Total graph coherent configuration: New graphs from Moore graphs - PowerPoint PPT Presentation

Total graph coherent configuration: New graphs from Moore graphs Leif K. Jørgensen Aalborg University Denmark Joint work with M. Klin and M. Ziv-Av, BGU, Israel

Our goal is to construct nice graphs from complements of Moore graphs, i.e., regular graphs with • large automorphism group, or • relation of a homogeneous coherent configuration with low rank.

A Moore graph (of diameter 2) is a graph M with diameter 2 and girth 5. It is known that M is regular of some valency k and that (Hoffman and Singleton 1960) either • k = 2 and M is C 5 , (Automorphism group: Dihedral group of order 10) • k = 3 and M is the Petersen graph, P (Automorphism group: S 5 of order 120) • k = 7 and M is the Hoffman-Singleton graph or (Automorphism group: P Σ U ( 3, 5 2 ) of order 252 000) • k = 57 where existence is unknown.

For each of the three known Moore graphs, the automorphism group G has rank 3, i.e., G acts transitively of vertices and sta- bilizer of a vertex v has 3 orbits: { v } , the set of neighbours of v , the set of non-neighbours of v . For a Moore graph of valency 57 the aut. group G satisfies: • (Aschbacher 1971) G is not rank 3 • (G. Higman, see Cameron 1999) G is not vertex-transitive • (Makhnev and Paduchikh 2001) caj and ˇ • (Maˇ Sir´ aˇ n 2010) G has order at most 375.

For a graph Γ , the total graph of Γ , denoted by T ( Γ ) , has vertex set V ( Γ ) ∪ E ( Γ ) and two vertices in T ( Γ ) are adjacent if they are adjacent or incident in Γ . We assume that Γ is regular of valency k . Then T ( Γ ) is regular of valency 2 k .

v 1 t v 2 t v k t v 1 t v 2 t v k t ❏ ✡ ❏ ✡ ❏ ✡ ❏ ✡ ❏ ✡ ❏ ✡ t ❏ ✡ ❏ ✡ v ❏ ✡ ❏ ✡ ❏ t ✡ ✡ ❏ ❏ ✡ ✡ ❏ v ✡ ❏ ✡ ❏ t ✡ t ❏ t ✡ ❏ vv 1 vv 2 vv k N Γ ( v ) and N T ( Γ ) ( v ) for v ∈ V ( Γ ) v 1 v k − 1 vv 1 vv k − 1 t t t t ✥✥✥✥✥✥✥✥✥✥ ✟✟✟✟ t ❆ � ❇ ✡ v ❇ ✡ ❆ � ◗◗◗◗◗◗ ❆ t � ❇ ✡ v ❆ � ❇ ✡ ❇ t ✡ ❇ ✡ uv ✑✑✑✑✑✑ t ✂ ❏ u ✂ ❏ ✁ ❅ t ✂ ❏ ✁ ❅ u ✂ ❏ ❵❵❵❵❵❵❵❵❵❵ ❍❍❍❍ t ✁ ❅ t t ✂ ❏ t ✁ ❅ ✂ ❏ u 1 u k − 1 uu 1 uu k − 1 N Γ ( uv ) and N T ( Γ ) ( uv ) for uv ∈ E ( Γ )

If Γ is connected and regular of valency k ≥ 3 and Γ is not a complete graph then the automorphism groups satisfy Aut ( T ( Γ )) ≃ Aut ( Γ ) . For a complete graph we have T ( K n ) ≃ L ( K n + 1 ) and so Aut ( T ( K n )) ≃ Aut ( L ( K n + 1 )) ≃ Aut ( K n + 1 ) ≃ S n + 1 �≃ Aut ( K n ) ≃ S n .

