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 V 1 , V 2 ⊆ V with | V 1 | = | V 2 | ≤ k and for each isomorphism ψ : Γ( V 1 ) → Γ( V 2 ) there exists an automorphism ϕ of Γ such that ϕ | V 1 = ψ . Ch. Pech On highly regular strongly regular graphs June 2014 2 / 20
Types Category of Graphs A graph-homomorphism h : Γ 1 → Γ 2 is a function from V (Γ 1 ) to V (Γ 2 ) 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 : Γ 1 ֒ → Γ 2 . Order of a Type → Γ 2 has order ( n , m ) if | V (Γ 1 ) | = n , and | V (Γ 2 ) | = m T : Γ 1 ֒ 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 ˆ ι : ∆ 2 ֒ → Γ that make the following diagram commute: ι ∆ 1 Γ T ˆ ι ∆ 2 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( J 2 ) 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 , q 2 ) 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 , q 2 ) fulfill the 7-vc, CP (2007): point graphs of PQ ( q − 1 , q 2 , q 2 − 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: → ∆ 4 , e : ∆ 3 ֒ T 1 : ∆ 1 ֒ → ∆ 2 , T 2 : ∆ 3 ֒ → ∆ 2 . Consider the following diagram: ∆ 4 T 2 ∆ 2 ∆ 3 e T 1 ∆ 1 Ch. Pech On highly regular strongly regular graphs June 2014 9 / 20
Composing types Given: → ∆ 4 , e : ∆ 3 ֒ T 1 : ∆ 1 ֒ → ∆ 2 , T 2 : ∆ 3 ֒ → ∆ 2 . Consider the following diagram: ι 4 Λ ∆ 4 ι 2 ι 3 T 2 ∆ 2 ∆ 3 e ι 1 T 1 ∆ 1 Let Λ be a colimes. Ch. Pech On highly regular strongly regular graphs June 2014 9 / 20
Composing types Given: → ∆ 4 , e : ∆ 3 ֒ T 1 : ∆ 1 ֒ → ∆ 2 , T 2 : ∆ 3 ֒ → ∆ 2 . Consider the following diagram: ι 4 Λ ∆ 4 ι 2 ι 3 T 2 ∆ 2 ∆ 3 e ι 1 T 1 ∆ 1 Let Λ be a colimes. Then ι 1 is a type. It is denoted by T 1 ⊕ e T 2 . Ch. Pech On highly regular strongly regular graphs June 2014 9 / 20
Comparison of Types Given: T 1 : ∆ 1 ֒ → ∆ 2 , T 2 : ∆ 1 ֒ → ∆ 3 . Definition We define T 1 � T 2 if there is an epimorphism τ : ∆ 2 ։ ∆ 3 that makes the following diagram commute: ∆ 2 τ T 1 ∆ 1 ∆ 3 T 2 If τ is not an isomorphism, then we write T 1 ≺ T 2 . Ch. Pech On highly regular strongly regular graphs June 2014 10 / 20
Type-Counting Lemma Given: → ∆ 4 , e : ∆ 3 ֒ T 1 : ∆ 1 ֒ → ∆ 2 , T 2 : ∆ 3 ֒ → ∆ 2 , a graph Γ . Lemma If Γ is T 1 - and T 2 -regular, and if Γ is T -regular for all T 1 ⊕ e T 2 ≺ T , then Γ is also T 1 ⊕ e T 2 -regular. Ch. Pech On highly regular strongly regular graphs June 2014 11 / 20
First consequence of the type-counting lemma Definition → Θ be a regularity-type of order ( m , n ) . Suppose Let T : ∆ ֒ ∆ = ( B , D ) , Θ = ( T , E ) . Let S ⊆ T be the image of T . Then we define � S � T := ( T , E ∪ � ) , 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 � T 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 P is a set of points, 1 L is a set of lines, 2 I ⊆ P × L . 3 Definition A partial linear space of order ( s , t ) is an incidence structure ( P , L , I ) such that each line is incident with s + 1 points, 1 each point is incident with t + 1 lines, 2 every pair of points is incident with at most one line. 3 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 P is a set of points, 1 L is a set of lines, 2 I ⊆ P × L . 3 Definition A partial linear space of order ( s , t ) is an incidence structure ( P , L , I ) such that each line is incident with s + 1 points, 1 each point is incident with t + 1 lines, 2 every pair of points is incident with at most one line. 3 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 Quadrangles Definition (Cameron 1975) A partial quadrangle of order ( s , t , µ ) is a partial linear space of order ( s , t ) such that if three points are pairwise collinear, then they are on one 1 line, if two points are non-collinear, then exactly t points are 2 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 K 4 − 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
Generalized quadrangles 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 , s 2 ) if and only if every triad (i.e. triple of pairwise non-collinear points) has the same number of centers. In a GQ ( s , s 2 ) every triad has s + 1 centers. Corollary The point-graph of a GQ ( s , s 2 ) is ( 3 , 4 ) -regular. Ch. Pech On highly regular strongly regular graphs June 2014 15 / 20
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