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Geometric clique structures in association schemes Xiaorui Sun 1 John Wilmes 2 1 Department of Computer Science Columbia University xiaoruisun@cs.columbia.edu 2 Department of Mathematics University of Chicago wilmesj@math.uchicago.edu Modern

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Geometric clique structures in association schemes

Xiaorui Sun 1 John Wilmes 2

1Department of Computer Science

Columbia University xiaoruisun@cs.columbia.edu

2Department of Mathematics

University of Chicago wilmesj@math.uchicago.edu

Modern Trends in Algebraic Graph Theory Combinatorics 2–5 June 2014

John Wilmes Geometric clique structures in association schemes

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Coherent configurations

Definition Coherent configuration X on vertices V is partition R0 ∪ · · · ∪ Rr−1 of V × V satisfying

1 ∆ = {(x, x) : x ∈ V } is union of some relations Ri 2 (∀j)(∃i)(R−1

j

= Ri)

3 (∀i, j, k)(∃pk

ij)(∀(x, y) ∈ Rk) there are exactly pk ij verts. z ∈ V

s.t. (x, z) ∈ Ri and (z, y) ∈ Rj x y z1 z2 z3

John Wilmes Geometric clique structures in association schemes

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Coherent configurations

John Wilmes Geometric clique structures in association schemes

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Coherent configurations

Definition CC is primitive if each constituent digraph (V , Ri) connected for i ≥ 1 (in particular ∆ = R0)

John Wilmes Geometric clique structures in association schemes

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Coherent configurations

Definition CC is primitive if each constituent digraph (V , Ri) connected for i ≥ 1 (in particular ∆ = R0)

John Wilmes Geometric clique structures in association schemes

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Coherent configurations

Definition CC is primitive if each constituent digraph (V , Ri) connected for i ≥ 1 (in particular ∆ = R0) Definition Rank of CC is r (number of colors/relations)

John Wilmes Geometric clique structures in association schemes

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Coherent configurations from groups

Given group action G V define CC X(G) by c(u, v) = c(x, y) iff (∃g ∈ G)(gu, gv) = (x, y) Example: X(S(2)

n ) is L(Kn)

(i.e., action of Sn on 2-element subsets of {1, . . . , n}) Proposition X(G) primitive iff G V primitive group action (Primitive action = no “blocks of imprimitivity”)

John Wilmes Geometric clique structures in association schemes

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Babai’s conjecture

Conjecture (Babai) (∀ε > 0)(∃nε) every primitive CC with n ≥ nε verts. and ≥ exp(nε) automorphisms has primitive automorphism group Established: for ε > 1/2 in (Babai, Annals of Math. 1981) ε > 1/3 if rank r = 3 in (Spielman, STOC 1996) ε > 9/37 if rank r = 3 in (Chen–Sun–Teng, 2013)

John Wilmes Geometric clique structures in association schemes

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Babai’s conjecture

Conjecture (Babai) (∀ε > 0)(∃nε) every primitive CC with n ≥ nε verts. and ≥ exp(nε) automorphisms has primitive automorphism group Established: for ε > 1/2 in (Babai, Annals of Math. 1981) ε > 1/3 if rank r = 3 in (Spielman, STOC 1996) ε > 9/37 if rank r = 3 in (Chen–Sun–Teng, 2013) Main Result (Sun–W, 2014) Conjecture holds for ε > 4/9 for ε > 1/3 assuming bounded rank

John Wilmes Geometric clique structures in association schemes

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Additional motivation: isomorphism testing

Current Graph Isomorphism time complexity: exp( O(n1/2)) (tilde hides polylog(n) factors) Primitive CC are obstacle to combinatorial divide-and-conquer for Graph Isomorphism testing Previous progress on Babai’s conjecture also gave algorithms for testing primitive CC isomorphism

John Wilmes Geometric clique structures in association schemes

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Classification from the conjecture

