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 Trends in Algebraic Graph Theory Combinatorics 2–5 June 2014 John Wilmes Geometric clique structures in association schemes
Coherent configurations Definition Coherent configuration X on vertices V is partition R 0 ∪ · · · ∪ R r − 1 of V × V satisfying 1 ∆ = { ( x , x ) : x ∈ V } is union of some relations R i 2 ( ∀ j )( ∃ i )( R − 1 = R i ) j 3 ( ∀ i , j , k )( ∃ p k ij )( ∀ ( x , y ) ∈ R k ) there are exactly p k ij verts. z ∈ V s.t. ( x , z ) ∈ R i and ( z , y ) ∈ R j z 1 z 2 z 3 x y John Wilmes Geometric clique structures in association schemes
Coherent configurations John Wilmes Geometric clique structures in association schemes
Coherent configurations Definition CC is primitive if each constituent digraph ( V , R i ) connected for i ≥ 1 (in particular ∆ = R 0 ) John Wilmes Geometric clique structures in association schemes
Coherent configurations Definition CC is primitive if each constituent digraph ( V , R i ) connected for i ≥ 1 (in particular ∆ = R 0 ) John Wilmes Geometric clique structures in association schemes
Coherent configurations Definition CC is primitive if each constituent digraph ( V , R i ) connected for i ≥ 1 (in particular ∆ = R 0 ) Definition Rank of CC is r (number of colors/relations) John Wilmes Geometric clique structures in association schemes
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 ( K n ) (i.e., action of S n 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
Babai’s conjecture Conjecture (Babai) ( ∀ ε > 0)( ∃ n ε ) every primitive CC with n ≥ n ε verts. and ≥ exp( n ε ) 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
Babai’s conjecture Conjecture (Babai) ( ∀ ε > 0)( ∃ n ε ) every primitive CC with n ≥ n ε verts. and ≥ exp( n ε ) 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
Additional motivation: isomorphism testing Current Graph Isomorphism time complexity: exp( � O ( n 1 / 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
Classification from the conjecture Theorem (Cameron, 1981) ( ∀ ε > 0)( ∃ n ε ) every primitive group G of degree n ≥ n ε ≤ G ≤ S ( k ) and order ≥ exp( n ε ) satisfies A n × · · · × A n m ≀ S r � �� � r for some r , k , m � [ m ] � ( S ( k ) denotes S m � ) m k Corollary (assuming conjecture) All suff. large primitive CCs with ≥ exp( n ε ) automorphisms have form X ( G ) with G as above John Wilmes Geometric clique structures in association schemes
Main result, restated Theorem (Sun–W, 2014) For ε > 4 / 9 , (or ε > 1 / 3 assuming bounded rank) if primitive CC X on n ≥ n ε vertices has ≥ exp( n ε ) automorphisms, then X is one of K n , L ( K n ) , or L ( K n , n ) (Johnson graph J ( m , 3) and Hamming graph H (3 , q ) have rank r = 4 and exp(Ω( n 1 / 3 log n )) automorphisms) Goal: separate L ( K n ) and L ( K n , n ) from other primitive CCs John Wilmes Geometric clique structures in association schemes
Dominant color Let n i = | R i | / n (degree of color i ) Assume WLOG that n i maximized for i = 1 By following, we assume n − n 1 = O ( n 2 / 3 ) (color 1 is dominant ) Theorem (Babai, 1981) Primitive CC X of rank r ≥ 3 has � � rn log n �� | Aut( X ) | ≤ exp O n − n 1 Lemma If n − n 1 = O ( n 2 / 3 ) then | Aut( X ) | ≤ exp( � O ( n 4 / 9 )) John Wilmes Geometric clique structures in association schemes
Notation X a primitive CC on n verts � � � p 1 p i µ = λ i = k = n − n 1 − 1 = n i jk ij i , j > 1 i > 1 j > 1 (total nondominant degree) John Wilmes Geometric clique structures in association schemes
