Restrictions on classical distance-regular graphs Aleksandar Juri - - PowerPoint PPT Presentation

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Restrictions on classical distance-regular graphs

Aleksandar Juriˇ si´ c

Joint work with Janoˇ s Vidali University of Ljubljana Faculty of Computer and Information Science IMFM

October 4, 2016

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Introduction Classical DRGs Tight DRGs Tight & classical Regulariy Strongly Regular Graphs Distance-regular graphs Intersection array

Hilbert

geometry: points/lines, where all the points look the same (and similarly for lines). ”Look”?

• algebra:

∃ an authomorphism ...

• topology:

something else (shape is not important)

• graph theory:

relation preserving adjacency

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Introduction Classical DRGs Tight DRGs Tight & classical Regulariy Strongly Regular Graphs Distance-regular graphs Intersection array

Our objects

a graph: vertices/edges regularity:

◮ vertices have the same valency

(k) (i.e., locally vertices look the same)

◮ on any edge there is the same number of triangles

(λ)

◮ any two nonadjacent vertices have

the same number of common neighbours (µ) We have introduced a strongly regular graph Γ: SRG(k, λ, µ)

• r SRG(n, k, λ, µ), in which case the complement graph Γ is

SRG(n, n − k − 1, n − 2k + µ − 2, n − 2k + λ).

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Introduction Classical DRGs Tight DRGs Tight & classical Regulariy Strongly Regular Graphs Distance-regular graphs Intersection array

Examples

trivial: Kn, t · Kn, and the complement Kt×n, i.e., SRG(tn, (t−1)n, (t−2)n, (t−1)n)). For n = 2 we obtain the Cocktail Party graph CP(t): The cycle C5 is SRG(5, 2, 0, 1), the Petersen graph is SRG(10, 3, 0, 1).

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Introduction Classical DRGs Tight DRGs Tight & classical Regulariy Strongly Regular Graphs Distance-regular graphs Intersection array

More examples of strongly regular graphs

L(Kv) is strongly regular with parameters n = v 2

• ,

k = 2(v − 1), λ = v − 2, µ = 4. For v = 8 this is the unique srg with these parameters. Similarly, L(Kv,v) = Kv × Kv is strongly regular, with parameters n = v2, k = 2(v − 2), λ = v − 2, µ = 2. For v = 4 this is the unique srg with these parameters.

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Introduction Classical DRGs Tight DRGs Tight & classical Regulariy Strongly Regular Graphs Distance-regular graphs Intersection array

Steiner graphs

Steiner graph is the block (line) graph of a 2-(v, s, 1) design with v − 1 > s(s − 1), and it is strongly regular with parameters n = v

2

• s

2

, k = s v −1 s−1 − 1

• ,

λ = v −1 s−1 − 2 + (s − 1)2, µ = s2. A point graph of a Steiner system is a complete graph, thus a line graph of a Steiner system S(2, v) is the triangular graph L(Kv). (If D is a square design, i.e., v − 1 = s(s − 1), then its line graph is the complete graph Kv.)

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Introduction Classical DRGs Tight DRGs Tight & classical Regulariy Strongly Regular Graphs Distance-regular graphs Intersection array

The fact that Steiner triple system with v points exists for all v ≡ 1 or 3 (mod 6) goes back to Kirkman in 1847. More recently Wilson showed that the number n(v) of Steiner triple systems on an admissible number v of points satisfies n(v) ≥ exp(v2 log v/6 − cv2). A Steiner triple system of order v > 15 can be recovered uniquely from its block graph, so we conclude: there are super-exponentially many SRG(n, 3s, s + 3, 9), for n = (s + 1)(2s + 3)/3 and s ≡ 0 or 2 (mod 3).

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Introduction Classical DRGs Tight DRGs Tight & classical Regulariy Strongly Regular Graphs Distance-regular graphs Intersection array

Transversal Design graphs (OA(s, v))

For 2 ≤ s ≤ v the block graph of a transversal design TD(s, v) (two blocks being adjacent iff they intersect) is strongly regular with parameters n = v2, k = s(v −1), λ = (v −2)+(s−1)(s−2), µ = s(s−1). Note that a line graph of TD(s, v) is a conference graph when v = 2s − 1. For s = 2 we obtain the lattice graph Kv × Kv. The number of Latin squares of order k is asymptotically equal to exp(k2 log k − 2k2).

