Vertex subsets with minimal width and dual width

Transcription

1 Vertex subsets with minimal width and dual width in Q-polynomial distance-regular graphs University of Wisconsin & Tohoku University February 2, 2011

2 Every face (or facet) of a hypercube is a hypercube...

3 Goal Generalize this situation to Q-polynomial distance-regular graphs. Discuss its applications.

4 Distance-regular graphs Γ = (X, R) : a finite connected simple graph with diameter d : the path-length distance function Define A 0, A 1,..., A d R X X by { 1 if (x, y) = i (A i ) xy = 0 otherwise For x X, set Γ i (x) = {y X : (x, y) = i}. Γ is distance-regular if there are integers a i, b i, c i such that A 1 A i = b i 1 A i 1 + a i A i + c i+1 A i+1 (0 i d) where A 1 = A d+1 = 0.

5 A 1 A i = b i 1 A i 1 + a i A i + c i+1 A i+1 (0 i d)

6 Example: hypercubes X = {0, 1} d x R y {i : x i y i } = 1 Γ = Q d = (X, R) : the hypercube Q 3 : Q d = the binary Hamming graph

7 Example: Johnson graphs Ω : a finite set with Ω = v 2d X = {x Ω : x = d} x R y x y = d 1 (x, y Ω) Γ = J(v, d) = (X, R) : the Johnson graph The complement of J(5, 2) with Ω = {1, 2, 3, 4, 5}: {3,4} {1,2} {1,5} {2,3} {1,4} {2,5} {3,5} {4,5} {2,4} {1,3}

8 A 1 A i = b i 1 A i 1 + a i A i + c i+1 A i+1 (0 i d) Γ = (X, R) : a distance-regular graph with diameter d A 0, A 1,..., A d : the distance matrices of Γ We set A := A 1 (the adjacency matrix of Γ). θ 0, θ 1,..., θ d : the distinct eigenvalues of A E i : the orthogonal projection onto the eigenspace of A with eigenvalue θ i R[A] = A 0,..., A d = E 0,..., E d : the Bose Mesner algebra of Γ

9 θ 0, θ 1..., θ d Γ is regular with valency k := b 0 : We always set θ 0 = k = b 0. E 0 R X = 1 where 1 : the all-ones vector E 0 = 1 X J where J : the all-ones matrix in RX X

10 θ 0, θ 1..., θ d Recall A 1 A i = b i 1 A i 1 + a i A i + c i+1 A i+1 (0 i d). Γ is Q-polynomial with respect to {E i } d i=0 if there are scalars a i, b i, c i (0 i d) such that b i 1 c i 0 (1 i d) and X E 1 E i = b i 1E i 1 + a i E i + c i+1e i+1 (0 i d) where E 1 = E d+1 = 0 and is the Hadamard product. The ordering {E i } d i=0 is uniquely determined by E 1.

11 Hypercubes and binary Hamming matroids {0, 1, } : the claw semilattice of order 3 : 0 1 (P, ) : the direct product of d claw semilattices: P = {0, 1, } d u v u i = or u i = v i (1 i d) H(d, 2) = (P, ) : the binary Hamming matroid rank(u) = {i : u i } (u P) X = {0, 1} d = top(p) : the top fiber of H(d, 2)

12 Hypercubes and binary Hamming matroids u P : rank i χ u R X : the characteristic vector of Y u := {x X : u x} rank d u rank i Remark There is an ordering E 0, E 1,..., E d such that i E i R X = χ u : u P, rank(u) = i (0 i d). h=0 Moreover, Q d is Q-polynomial with respect to {E i } d i=0.

13 Y u = {x X : u x} is a facet of Q d If d = 3 and u = (, 0, ) then: Y u Every facet of Q d is of this form. The induced subgraph on Y u is Q d rank(u).

14 Johnson graphs and truncated Boolean algebras Recall Ω : a finite set with Ω = v 2d P = {u Ω : u d} u v u v B(d, v) = (P, ) : the truncated Boolean algebra rank(u) = u (u P) X = {x Ω : x = d} = top(p) : the top fiber of B(d, v)

15 Johnson graphs and truncated Boolean algebras u P : rank i χ u R X : the characteristic vector of Y u := {x X : u x} rank d u rank i Remark There is an ordering E 0, E 1,..., E d such that i E i R X = χ u : u P, rank(u) = i (0 i d). h=0 Moreover, J(v, d) is Q-polynomial with respect to {E i } d i=0.

16 Y u = {x X : u x} induces J(v rank(u), d rank(u)) Ω Ω\u x u x\u Remark H(d, 2) and B(d, v) are examples of regular quantum matroids (Terwilliger, 1996).

17 Width and dual width (Brouwer et al., 2003) Γ = (X, R) : a distance-regular graph with diameter d A 0, A 1,..., A d : the distance matrices E 0, E 1,..., E d : the primitive idempotents of R[A] Suppose Γ is Q-polynomial with respect to {E i } d i=0. Y X : a nonempty subset of X χ R X : the characteristic vector of Y w = max{i : χ T A i χ 0} : the width of Y w = max{i : χ T E i χ 0} : the dual width of Y Y w

18 w = max{i : χ T A i χ 0}, w = max{i : χ T E i χ 0} Theorem (Brouwer Godsil Koolen Martin, 2003) We have w + w d. If equality holds then the induced subgraph Γ Y on Y is a Q-polynomial distance-regular graph with diameter w provided that it is connected. Definition We call Y a descendent of Γ if w + w = d.

