quadrangle (P,L,I) of order(s, s2).
Step 2. Take some point V ∈ P and define
PV := {P ∈ P | P 6= V , P,V not collinear},
LV := {l ∈ L | V not on l},
IV := I ∩ (PV × LV ).
Proposition
The incidence structure (PV ,LV ,IV ) is a partial quadrangle oforder (s − 1, s2, s2 − s)
Remarks
In fact, every such PQ(s − 1, s2, s2 − s) has an extensionto a GQ(s, s2) (Ivanov, Shpektorov 1991).Thus, every partial quadrangle of order (s − 1, s2, s2 − s)can be obtained this way.
Ch. Pech On highly regular strongly regular graphs June 2014 16 / 20
On the (2,6)-regularity of PQ(s − 1, s2, s2 − s)
Proving (2,6)-regularity of PQ(s − 1, s2, s2 − s) so farmade heavily use of the (2,7)-regularity and the(3,4)-regularity of the associated GQ(s, s2).
A close analysis of the (very technical) proof revealed thatin fact for many types T of order (3,7), it was shown thatGQ(s, s2) is T-regular.
Goal:
Simplify the proof of the (2,6)-regularity of PQ(s − 1, s2, s2 − s)by studying types of order (3,7) in the associated GQ(s, s2).
Ch. Pech On highly regular strongly regular graphs June 2014 17 / 20
Main result
Question
Which types have to be counted in order to show that aGQ(s, s2) is (3,7)-regular?
Answer
Only the types T1 : K3 →֒ K5, T2 : K3 →֒ K6, T3 : K3 →֒ K7 needto be counted.
Theorem
GQ(s, s2) is (3,7)-regular.
Proof.
Needed in the proof:1 (3,4)-regularity of GQ(s, s2),2 the type-counting lemma,3 a computer for finding all indecomposable types.
Ch. Pech On highly regular strongly regular graphs June 2014 18 / 20
Concluding remarks
Consequences
1 The presented result strengthens Reichard’s theorem onthe 7-vertex condition for GQ(s, s2).
2 The proof of the presented result drastically simplifies theproof of the 6-vertex condition for PQ(s − 1, s2, s2 − s).
Smallest non-classical (3,7)− regular example
The smallest non-classical example is GQ(5,25).
Its point-graph has parameters(v , k , λ, µ) = (756,130,4,26).
its automorphism group acts intransitively on the vertices.
Ch. Pech On highly regular strongly regular graphs June 2014 19 / 20
Conjecture (Klin 1994?)
There exists a t0, such that every strongly regular graph thatsatisfies the t0-vertex condition is in fact already2-homogeneous (i.e., is a rank-3-graph).
if Klin’s conjecture is true, then t0 ≥ 8 (by Reichard’sresult).Morevover, if a Moore-graph of valency 57 exists, thent0 ≥ 10 (Reichard, CP 2014).
Problem
Does there exist a t0, such that every (3, t0)-regular graph is3-homogeneous?
If so, then t0 ≥ 8.
Problem
Does there exists a t0, such that every (4, t0)-regular graph is4-homogeneous?
If so, then t0 ≥ 6, as the only known (4,5)-regular graph that isnot 4-homogeneous is the McLaughlin-graph (on 275 vertices).It is not (4,6)-regular.
Ch. Pech On highly regular strongly regular graphs June 2014 20 / 20
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