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Introduction Classical DRGs Tight DRGs Tight & classical Regulariy Strongly Regular Graphs Distance-regular graphs Intersection array

Theorem [Neumaier]. A SRG with the smallest eigenvalue −m, m ≥ 2 integral, is with finitely many exceptions, either a complete multipartite graph, a Steiner graph, or the block graph of a transversal design. Feasibility conditions and a table

• divisibility conditions
• integrality of eigenvalues
• integrality of multiplicities
• Krein conditions
• Absolute bounds

n ≤ 1

2 mσ(mσ + 3),

and if q1

11 = 0 also

n ≤ 1

2 mσ(mσ + 1).

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Introduction Classical DRGs Tight DRGs Tight & classical Regulariy Strongly Regular Graphs Distance-regular graphs Intersection array

Paley graph P(13)

1 4 3 2 2 1 5 6 6 6 6 5 5 4 3 2 −6 −1 −3 −6 −6 −5 −4 −3 −2 −4 −5 −3 −2

imbedded on a torus The Shrikhande graph and P(13) are the only distance-regular graphs which are locally C6 (one has µ = 2 and the other µ = 3).

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Introduction Classical DRGs Tight DRGs Tight & classical Regulariy Strongly Regular Graphs Distance-regular graphs Intersection array

Tutte 8-age

The Tutte’s 8-cage is the GQ(2, 2) = W (2). A cage is the smallest possible regular graph (here degree 3) that has a prescribed girth.

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Introduction Classical DRGs Tight DRGs Tight & classical Regulariy Strongly Regular Graphs Distance-regular graphs Intersection array

Clebsch graph

Two drawings of the complement of the Clebsch graph.

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Introduction Classical DRGs Tight DRGs Tight & classical Regulariy Strongly Regular Graphs Distance-regular graphs Intersection array

Shrikhande graph

The Shrikhande graph drawn on two ways: (a) on a torus, (b) with imbedded four-cube. The Shrikhande graph is not distance transitive, since some µ-graphs, i.e., the graphs induced by common neighbours of two vertices at distance two, are K2 and some are 2.K1.

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Introduction Classical DRGs Tight DRGs Tight & classical Regulariy Strongly Regular Graphs Distance-regular graphs Intersection array

Schl¨ afli graph

How to construct the Schl¨ afli graph: make a cyclic 3-cover corresponding to arrows, and then join vertices in every antipodal class.

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Introduction Classical DRGs Tight DRGs Tight & classical Regulariy Strongly Regular Graphs Distance-regular graphs Intersection array

Moore graphs

Let Γ be a graph of diameter d. Then Γ has girth at most 2d + 1, while in the bipartite case the girth is at most 2d. Graphs with diameter d and girth 2d + 1 are called Moore graphs (Hoffman and Singleton). Bipartite graphs with diameter d and girth 2d are known as generalized polygons (Tits).

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Introduction Classical DRGs Tight DRGs Tight & classical Regulariy Strongly Regular Graphs Distance-regular graphs Intersection array

Moore graphs of diameter 2

A Moore graph of diameter two is a regular graph with girth five and diameter two. The only Moore graphs are

◮ the pentagon, ◮ the Petersen graph, ◮ the Hoffman-Singleton graph, and ◮ possibly a strongly regular graph on 3250 vertices.

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Introduction Classical DRGs Tight DRGs Tight & classical Regulariy Strongly Regular Graphs Distance-regular graphs Intersection array

Distance-regular graphs

◮ A graph Γ, diameter d, vertex set V Γ, and

Γi(u) ⊂ V Γ the subset of vertices at distance i from u ∈ V Γ.

◮ u, v ∈ V Γ with ∂(u, v) = h, the intersection numbers are

ph

ij(u, v) := |Γi(u) ∩ Γj(v)| . ◮ Γ is distance-regular if the values of

ph

ij := ph ij(u, v)

(a cubic number of them – O(d3))

• nly depend on the choice of distances h, i, j

and not on the particular vertices u, v.

