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 C5,

(Automorphism group: Dihedral group of order 10)

• k = 3 and M is the Petersen graph, P

(Automorphism group: S5 of order 120)

• k = 7 and M is the Hoffman-Singleton graph or

(Automorphism group: PΣU(3, 52) 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)
• (Maˇ

caj and ˇ 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 2k.

t t t t t t t ✡ ✡ ✡ ✡ ✡ ✡ ❏ ❏ ❏ ❏ ❏ ❏ t t t t ❏ ❏ ❏ ❏ ❏ ❏ ✡ ✡ ✡ ✡ ✡ ✡

v v1 v2 vk

❏ ❏ ❏ ❏ ❏ ❏ ✡ ✡ ✡ ✡ ✡ ✡

v1 v2 vk v vv1 vv2 vvk NΓ(v) and NT(Γ)(v) for v ∈ V(Γ)

t t t t t t

v1 vk−1 v u u1 uk−1

❆ ❆ ❆ ❆

• ✁

✁ ✁ ✁ ❅ ❅ ❅ ❅ ✟✟✟✟ ✥✥✥✥✥✥✥✥✥✥ ◗◗◗◗◗◗ ❍❍❍❍ ❵❵❵❵❵❵❵❵❵❵ ✑✑✑✑✑✑ t t

v u

t t t t t ❇ ❇ ❇ ❇ ❇ ❇ ✡ ✡ ✡ ✡ ✡ ✡ ✂ ✂ ✂ ✂ ✂ ✂ ❏ ❏ ❏ ❏ ❏ ❏

uu1 uuk−1 vv1 vvk−1 uv NΓ(uv) and NT(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(Kn) ≃ L(Kn+1) and so Aut(T(Kn)) ≃ Aut(L(Kn+1)) ≃ Aut(Kn+1) ≃ Sn+1 ≃ Aut(Kn) ≃ Sn.

✈ ✈ ✈ ✈ ✈

v1 v5 v4 v3 v2

✈ ✈ ✈ ✈ ✈

v1 v5 v4 v3 v2

✈ ✈ ✈ ✈ ✈

v3v4 v2v3 v1v2 v5v1 v4v5 C5 and T(C5) This picture is easily generalized to the total graph of Cn.

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 = C5. On the next page we will construct a new graph with the same set of vertices as T(C5), shown on the previous page, but with another construction of the edge set.

✈ ✈ ✈ ✈ ✈

v1 v5 v4 v3 v2 M = C5 v1 v5 v4 v3 v2 v3v4 v2v3 v1v2 v5v1 v4v5

✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈

Γ Adjacent pairs in new graph Γ isomorphic to Petersen graph: . R2:

{vi, vj} where vi and vj are adjacent in M

. R15: {vivj, vℓvh} where vi = vℓ and vj and vh are adjacent in M . R5:

(vℓ, vivj) and

. R10: (vivj, vℓ) where vℓ is adjacent to vi and vj in M.

The automorphism group of T(C5) is vertex transitive and has

• rder 20.

Consider the subgroup G of Aut(T(C5)) mapping (blue) vertices

• f C5 to (blue) vertices of C5.

The above are 4 orbits under the action of G on ordered pairs

• f vertices.

However, we will now focus on the combinatorial description of types of adjacency.

A coherent configuration (X, {R1, . . . , Rr} consists of a finite set X (of points or vertices) and a partition of X × X in relations Ri, i ∈ I = {1, . . . , r} satisfying that

• 1. There is a subset I′ ⊂ I such that
• i∈I′

Ri = ∆ := {(x, x) | x ∈ X}.

• 2. For each i ∈ I there is i′ ∈ I such that RT

i

= Ri′, where

RT

i = {(y, x) | (x, y) ∈ Ri}.

• 3. For all h, i, j ∈ I there is a constant ph

ij such that for every

x, y ∈ X where (x, y) ∈ Rh the number of points z ∈ X with

(x, z) ∈ Ri and (z, y) ∈ Rj is exactly ph

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 R1 = ∆, R2 = E(M), R3 = 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 = n2, n = |X|, where each Ri consists of one element from X × X. For a graph Γ with V(Γ) = X the partition of X × X in R1 = ∆, R2 = E(Γ), R3 = E(Γ) is in general not a coherent configuration. But there is a refinement {R′

1, . . . , R′ r} of {R1, R2, R3} 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. ∆ = R1 ∪ R2, R3T = R4, R5, R6.