Theorem (Cameron, 1981) (∀ε > 0)(∃nε) every primitive group G of degree n ≥ nε and order ≥ exp(nε) satisfies An × · · · × An

  • r

≤ G ≤ S(k)

m ≀ Sr

for some r, k, m (S(k)

m

denotes Sm [m]

k

  • )

Corollary (assuming conjecture) All suff. large primitive CCs with ≥ exp(nε) automorphisms have form X(G) with G as above

John Wilmes Geometric clique structures in association schemes

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Main result, restated

Theorem (Sun–W, 2014) For ε > 4/9, (or ε > 1/3 assuming bounded rank) if primitive CC X on n ≥ nε vertices has ≥ exp(nε) automorphisms, then X is one of Kn, L(Kn), or L(Kn,n) (Johnson graph J(m, 3) and Hamming graph H(3, q) have rank r = 4 and exp(Ω(n1/3 log n)) automorphisms) Goal: separate L(Kn) and L(Kn,n) from other primitive CCs

John Wilmes Geometric clique structures in association schemes

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Dominant color

Let ni = |Ri|/n (degree of color i) Assume WLOG that ni maximized for i = 1 By following, we assume n − n1 = O(n2/3) (color 1 is dominant) Theorem (Babai, 1981) Primitive CC X of rank r ≥ 3 has | Aut(X)| ≤ exp

  • O

rn log n n − n1

  • Lemma

If n − n1 = O(n2/3) then | Aut(X)| ≤ exp( O(n4/9))

John Wilmes Geometric clique structures in association schemes

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Notation

X a primitive CC on n verts µ =

  • i,j>1

p1

jk

k = n − n1 − 1 =

  • i>1

ni (total nondominant degree) λi =

  • j>1

pi

ij

John Wilmes Geometric clique structures in association schemes

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Proof outline

Theorem (Sun–W, 2014) For ε > 1/3, if bounded rank primitive CC X on n ≥ nε verts. has ≥ exp(nε) automorphisms, then X is one of Kn, L(Kn), or L(Kn,n) Cases:

1 No overwhelmingly dominant color

(k = Ω(n2/3))

2 Neighborhoods don’t overlap much for nbrs. in some color

(k = o(n2/3) and λi = o(√n) for some i > 1)

3 Neighborhoods in nondominant colors always overlap

(k = o(n2/3) and λi = Ω(√n) for all i > 1)

John Wilmes Geometric clique structures in association schemes

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Proof outline

Theorem (Sun–W, 2014) For ε > 1/3, if bounded rank primitive CC X on n ≥ nε verts. has ≥ exp(nε) automorphisms, then X is one of Kn, L(Kn), or L(Kn,n) Cases:

1 No overwhelmingly dominant color

(k = Ω(n2/3)) (case includes Kn)

2 Neighborhoods don’t overlap much for nbrs. in some color

(k = o(n2/3) and λi = o(√n) for some i > 1)

3 Neighborhoods in nondominant colors always overlap

(k = o(n2/3) and λi = Ω(√n) for all i > 1) (case includes L(Kn) and L(Kn,n))

John Wilmes Geometric clique structures in association schemes

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Proof outline

Theorem (Sun–W, 2014) For ε > 1/3, if bounded rank primitive CC X on n ≥ nε verts. has ≥ exp(nε) automorphisms, then X is one of Kn, L(Kn), or L(Kn,n) Cases:

1 No overwhelmingly dominant color

(k = Ω(n2/3)) (already done by (Babai,1981))

2 Neighborhoods don’t overlap much for nbrs. in some color

(k = o(n2/3) and λi = o(√n) for some i > 1)

3 Neighborhoods in nondominant colors always overlap

(k = o(n2/3) and λi = Ω(√n) for all i > 1)

John Wilmes Geometric clique structures in association schemes

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Almost all pairs have distance 2