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Introduction Classical DRGs Tight DRGs Tight & classical Regulariy Strongly Regular Graphs Distance-regular graphs Intersection array

Intersection array

◮ Distance-regular graphs are regular with valency k := p0 11

and have subconstituents Γi(u) (0 ≤ i ≤ d) of size ki := p0

ii

and valency ai := pi

1,i. u k a1 k2 a2 k3 a3 · · · kd ad k 1 b1 c2 b2 c3 b3 cd ◮ Set bi := pi 1,i+1, ci := pi 1,i−1; note ai +bi +ci = k (0≤i ≤d). ◮ All ph ij can be determined from the intersection array

{k, b1, . . . , bd−1; 1, c2, . . . , cd} (2d − 1 parameters).

◮ Eigenvalues and their multiplicities

can be computed directly from the intersection array.

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Introduction Classical DRGs Tight DRGs Tight & classical Classical parameters Known families Open cases Bounds for β

Distance-regular graphs with classical parameters

The largest known infinite families of distance-regular graphs are the families of bilinear forms graphs Hq(d, e) (e ≥ d) and Grassmann graphs Jq(e, d) (e ≥ 2d), each of which is parameterized with 3 unbounded parameters, It seems that one could reduce the number of parameters considerably. Indeed, in 1984 Leonard succeeded to parametrize Q-polynomial distance-regular graphs with only 5 parameters, d, k, cd, b := b1/(1+θ) and b′ := b2/(θ−k +b1 +c2 −b), where θ = θ1 is the second eigenvalue in the Q-polynomial

• rdering, cf. [BCN89, Prop. 8.1.5].

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Introduction Classical DRGs Tight DRGs Tight & classical Classical parameters Known families Open cases Bounds for β

Distance-regular graphs with classical parameters

◮ A. Neumaier [BCN89] observed that the intersection arrays

• f many known distance-regular graphs can be expressed in

terms of just four parameters. (adding the relation b = b′ to Q-polynomial distance-regular graphs).

◮ A distance-regular graph of diameter d has

classical parameters (d, b, α, β) when bi = ([d] − [i])(β − α[i]), ci = [i](1 + α[i − 1]) (0 ≤ i ≤ d), where [n] := [n]b = n−1

i=0 bi

(e.g. [0]=0, [1]=1, [2]=1+b).

◮ If d ≥ 3, the parameter b is an integer distinct from 0 and −1.

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Introduction Classical DRGs Tight DRGs Tight & classical Classical parameters Known families Open cases Bounds for β name d b α + 1 β + 1 Johnson graphs J(e, d), e ≥ 2d d 1 2 e − d + 1 Grassmann graphs Jq(e, d), e ≥ 2d d q q + 1 [e − d + 1] Twisted Grassmann graphs ˆ Jq(2d + 1, d) d q q + 1 [d + 2] Hamming graphs H(d, e) d 1 1 e Doob graphs ˆ Hi (d, 4), 1 ≤ i ≤ d/2 d 1 1 4 Halved cubes 1

2 H(n, 2)

d 1 3 m + 1 Dual polar graphs Bd (q) d q 1 q + 1 Dual polar graphs Cd (q) d q 1 q + 1 Dual polar graphs Dd (q) d q 1 2 Hemmeter graphs ˆ Dd (q) d q 1 2 Halved dual polar graphs Dn,n(q) d q2 [3]q [m + 1]q Ustimenko graphs ˆ Dn,n(q) d q2 [3]q [m + 1]q Dual polar graphs 2Dd+1(q) d q 1 q2 + 1 Dual polar graphs 2A2d (q) d q2 1 q3 + 1 Dual polar graphs 2A2d−1(q) d q2 1 q + 1

• r

d −q

1+q2 1−q 1−(−q)d+1 1−q

Bilinear forms graphs Hq(d, e), e ≥ d d q q qe Alternating forms graphs Altn(q) d q2 q2 qm Quadratic forms graphs Qn−1(q) d q2 q2 qm Hermitean forms graphs Herd (q) d −q −q −(−q)d Triality graphs 3D4,2(q) 3 −q

1 1−q

[3]q Affine E6(q) graphs 3 q4 q4 q9 Exceptional Lie graphs E7,7(q) 3 q4 [5]q [10]q Gosset graph E7(1) 3 1 5 10 Witt graph M23 3 −2 −1 6 Witt graph M24 3 −2 −3 11 Coset graph of the extended ternary Golay code 3 −2 −2 9 q is a prime power; m = n = 2d + 1 or m + 1 = n = 2d