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(Kn) (triangular graph) then interesting mergings exist

• nly for n = 5 and n = 7.

. For n = 7: Zara graph. . L(K5) is the complement of the Petersen graph, see later. If Γ = L(Kn,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

• f 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 = C5, 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

• rder 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 Relation so that (v, x) ∈ Ri valency of v in Ri Restrictions R1 x = v 1 R2 x ∼ v k R3 x ≁ v k(k − 1)

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

• f x and y in M

Description of xy Relation so that (v, xy) ∈ Ri valency of v in Ri Restrictions R4 = RT

9

v ∈ {x, y} k(k − 1) R5 = RT

10

x ∼ v and y ∼ v

1 2k(k − 1)

R6 = RT

12

x ∼ v ≁ y or x ≁ v ∼ y k(k − 1)2 R7 = RT

11

x ≁ v, y ≁ v and z ∼ v

1 2k(k − 1)(k − 2)

k ≥ 3 R8 = RT

13

x ≁ v, y ≁ v and z ≁ v

1 2k(k − 1)2(k − 2)

k ≥ 3

Let v1 and v2 be non-adjacent vertices in M. Let w be the common neighbour of v1 and v2. Let A = N(w) \ {v1, v2}, B1 = N(v1) \ {w}, B2 = N(v2) \ {w} and C = V(M) \ ({v1, v2, w} ∪ A ∪ B1 ∪ B2). Then |A| = k − 2, |B1| = |B2| = k − 1, and

|C| = k2 + 1− (3+ (k − 2) + (k − 1) + (k − 1)) = k2 − 3k + 2 = (k − 1)(k − 2).

① ① ① ❳❳❳❳❳❳❳❳❳❳❳ ❳ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆

w v1 v2 B1 B2 A C

Now we consider relations of the form (v1v2, x) where v1v2 is a non-edge and x ∈ V(M). Description of x Relation so that (v1v2, x) ∈ Ri valency of v1v2 in Ri Restrictions R9 = RT

4

x ∈ {v1, v2} 2 R10 = RT

5

x = w 1 R11 = RT

7

x ∈ A k − 2 k ≥ 3 R12 = RT

6

x ∈ B1 ∪ B2 2(k − 1) R13 = RT

8

x ∈ C

(k − 1)(k − 2)

k ≥ 3

Finally we consider relations of the form (v1v2, xy) where v1v2 and xy are non-edges. We let z denote the common neighbour of x and y.

Description of xy Relation so that (v1v2, xy) ∈ Ri valency of v1v2 in Ri Restrictions R14

{x, y} = {v1, v2}

1 R15 x = v1, y ∈ B2 2(k − 1) R16 x ∈ {v1, v2}, y ∈ A 2(k − 2) k ≥ 3 R17 x ∈ {v1, v2}, y ∈ C 2(k − 1)(k − 2) k ≥ 3 R18 x = w, y ∈ B1 ∪ B2 2(k − 1) R19 = RT

23

x = w, y ∈ C

(k − 1)(k − 2)

k ≥ 3 R20 = RT

25

x ∈ A, y ∈ B1 ∪ B2 2(k − 2)(k − 1) k ≥ 3 R21 x, y ∈ A

(k−2

2 )

k ≥ 4 R22 = RT

28

x ∈ A, y ∈ C

(k − 1)(k − 2)(k − 3)

k ≥ 4 R23 = RT

19

x, y ∈ B1 or x, y ∈ B2 2(k−1

2 )

k ≥ 3 R24 x ∈ B1, y ∈ B2

(k − 1)(k − 2)

k ≥ 3 R25 = RT

20

x ∈ B1, y ∈ C, z ∈ B2 or 2(k − 1)(k − 2) k ≥ 3 x ∈ B2, y ∈ C, z ∈ B1 R26 x ∈ B1 ∪ B2, y ∈ C, z ∈ C 2(k − 1)(k − 2)(k − 3) k ≥ 4 R27 x ∈ C, y ∈ C, z ∈ A