19 Examples: Γ = Q d or J(v, d) u P : rank i Y u := {x X : u x} satisfies w = d i and w = i. rank d u rank i Theorem (Brouwer et al., 2003; T., 2006) If Γ is associated with a regular quantum matroid, then every descendent of Γ is isomorphic to some Y u under the full automorphism group of Γ.

20 Observation Y u Y u is convex (geodetically closed).

21 AA i = b i 1 A i 1 + a i A i + c i+1 A i+1 (0 i d) We say Γ has classical parameters (d, q, α, β) if ([ ] [ ] )( [ ] ) [ ] ( [ ] ) d i i i i 1 b i = β α, c i = 1+α Example q q for 0 i d, where [ ] i j is the q-binomial coefficient. q If Γ = Q d then b i = d i and c i = i, so Γ has classical parameters (d, 1, 0, 1). q q q Currently, there are 15 known infinite families of distance-regular graphs with classical parameters and with unbounded diameter.

22 The families related to Hamming graphs Doob graphs halved cubes bilinear forms graphs pseudo halving q-analog Hamming graphs (hypercubes) half dual polar graphs dual polar graphs with second Q-polynomial ordering Ustimenko graphs distance 1-or-2 q-analog dual polar graphs pseudo halving alternating forms graphs last subconst. Hermitean forms graphs halving Hemmeter graphs last subconst. quadratic forms graphs

23 The families related to Johnson graphs Johnson graphs q-analog Grassmann graphs pseudo twisted Grassmann graphs

24 b i = ( [ d 1 ]q [ i 1 ]q )(β α[ i 1 ]q ), c i = [ i 1 ]q (1 + α[ ] i 1 1 q ) Y X : a descendent of Γ, i.e., w + w = d Γ Y : the induced subgraph on Y Theorem (T.) Suppose 1 < w < d. Then Y is convex precisely when Γ has classical parameters. Theorem (T.) If Γ has classical parameters (d, q, α, β) then Γ Y has classical parameters (w, q, α, β). The converse also holds, provided w 3. Classification of descendents is complete for all 15 families (T.).

25 The Erdős Ko Rado theorem (1961) Ω : a finite set with Ω = v 2d X = {x Ω : x = d} Theorem (Erdős Ko Rado, 1961) Let v (t + 1)(d t + 1) and let Y X be a t-intersecting family, i.e., x y t for all x, y Y. Then ( ) v t Y. d t If v > (t + 1)(d t + 1) and if Y = ( v t d t) then for some u Ω with u = t. Y = {x X : u x}

26 A modern treatment of the E K R theorem This is in fact a result about the Johnson graph J(v, d) and the truncated Boolean algebra B(d, v) = (P, ). Theorem (Erdős Ko Rado, 1961) Let v (t + 1)(d t + 1) and let Y X be a t-intersecting family, i.e., w(y) d t. Then ( ) v t Y. d t If v > (t + 1)(d t + 1) and if Y = ( v t d t) then for some u P with rank(u) = t. Y = Y u

27 Delsarte s linear programming method Define Q = (Q ij ) 0 i,j d by E j = 1 X d Q ij A i i=0 (0 j d), or equivalently (E 0, E 1,..., E d ) = 1 X (A 0, A 1,..., A d )Q. Since E 0 = 1 X J = 1 X (A 0 + A A d ) we find Q 00 = Q 10 = = Q d0 = 1.

28 E j = 1 X d i=0 Q ija i, Q 00 = Q 10 = = Q d0 = 1 Y X : w(y) d t χ R X : the characteristic vector of Y e = (e 0, e 1,..., e d ) : the inner distribution of Y : e i = 1 Y χt A i χ (0 i d) Then (P0) (P1) (P2) (P3) (eq) 0 = e 0 + e e d = 1 Y χt Jχ = Y, (eq) j = e 0 = 1, e d t+1 = = e d = 0, d e i Q ij = X Y χt E j χ 0 (1 j d). i=0

29 (eq) 0 = Y, e 0 = 1, e d t+1 = = e d = 0, (eq) j 0 ( j) A vector f (unique, if any) satisfying the following conditions gives a feasible solution to the dual problem: (D1) (D2) (D3) (D4) f 0 = 1, f 1 = = f t = 0, f t+1 > 0,..., f d > 0, (fq T ) 1 = = (fq T ) d t = 0. By the duality of linear programming, we have Y (fq T ) 0 and equality holds if and only if (eq) j f j = 0 (1 j d) (eq) t+1 = = (eq) d = 0 w (Y) t.

30 Y (fq T ) 0 ; Y = (fq T ) 0 w (Y) t Since w(y) d t and w(y) + w (Y) d, we find Y = (fq T ) 0 if and only if Y is a descendent of J(v, d). Under certain conditions, the vector satisfying (D1) (D4) was constructed in each of the following cases: Γ f (fq ( T 0 Johnson J(v, d) Wilson (1984) v t ) d t Hamming H(d, q) MDS weight enumerators q d t ] Grassmann J q (v, d) Frankl Wilson (1986) bilinear forms Bil q (d, e) (d, e, t, q)-singleton systems, Delsarte (1978) [ v t d t q q (d t)e Since J q (2d + 1, d) and the twisted Grassmann graph J q (2d + 1, d) have the same Q, we now also get the Erdős Ko Rado theorem for J q (2d + 1, d).