Vertex subsets with minimal width and dual width in Q-polynomial distance-regular graphs University of Wisconsin & Tohoku University February 2, 2011
Every face (or facet) of a hypercube is a hypercube...
Goal Generalize this situation to Q-polynomial distance-regular graphs. Discuss its applications.
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.
A 1 A i = b i 1 A i 1 + a i A i + c i+1 A i+1 (0 i d)
Example: hypercubes X = {0, 1} d x R y {i : x i y i } = 1 Γ = Q d = (X, R) : the hypercube Q 3 : 011 001 101 111 000 010 110 100 Q d = the binary Hamming graph
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}
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
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
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.
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) 0 1 0 1 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)
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.
Y u = {x X : u x} is a facet of Q d If d = 3 and u = (, 0, ) then: 001 101 011 111 000 Y u 010 110 100 Every facet of Q d is of this form. The induced subgraph on Y u is Q d rank(u).
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)
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.
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).
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
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.
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.
Observation 001 101 011 111 000 Y u 010 110 100 Y u is convex (geodetically closed).
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 1 1 1 1 1 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.
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
The families related to Johnson graphs Johnson graphs q-analog Grassmann graphs pseudo twisted Grassmann graphs
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.).
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}
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
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 1 + + A d ) we find Q 00 = Q 10 = = Q d0 = 1.
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 1 + + 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
(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.
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).