(k−1

2 )(k − 2)

k ≥ 3 R28 = RT

22

x ∈ C, y ∈ C, z ∈ B1 ∪ B2 2(k−2

2 )(k − 1)

k ≥ 4 R29 x ∈ C, y ∈ C, z ∈ C

(k − 1)(k − 2)(k−3

2 )

k ≥ 5

Relation R27:

① ① ① ❳❳❳❳❳❳❳❳❳❳❳ ❳ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ① ① ① ① ① ① ① ✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ❈ ❈ ❈ ❈ ❈❈ ✄ ✄ ✄ ✄ ✄ ✄ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊❊ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ☎ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉

• ❅

❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅

• w

v1 v2 B1 B2 A C z x y x1 y1 x2 y2

Relation R27 is divided into three relations: R27c: x1y2, x2y1 ∈ E(M), Petersen graph R27b: one of x1y2, x2y1 is in E(M), Petersen minus edge R27a: x1y2, x2y1 ∈ E(M)

Similarly, there are three cases for R29. R29c: x1y2, x2y1 ∈ E(M), Petersen graph minus edge R29b: one of x1y2, x2y1 is in E(M), R29a: x1y2, x2y1 ∈ E(M)

Consider the case k = 3 when M is the Petersen graph P. There are five interesting homogenous mergings of the total graph coherent configuration of P = L(K5):

• 1. Two isomorphic association schemes of rank 5, with valencies

1, 6, 6, 24, 3. Automorphism group has order 7680.

• 2. Two isomorphic association schemes of rank 5, with valencies

1, 12, 12, 12, 3. Automorphism group has order 1920.

• 3. One association scheme of rank 3, with valencies 1, 27, 12.

Automorphism group has order 51840.

Case 1 Theorem There is a unique homogeneous, symmetric, rank 5 coherent configuration (X, {S0, S1, S2, S3, S4}) on 40 points with p0

11 = p0 22 = 6,

p0

33 = 24, p0 44 = 3, p4 44 = 2, p1 14 = 1, p1 24 = 2, p3 11 = 1.

It can be constructed from the total graph coherent configuration

• f P:

S0 = R1 ∪ R14 S1 = R2 ∪ R5 ∪ R10 ∪ R15 ∪ R27c S2 = R8 ∪ R13 ∪ R19 ∪ R23 S3 = R3 ∪ R4 ∪ R6 ∪ R9 ∪ R12 ∪ R18 ∪ R16 ∪ R17 ∪ R20 ∪ R25 S4 = R7 ∪ R11 ∪ R24. (It can also be constructed from another merging.)

Let the vertex set of P be {v0, . . . , v9}. Let X = {xia | i = 0, . . . , 9, a = 0, . . . , 3}.

(X, S4) is the graph with 10 connected components

Xi = {xi0, xi1, xi2, xi3} isomorphic to K4.

(X, S3) is the 4-coclique extension of P

and (X, S1 ∪ S2) is the 4-coclique extension of P.

Let (X, S1) be the graph with the property that if the neighbours

• f vi in P are vj1, vj2, vj3 where j1 < j2 < j3 then

xi0xjℓ0, xiℓxjℓ0 ∈ S1, for ℓ = 1, 2, 3, and if vivj ∈ E(P) then subgraph of S1 spanned by Xi ∪ Xj has two connected components isomorphic to K2,2.

① ① ① ① ① ① ❍❍❍❍❍❍❍❍❍❍❍ ❍ ✟✟✟✟✟✟✟✟✟✟✟ ✟ ❍❍❍❍❍❍❍❍❍❍❍ ❍

• ①

① ① ① ① ① ① ①

v6 v7 v4 v5 v3 v8 x43 x42 x41 x40 x53 x52 x51 x50

✟ ✟ ✟ ✟ ✟ ✟ ❍❍❍❍❍ ❍ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ✟✟✟✟✟✟✟✟✟✟✟ ✟

Part of P and (X, S1).

Additional properties: S1 and S2 are cospectral, but S1 has 64 Petersen subgraphs, S2 has no Petersen subgraph. The automorphism group is E64 ⋊ S5. It is transitive on vertices and has rank 5.

Case 2+3 There two mergings both isomorphic an association scheme

(X, {S0, S1, S2, S3, S4}) with valencies 1, 12, 12, 12, 3 where

S1 ∪ S2 is 4-coclique extension of P S2 is srg(40,12,2,4) S3 is 4-coclique extension of P.