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Introduction Classical DRGs Tight DRGs Tight & classical Classical parameters Known families Open cases Bounds for β

Classical DRG d b α β Johnson graph d 1 1 n − d J(n, d), n ≥ 2d Grassmann graph d q q

• n−d +1

1

• −1

Jq(n, d), n ≥ 2d Hamming graph d 1 n − 1 H(d, n) Bilinear forms graph d q q − 1 qe − 1 Bil(d × e, q), d ≤ e Alternating forms graph ⌊n/2⌋ q2 q2 − 1 qm − 1 Alt(n, q), m = 2⌈n/2⌉−1 Hermitean forms graph d −q −q − 1 −(−q)d − 1 Her(d, q2) Quadratic forms graph n + 1 2

• q2

q2 − 1 qm − 1 Qua(n, q), m = 2⌊n/2⌋+1 Dual polar graph d q qe

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Introduction Classical DRGs Tight DRGs Tight & classical Classical parameters Known families Open cases Bounds for β

Open cases

◮ For many known graphs with classical parameters,

uniqueness is not known.

◮ There are also many open cases. ◮ All known open cases with diameter at least 4,

see [BCN89, Bro11], have either α = b − 1

• r

α = b

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Introduction Classical DRGs Tight DRGs Tight & classical Classical parameters Known families Open cases Bounds for β

Bounds for β

For a distance-regular graph Γ with classical parameters and d ≥ 3, we show the following bounds for the parameter β:

◮ b < 0, α = b − 1:

−bd + b + 1 ≤ β, for d = 3: β ≤ −b(b + 1)(b − 2), for d ≥ 4: Γ is a Hermitean forms graph

• r the lower bound is met – existence open!

◮ b < 0, α = b:

−b2[d − 1] + 1 ≤ β ≤ −b3[d − 2] + b, for d ≥ 4, the lower bound is met – existence open!

◮ b ≥ 2, α = b − 1, d = 3:

b2 + √ b + 1

• ≤

β with 6 ≤ β if b = 2.

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Introduction Classical DRGs Tight DRGs Tight & classical Classical parameters Known families Open cases Bounds for β

Corollaries/Classifications

◮ A graph with classical parameters (d, −2, −3, β) is

◮ the coset graph of the extended ternary Golay code, or ◮ a Hermitean forms graph.

◮ A graph with classical parameters (d, −2, −2, β) is

◮ the Witt graph associated to M24, or ◮ the generalized hexagon GH(2, 8).

◮ Nonexistence follows for the cases

◮ (d, b, α, β) = (3, 2, 1, 5) with 216 vertices [JV12], ◮ (d, b, α, β) = (3, 3, 2, 10) with 1331 vertices, and ◮ (d, b, α, β) = (3, 8, 7, 66) with 300763 vertices.

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Introduction Classical DRGs Tight DRGs Tight & classical Tight distance-regular graphs Local graphs Equitable partitions

Fundamental Bound and Tight distance-regular graphs

J., Koolen & Terwilliger [JKT00] Γ distance-regular, d ≥ 2, and eigenvalues θ0 > θ1 > · · · > θd.

• θ1 +

k a1 + 1 θd + k a1 + 1

• ≥ −

ka1b1 (a1 + 1)2 . A non-bipartite graph with equality in this bound is called tight. Such graphs can be parametrized with d + 1 parameters. The only known primitive tight graph is the Patterson graph with 22880 vertices, which is uniquely determined by its intersection array {280, 243, 144, 10; 1, 8, 90, 280} [BJK08].

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Introduction Classical DRGs Tight DRGs Tight & classical Tight distance-regular graphs Local graphs Equitable partitions

Local graphs of tight graphs

Characterization [JKT00]: A distance-regular graph is tight iff for any vertex u of a Γ, the local graph Γ(u) is strongly regular with nontrivial eigenvalues τ = −1 − b1 1 + θd and σ = −1 − b1 1 + θ1 and multiplicities mτ = a1(a1 − σ)(σ + 1) (a1 + στ)(σ − τ) and mσ = a1(a1 − τ)(τ + 1) (a1 + στ)(τ − σ) .