✈ ✈ v 5 v 1 v 1 v 2 ✈ ✈ v 1 v 1 ✈ ✈ ✈ ✈ v 2 v 2 v 5 v 5 ✈ ✈ v 4 v 5 v 2 v 3 ✈ v 4 v 3 ✈ ✈ v 4 v 3 ✈ ✈ v 3 v 4 C 5 and T ( C 5 ) This picture is easily generalized to the total graph of C n .

We want to consider the total graph of the complement of a Moore graph. First, let M be the Moore graph of valency k = 2 and M be its complement, i.e., M = C 5 . On the next page we will construct a new graph with the same set of vertices as T ( C 5 ) , shown on the previous page, but with another construction of the edge set.

✈ ✈ v 5 v 1 v 1 v 2 v 1 ✈ ✈ ✈ ✈ v 1 v 5 v 2 ✈ ✈ ✈ ✈ v 2 v 5 ✈ ✈ v 4 v 5 v 2 v 3 ✈ v 4 v 3 ✈ ✈ ✈ v 4 v 3 M = C 5 Γ ✈ v 3 v 4 Adjacent pairs in new graph Γ isomorphic to Petersen graph: { v i , v j } where v i and v j are adjacent in M . R 2 : R 15 : { v i v j , v ℓ v h } where v i = v ℓ and v j and v h are adjacent in M . . R 5 : ( v ℓ , v i v j ) and . R 10 : ( v i v j , v ℓ ) where v ℓ is adjacent to v i and v j in M .

The automorphism group of T ( C 5 ) is vertex transitive and has order 20 . Consider the subgroup G of Aut ( T ( C 5 )) mapping (blue) vertices of C 5 to (blue) vertices of C 5 . The above are 4 orbits under the action of G on ordered pairs of vertices. However, we will now focus on the combinatorial description of types of adjacency.

A coherent configuration ( X , { R 1 , . . . , R r } consists of a finite set X (of points or vertices) and a partition of X × X in relations R i , i ∈ I = { 1, . . . , r } satisfying that 1. There is a subset I ′ ⊂ I such that � R i = ∆ : = { ( x , x ) | x ∈ X } . i ∈ I ′ 2. For each i ∈ I there is i ′ ∈ I such that R T = R i ′ , where i R T i = { ( y , x ) | ( x , y ) ∈ R i } . 3. For all h , i , j ∈ I there is a constant p h ij such that for every x , y ∈ X where ( x , y ) ∈ R h the number of points z ∈ X with ( x , z ) ∈ R i and ( z , y ) ∈ R j is exactly p h ij .

Example: From a Moore graph M of diameter 2 (or any strongly regular graph), we get a coherent configuration with rank r = 3 : Let X = V ( M ) and let R 1 = ∆ , R 2 = E ( M ) , R 3 = E ( M ) . This coherent configuration is • homogeneous , i.e., ∆ is one the relations, • symmetric , i.e., i ′ = i for all i ∈ I . A homogeneous (and symmetric) coherent configuration is also called an association scheme.

Example: For any finite set X there is a coherent configuration with rank r = n 2 , n = | X | , where each R i consists of one element from X × X . For a graph Γ with V ( Γ ) = X the partition of X × X in R 1 = ∆ , R 2 = E ( Γ ) , R 3 = E ( Γ ) is in general not a coherent configuration. But there is a refinement { R ′ 1 , . . . , R ′ r } of { R 1 , R 2 , R 3 } such that ( X , { R ′ 1 , . . . , R ′ r } ) is a coherent configuration. The coarsest such partition (i.e., with least possible r ) is called the coherent configuration generated by Γ . A polynomial-time algorithm for computing this is Weisfeiler- Leman stabilization (1968).

③ ③ ③ ③ ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ ③ � ③ ❅ � � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ ③ ③ ③ ③ � ❅ � ❅ Graph Γ and the coherent configuration generated by Γ . The coherent configuration generated by Γ has rank 6 and 2 fibers. ∆ = R 1 ∪ R 2 , R 3 T = R 4 , R 5 , R 6 .