(X, {S0, S2, S1 ∪ S3 ∪ S4}) is a rank 3 association scheme.

Now consider the case k = 7 when M is the Hoffman-Singleton graph HS. Consider the following mergings of relations from the total graph coherent configuration of HS: S0 = R1 ∪ R14 S1 = R2 ∪ R5 ∪ R10 ∪ R15 ∪ R27c S2 = R8 ∪ R13 ∪ R19 ∪ R23 ∪ R22 ∪ R26 ∪ R27a ∪ R28 S3 = R3 ∪ R4 ∪ R6 ∪ R9 ∪ R12 ∪ R18 ∪ R16 ∪ R17 ∪ R20 ∪ R25 ∪ R29b S4 = R7 ∪ R11 ∪ R24 ∪ R21 ∪ R29a. Then (V(T(HS)), {S0, S1, S2, S3, S4}) is a homogenous, symmetric coherent configuration on 1100 vertices. Valencies: 1, 28, 630, 336, 105. It automorphism has order 88 704 000 and is isomorphic to the automorphism group of the Higman-Sims graph. It is vertex transitive and has rank 5.

The number of Petersen subgraphs of a Moore graph of valency k satisfies: 0 ≤ number of P’s ≤ (k2 + 1)k(k − 1)2(k − 2) 120 For the Hoffman-Singleton graph the number of Petersen sub- graphs is the upper bound: 525. Kov´ aˇ cikov´ a 2015: For k = 57, the upper bound on the number

• f Petersen subgraphs is 266 266 000

and the number of other subgraphs on 10 vertices can be deter- mined from the number of Petersen subgraphs.

If a Moore graph has has exactly 266 266 000 Petersen subgraphs (i.e., no Petersen minus edge) then the graphs constructed from the total graph of a Moore graph of valency 57 with the following edge sets then the fol- lowing graphs are regular S1 = R2 ∪ R5 ∪ R10 ∪ R15 ∪ R27c S2 = R8 ∪ R13 ∪ R19 ∪ R23 ∪ R22 ∪ R26 ∪ R29a ∪ R28 S3 = R3 ∪ R4 ∪ R6 ∪ R9 ∪ R12 ∪ R18 ∪ R16 ∪ R17 ∪ R20 ∪ R25 ∪ R29b S4 = R7 ∪ R11 ∪ R24 ∪ R21 ∪ R27a with valencies: 1 653, 4 915 680, 185 136, 87 780 Is it possible that a Moore graph of valency 57 contains 266 266 000 Petersen subgraphs ???

Junker (unpublished, 2005), Higman (Cameron & van Lint, 1991): Vertices of a Moore of valency k:

① ① ① ① ① ①

∞ k − 1 j i 1

• ❅

❅ ❅ ❅ ❅ ❅ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ✭✭✭✭✭✭✭ ✭ ❤❤❤❤❤❤❤ ❤ ✭✭✭✭✭✭✭ ✭ ❤❤❤❤❤❤❤ ❤ ❤❤❤❤❤❤❤ ❤ ❍❍❍❍❍❍❍ ❍ ✬ ✫ ✩ ✪ ✛ ✚ ✘ ✙ ✛ ✚ ✘ ✙ ①

. . .

①

(0, 1) (0, k − 1)

Common neighbour of i and (0, ℓ) is denoted by (i, ℓ).

For some permutation σij of {1, . . . , k − 1}: σij : ℓ → m

⇔ (i, ℓ) and (j, m) are adjacent.

σji = σ−1

ij .

For Hoffman-Singleton graph: every σij is an involution, and

(i j) → σij defines an outer automorphism of S6.

S56 has no outer automorphism and for a Moore graph of valency 57 not every σij can be an involution.

Junker: ”A Moore graph is involutional if and only if it is built up from Petersen graphs.” Theorem (Junker) σij is an involution if and only if for every 5-cycle containing i, ∞, j there is a Petersen subgraph containing this 5-cycle and 0. Corollary (Junker) A Moore graph of valency 57 does not have 266 266 000 Petersen subgraphs.

Problem Is there a strongly regular graph Γ for which the total graph coherent configuration of Γ is not equal to the partition in orbits under the automorphism group of T(Γ)? Theorem Maˇ caj and ˇ Sir´ aˇ n, 2010 The automorphism group of a Moore graph of valency 57 has

• rder at most 375.