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Introduction Classical DRGs Tight DRGs Tight & classical Tight distance-regular graphs Local graphs Equitable partitions

1-homogeneity

A partition {Ci}t

i=1 of V Γ is equitable if there exist parameters nij

such that every vertex in Ci has precisely nij neighbours in Cj. A graph is distance-regular iff the distance partition for every vertex is equitable with the same parameters. A graph Γ is 1-homogeneous [Nom94] if the distance partition corresponding to any edge is equitable with the same parameters.

u v a1 b1 b1 . . . . . . . . .

p1

i− 1, i− 1

p1

i− 1,i

p1

i,i− 1

p1

ii p1

i,i+ 1

p1

i+ 1,i

p1

i+ 1, i+ 1

. . . . . . . . .

p1

d − 2, d − 1

p1

d − 1, d − 2

p1

d − 1, d − 1

p1

d − 1,d

p1

d,d − 1

p1

dd

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Introduction Classical DRGs Tight DRGs Tight & classical Tight distance-regular graphs Local graphs Equitable partitions

The CAB property

A graph Γ has the CAB property [JK00] if for any u, v ∈ V Γ, the partition of the local graph Γ(u) corresponding to the distances from v is equitable with parameters depending only on the distance ∂(u, v).

v Γ(v) · · · Γh−1(v) Γh(v) Γh+1(v) · · · u ch ah bh

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Introduction Classical DRGs Tight DRGs Tight & classical Tight distance-regular graphs Local graphs Equitable partitions

Characterization

Characterization [JKT00, JK00]: Let Γ be a distance-regular graph with a1 = 0 and ad = 0. The following are equivalent:

◮ Γ is 1-homogeneous, ◮ Γ has the CAB property, ◮ Γ is tight.

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Introduction Classical DRGs Tight DRGs Tight & classical Condition Parameters of partitions Feasible family Local graphs

Tight distance-regular graphs with classical parameters

Theorem. Γ distance-regular, diameter d ≥ 3 with classical parameters (d, b, α, β) Γ is tight iff β = 1 + α[d − 1] and b, α > 0. For an imprimitive classical distance-regular graph Γ we have:

◮ Γ is bipartite ⇐

⇒ α = 0 and β = 1.

◮ Γ is antipodal ⇐

⇒ b = 1 and β = 1 + α[d − 1].

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Introduction Classical DRGs Tight DRGs Tight & classical Condition Parameters of partitions Feasible family Local graphs

Parameters of partitions

The parameters of the CAB and 1-homogeneous partitions

• f tight distance-regular graphs with classical parameters

can be computed explicitly:

ch γh ah a1 − αh − βh bh a1 − δh a1 − γh αh βh δh

αh = 1+α[h −1], βh = b(1+αbh[d −h −1]), γh = δh−1 = α(b +1)[h −1],

u v a1 b1 a1 −ρ2 b1 a1 −ρ2 . . . . . . . . . p1

i− 1, i− 1

p1

i− 1,i

ai−

1 −ρi

p1

i,i− 1

ai−

1 −ρi

p1

ii

p1

i,i+ 1

ai −ρi+

1

p1

i+ 1,i

ai −ρi+

1

p1

i+ 1, i+ 1

. . . . . . . . . p1

d − 2, d − 1

ad

− 2 −ρd − 1

p1

d − 1, d − 2

ad

− 2 −ρd − 1

p1

d − 1, d − 1

kd kd 1 1 a1 1 b1 1 a1 1 b1 1 ρ2 ρ2 b2 τ1 b2 ci−

1

ci−

1

σi−

1

τi−

1

σi ρi ρi bi ci bi ci τi σi+

1

ρi+

1

ρi+

1

bi+

1

bi+

1

τi+

1

cd

− 2

cd

− 2

σd

− 1

ρd

− 1

ρd

− 1

bd

− 1 cd − 1

bd

− 1 cd − 1

ρd ρd bd

− 1

bd

− 1

ρi = αbi−2(b+1)[i−1], σi = [i−1](1+α[i−1]), τi = bi+1[d−i−1](1+αbi[d−i−1]).