Our goal is to find new homogeneous coherent configurations from a given coherent configuration by merging some of the relations. Such merging can be computed by COCO (Faradˇ zev and Klin 1992).

For a graph Γ the coherent configuration generated by T ( Γ ) is called the total graph coherent configuration of Γ . For a given graph Γ we can perform the following computations 1. Compute T ( Γ ) 2. Compute coherent configuration genrated by T ( Γ ) , using Weisfeiler- Leman stabilization 3. Compute homogeneous coherent configurations by merging relations using COCO.

We are interested in the case when Γ is a strongly regular graph and in particular when Γ is the complement of a Moore graph (of diameter 2). Theorem Ziv-Av, 2009 If Γ = L ( K n ) (triangular graph) then interesting mergings exist only for n = 5 and n = 7 . . For n = 7 : Zara graph. . L ( K 5 ) is the complement of the Petersen graph, see later. If Γ = L ( K n , n ) (square lattice graph) then no homogenous merging exist. Proof is based on clever use of computers. But when possible we want to describe the results without use of computer.

Let M denote a Moore graph of diameter 2 and valency k . We will consider T ( M ) . We describe 33 relations that are naturally obtained from T ( M ) . For a particular Moore graph these 33 relations need not all appear. For k = 2 , i.e., M = C 5 , 12 of the relations appear. For k = 3 , i.e., M = Petersen graph, 24 of the relations appear. For k = 7 , i.e., M = Hoffman-Singleton graph, 31 relations appear. For k = 57 it may be that some further refinement is needed in order to get a coherent configuration.

A vertex in T ( M ) will be denoted by v , where v ∈ V ( M ) or vw where v and w are non-adjacent vertices in M . First consider relations of the form ( v , x ) , for v , x ∈ V ( M ) . Description of x so that ( v , x ) ∈ R i Relation valency of v in R i Restrictions x = v 1 R 1 x ∼ v R 2 k k ( k − 1 ) R 3 x ≁ v

Next consider relations of the form ( v , xy ) , where v ∈ V ( M ) and xy is a non-edge in M . Let z denote the unique common neighbour of x and y in M Description of xy Relation so that ( v , xy ) ∈ R i valency of v in R i Restrictions R 4 = R T v ∈ { x , y } k ( k − 1 ) 9 R 5 = R T 1 x ∼ v and y ∼ v 2 k ( k − 1 ) 10 R 6 = R T k ( k − 1 ) 2 x ∼ v ≁ y or x ≁ v ∼ y 12 R 7 = R T 1 x ≁ v , y ≁ v and z ∼ v 2 k ( k − 1 )( k − 2 ) k ≥ 3 11 R 8 = R T 1 2 k ( k − 1 ) 2 ( k − 2 ) x ≁ v , y ≁ v and z ≁ v k ≥ 3 13

Let v 1 and v 2 be non-adjacent vertices in M . Let w be the common neighbour of v 1 and v 2 . Let A = N ( w ) \ { v 1 , v 2 } , B 1 = N ( v 1 ) \ { w } , B 2 = N ( v 2 ) \ { w } and C = V ( M ) \ ( { v 1 , v 2 , w } ∪ A ∪ B 1 ∪ B 2 ) . Then | A | = k − 2 , | B 1 | = | B 2 | = k − 1 , and | C | = k 2 + 1 − ( 3 + ( k − 2 ) + ( k − 1 ) + ( k − 1 )) = k 2 − 3 k + 2 = ( k − 1 )( k − 2 ) .

w ① ✘ ❳❳❳❳❳❳❳❳❳❳❳ ✘ ✘ ✘ ✘ ✘ ① ✘ ① ✬ ✩ ✘ ✘ ✘ ✘ ✘ ❳ v 1 v 2 A ✬ ✩ ✬ ✩ ✫ ✪ B 1 B 2 ✫ ✪ ✫ ✪ ❆ ✁ ❆ ✁ ❆ ✁ ✬ ✩ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ C ✫ ✪