Lemma If k = o(n2/3) then p1

ii > 0 for all i > 1

John Wilmes Geometric clique structures in association schemes

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Almost all pairs have distance 2

Lemma If k = o(n2/3) then p1

ii > 0 for all i > 1

John Wilmes Geometric clique structures in association schemes

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Almost all pairs have distance 2

Lemma If k = o(n2/3) then p1

ii > 0 for all i > 1

degree = µi

John Wilmes Geometric clique structures in association schemes

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Almost all pairs have distance 2

Lemma If k = o(n2/3) then p1

ii > 0 for all i > 1

≤ µik edges

John Wilmes Geometric clique structures in association schemes

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Almost all pairs have distance 2

Lemma If k = o(n2/3) then p1

ii > 0 for all i > 1

degree ≤ µik2/n

John Wilmes Geometric clique structures in association schemes

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Almost all pairs have distance 2

Lemma If k = o(n2/3) then p1

ii > 0 for all i > 1

µin µik3/n

John Wilmes Geometric clique structures in association schemes

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Case 2: small λi

Lemma Suppose k = o(n2/3) and λi = o(√n) for some i > 1. Then | Aut(X)| ≤ exp(n1/4 log n) Proof idea. Xi looks like strongly regular graph (rank 3 CC) with small λ: Most vertices are at distance 2 Vertices at distance 2 have ≤ µ common nbrs.

  • Adj. vertices have o(√n) common nbrs.

Spielman’s proof for SR graphs generalizes

John Wilmes Geometric clique structures in association schemes

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Case 3: large λi

Need to separate L(Kn) and L(Kn,n) Xi still “looks like” SR graph If Xi was SR, it would be “geometric:” line graph of Steiner

  • r transversal design

Plan: recover geometry from CC

John Wilmes Geometric clique structures in association schemes

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Geometric clique structure

Definition An (asymptotically uniform) geometric clique structure of order ν ≥ 3 in a graph G is a collection C of cliques satisfying

1 every pair of adj. verts. belongs to a unique clique in C 2 every C ∈ C has |C| ∼ ν John Wilmes Geometric clique structures in association schemes

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Geometric clique structure

Definition An (asymptotically uniform) geometric clique structure of order ν ≥ 3 in a graph G is a collection C of cliques satisfying

1 every pair of adj. verts. belongs to a unique clique in C 2 every C ∈ C has |C| ∼ ν John Wilmes Geometric clique structures in association schemes

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Geometric clique structure

Definition An (asymptotically uniform) geometric clique structure of order ν ≥ 3 in a graph G is a collection C of cliques satisfying

1 every pair of adj. verts. belongs to a unique clique in C 2 every C ∈ C has |C| ∼ ν

Example: point-graph of partial geometry: cliques are lines in geometry

John Wilmes Geometric clique structures in association schemes

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Geometric clique structure

Definition An (asymptotically uniform) geometric clique structure of order ν ≥ 3 in a graph G is a collection C of cliques satisfying

1 every pair of adj. verts. belongs to a unique clique in C 2 every C ∈ C has |C| ∼ ν

Example: point-graph of partial geometry: cliques are lines in geometry Recovering clique geometry will let us separate L(Kn) and L(Kn,n) from other CCs

John Wilmes Geometric clique structures in association schemes

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Geometric clique structures in amply-regular graphs

Theorem (Metsch, 1991) G amply regular on n vertices: valency k two adj. vertices have λ common nbrs two vertices at dist. 2 have µ common nbrs Suppose kµ = o(λ2). Then: G has geometric clique structure of order ∼ λ all other maximal cliques have order o(λ) Goal: Generalize to coherent configurations