Proof outline Theorem (Sun–W, 2014) For ε > 1 / 3 , if bounded rank primitive CC X on n ≥ n ε verts. has ≥ exp( n ε ) automorphisms, then X is one of K n , L ( K n ) , or L ( K n , n ) Cases: 1 No overwhelmingly dominant color ( k = Ω( n 2 / 3 )) 2 Neighborhoods don’t overlap much for nbrs. in some color ( k = o ( n 2 / 3 ) and λ i = o ( √ n ) for some i > 1) 3 Neighborhoods in nondominant colors always overlap ( k = o ( n 2 / 3 ) and λ i = Ω( √ n ) for all i > 1) John Wilmes Geometric clique structures in association schemes
Proof outline Theorem (Sun–W, 2014) For ε > 1 / 3 , if bounded rank primitive CC X on n ≥ n ε verts. has ≥ exp( n ε ) automorphisms, then X is one of K n , L ( K n ) , or L ( K n , n ) Cases: 1 No overwhelmingly dominant color ( k = Ω( n 2 / 3 )) (case includes K n ) 2 Neighborhoods don’t overlap much for nbrs. in some color ( k = o ( n 2 / 3 ) and λ i = o ( √ n ) for some i > 1) 3 Neighborhoods in nondominant colors always overlap ( k = o ( n 2 / 3 ) and λ i = Ω( √ n ) for all i > 1) (case includes L ( K n ) and L ( K n , n ) ) John Wilmes Geometric clique structures in association schemes
Proof outline Theorem (Sun–W, 2014) For ε > 1 / 3 , if bounded rank primitive CC X on n ≥ n ε verts. has ≥ exp( n ε ) automorphisms, then X is one of K n , L ( K n ) , or L ( K n , n ) Cases: 1 No overwhelmingly dominant color ( k = Ω( n 2 / 3 )) (already done by (Babai,1981)) 2 Neighborhoods don’t overlap much for nbrs. in some color ( k = o ( n 2 / 3 ) and λ i = o ( √ n ) for some i > 1) 3 Neighborhoods in nondominant colors always overlap ( k = o ( n 2 / 3 ) and λ i = Ω( √ n ) for all i > 1) John Wilmes Geometric clique structures in association schemes
Almost all pairs have distance 2 Lemma If k = o ( n 2 / 3 ) then p 1 ii > 0 for all i > 1 John Wilmes Geometric clique structures in association schemes
Almost all pairs have distance 2 Lemma If k = o ( n 2 / 3 ) then p 1 ii > 0 for all i > 1 John Wilmes Geometric clique structures in association schemes
Almost all pairs have distance 2 Lemma If k = o ( n 2 / 3 ) then p 1 ii > 0 for all i > 1 degree = µ i John Wilmes Geometric clique structures in association schemes
Almost all pairs have distance 2 Lemma If k = o ( n 2 / 3 ) then p 1 ii > 0 for all i > 1 ≤ µ i k edges John Wilmes Geometric clique structures in association schemes
Almost all pairs have distance 2 Lemma If k = o ( n 2 / 3 ) then p 1 ii > 0 for all i > 1 degree ≤ µ i k 2 / n John Wilmes Geometric clique structures in association schemes
Almost all pairs have distance 2 Lemma If k = o ( n 2 / 3 ) then p 1 ii > 0 for all i > 1 µ i n � µ i k 3 / n John Wilmes Geometric clique structures in association schemes
Case 2: small λ i Lemma Suppose k = o ( n 2 / 3 ) and λ i = o ( √ n ) for some i > 1 . Then | Aut( X ) | ≤ exp( n 1 / 4 log n ) Proof idea. X i 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
Case 3: large λ i Need to separate L ( K n ) and L ( K n , n ) X i still “looks like” SR graph If X i was SR, it would be “geometric:” line graph of Steiner or transversal design Plan: recover geometry from CC John Wilmes Geometric clique structures in association schemes
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
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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