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Introduction Classical DRGs Tight DRGs Tight & classical Condition Parameters of partitions Feasible family Local graphs

Classical distance-regular graphs with b = 1

Classification. Γ distance-regular, diameter d ≥ 2 with classical parameters. Then Γ is antipodal iff Γ is

• 1. the Cocktail Party graph K(α

+ 2)×2 (d =2, b=1, α, β =α+1),

• 2. the Gosset graph {27,10,1

;1,10,27} (d =3, b=1, α=4, β =9),

• 3. the d-cube (Qd)

(d, b=1, α=0, β =1),

• 4. the Johnson graph J(2d, d)

(d, b=1, α=1, β =d),

• 5. the halved (2d)-cube ( 1

2Q2d)

(d, b=1, α=2, β =2d −1). Our idea: Γ is tight(d, α) and µ-graphs are K(α+1)×2, but cf. [BCN. Thm.6.1.1], where they note that for b = 1 (not necessarily antipodal) one has θ = b1 − 1 is an eigenvalue with the cosine sequence σi = 1 − i(λ + 2)/k (0 ≤ i ≤ d).

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Introduction Classical DRGs Tight DRGs Tight & classical Condition Parameters of partitions Feasible family Local graphs

A feasible family

We find a two-parametrical family of classical parameters for tight distance-regular graphs: (d, b, α, β) = (d, b, b − 1, bd−1). (1)

◮ α = b − 1 implies that corresponding graphs are formally self-dual. ◮ For b = 1 we get d-cubes, which are bipartite and thus not tight. ◮ For b < 0 or d = 3 and b ≥ 2, the parameters are not feasible. ◮ For b ≥ 2 and d ≥ 4 we have a feasible parameter set (primitive). ◮ The smallest open case is {120, 98, 60, 8; 1, 6, 28, 120}.

Theorem: Family (1) for d ≥ 3 is realized precisely by d-cubes. Idea of proof: for b ≥ 2, local graphs are strongly regular, but their eigenvalues have nonintegral multiplicities.

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Introduction Classical DRGs Tight DRGs Tight & classical Condition Parameters of partitions Feasible family Local graphs

Local graphs

Let Γ be a tight distance-regular graph with classical parameters (d, b, α, β), diameter d ≥ 3 and parameter b ≥ 2 such that its local graph is an orthogonal array or Steiner system graph.

◮ Then, α is b or b + 1, respectively. ◮ We have checked that the multiplicity

• f the smallest eigenvalue is never integral when

α = b and d ≤ 17,

• r

α = b + 1 and d ≤ 7. Conjecture: The local graph of a tight distance-regular graph with classical parameters (d, b, α, β), where d ≥ 3 and b ≥ 2, is not a orthogonal array or Steiner system graph.

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A.E. Brouwer, A.M. Cohen, and A. Neumaier. Distance-regular graphs. Springer-Verlag, Berlin, 1989. A.E. Brouwer, A. Juriˇ si´ c, and J. Koolen. Characterization of the Patterson graph.

• J. Algebra, 320(5):1878–1886, 2008.

A.E. Brouwer. Parameters of distance-regular graphs, 2011. http://www.win.tue.nl/~aeb/drg/drgtables.html.

• A. Juriˇ

si´ c and J. Koolen. A local approach to 1-homogeneous graphs.

• Des. Codes Cryptogr., 21(1–3):127–147, 2000.
• A. Juriˇ

si´ c, J. Koolen, and P. Terwilliger. Tight distance-regular graphs.

• J. Algebraic Combin., 12(2):163–197, 2000.
• A. Juriˇ

si´ c and J. Vidali. Extremal 1-codes in distance-regular graphs of diameter 3.

• Des. Codes Cryptogr., 65(1–2):29–47, 2012.
• K. Nomura.

Homogeneous graphs and regular near polygons.

• J. Combin. Theory Ser. B, 60(1):63–71, 1994.

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Introduction Classical DRGs Tight DRGs Tight & classical Condition Parameters of partitions Feasible family Local graphs

Bounds for β

For a distance-regular graph Γ with classical parameters, diameter d ≥ 3 and α = b ≥ 2, we show the following bounds for the parameter β: min

• (b + 1)2, (b + 1)2 − b
• 1 −
• 2

b

• + 1

2 + 25 8 √ 2b

• ≤

β A feasible family arises when β = (b + 1)2 (b ≥ 1).

◮ It is realized by the Johnson graph J(7, 3) when b = 1. ◮ The smallest open case is {63, 42, 12; 1, 9, 49} (b = 2).