John Wilmes Geometric clique structures in association schemes

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Proof of clique structure

x y

John Wilmes Geometric clique structures in association schemes

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Proof of clique structure

x y

John Wilmes Geometric clique structures in association schemes

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Proof of clique structure

µ − 2 x y

John Wilmes Geometric clique structures in association schemes

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Proof of clique structure

≤ µ − 2 x y

John Wilmes Geometric clique structures in association schemes

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Proof of clique structure

≤ µ − 2 x y

John Wilmes Geometric clique structures in association schemes

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Proof of clique structure

≤ µ − 2 x y

John Wilmes Geometric clique structures in association schemes

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Proof of clique structure

degree ≤ (λ − µ + 2)/2 x y

John Wilmes Geometric clique structures in association schemes

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Proof of clique structure

set with degree ≤ (λ − µ + 2)/2 D x y

John Wilmes Geometric clique structures in association schemes

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Proof of clique structure

set with degree ≤ (λ − µ + 2)/2 D an endpoint of each nonedge lies in D x y clique!

John Wilmes Geometric clique structures in association schemes

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Proof of clique structure

set with degree ≤ (λ − µ + 2)/2 D Count nonedges: |D|(λ − (λ − µ + 2)/2) ≤ (k − λ − 1)(µ − 1) ≤ kµ = o(λ2) so |D| = o(λ) x y clique!

John Wilmes Geometric clique structures in association schemes

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Proof of clique structure

set with degree ≤ (λ − µ + 2)/2 D Count nonedges: |D|(λ − (λ − µ + 2)/2) ≤ (k − λ − 1)(µ − 1) ≤ kµ = o(λ2) so |D| = o(λ)

  • rder ∼ λ

x y clique!

John Wilmes Geometric clique structures in association schemes

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Proof of clique structure

set with degree ≤ (λ − µ + 2)/2 D Count nonedges: |D|(λ − (λ − µ + 2)/2) ≤ (k − λ − 1)(µ − 1) ≤ kµ = o(λ2) so |D| = o(λ)

  • rder ∼ λ

maximal clique C(x,y)! x y

John Wilmes Geometric clique structures in association schemes

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Proof of clique structure

set with degree ≤ (λ − µ + 2)/2 D

  • rder ∼ λ

maximal clique C(x,y)! All other max. cliques small: If z ∈ D, then z has ≤ µ neighbors in C(x, y). Hence clique with x, y, z has

  • rder ≤ |D| + µ = o(λ)

x y

John Wilmes Geometric clique structures in association schemes

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Geometric clique structures in coherent configurations

Theorem (Sun–W, 2014) X a primitive CC s.t. for all i, j > 1 niµ = o

  • λi min

λj r , λinj ni

  • Then ∃ partition I1 ∪ · · · ∪ It of {2, . . . , n} s.t.

(∀1 ≤ s ≤ t) union of constituent digraphs in Is has geometric clique structure of order ≥ min{λi : i ∈ Is}

John Wilmes Geometric clique structures in association schemes

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Proof of clique structure in CC

Metsch guarantees “local” clique structure in each color Inequalities from our theorem let us combine several colors for globally consistent structure

John Wilmes Geometric clique structures in association schemes

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Summary

Main result: every suff. large primitive CC with ≥ exp(nε) automorphisms has primitive automorphism group ∀ε > 4/9 ∀ε > 1/3 assuming bounded rank Previous best: ε > 1/2 general, ε > 9/37 ≈ 0.24 for rank 3 Proof separates L(Kn) and L(Kn,n) (many automorphisms) from

  • ther primitive CCs by finding clique geometry

John Wilmes Geometric clique structures in association schemes

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Open questions

∃ SR graphs w/ asymp. unif. geometric clique structure

  • ther than line-graphs of partial geometries?

(i.e., asymptotic but not exact uniformity of cliques) Extend verification of Babai’s conjecture:

◮ all ε > 0 for r = 3 ◮ extend this proof for ε > 1/3 to arbitrary rank r

Separate J(n, 3) and H(3, q) from other CCs by parameters

◮ find more general clique structure in wider range of parameters

John Wilmes Geometric clique structures in association schemes