On strongly regular graphs attaining the claw bound M. Ma caj - - PowerPoint PPT Presentation

On strongly regular graphs attaining the claw bound

• M. Maˇ

caj Comenius University, Bratislava, Slovakia Villanova, June 5th 2014

Strongly regular graphs A k-regular graph on n vertices is strongly regular with parame- ters (n, k, λ, µ) if a ∼ b ⇔ |N(a) ∩ N(b)| = λ, a ∼ b ⇔ |N(a) ∩ N(b)| = µ. (Usually) a SRG has three distinct eigenvalues k > r > s, r > 0 and s < −1 and all the eigenvalues are integers.

Partial geometries (according to Bose) A partial geometry PG(R, K, T) is a triple (V, B, I) of sets V and B, and the relation I of incidence on V × B such that A1 Any two points lie on at most one line. A2 Each point lies on exactly R lines. A3 Each line has exactly K points. A4 If the point P does not lie on line l, then there are exactly T lines containing P which intersects l.

An example A Steiner triple system of order n is a pair (V, B) of sets such that |V | = n, B ⊆

V

3

• , and each pair of vertices lies exactly in
• ne triple of B.

A STS(n) exists iff n mod 6 ∈ {1, 3}. Each STS(n) is a partial geometry PG((n − 1)/2, 3, 3) (I is the ∈ relation).

The dual geometry If (V, B, I) is a PG(R, K, T), then (B, V, IR) is a PG(K, R, T). It is the dual geometry of (V, B, I). The dual geometry of a STS(n) has parameters (3, (n − 1)/2, 3).

The point graph of a geometry The point graph of a partial geometry (V, B, I) is the graph on V in which two points are adjacent iff they are collinear. The block graph of a partial geometry is the point graph of the dual geometry.

The point graph of a geometry Theorem (Bose, 1963). The point graph of a PG(R, K, T) is strongly regular with parameters n = K(1 + (K − 1)(R − 1) T ), k = R(K − 1), λ = K − 2 + (R − 1)(T − 1), µ = RT, and eigenvalues r = K − 1 − T, s = −R.

An example The point graph of a STS(v) is complete. The block graph of a STS(v) has parameters n = v(v − 1) 6 , k = 3v − 9 2 , λ = v + 3 2 , µ = 9.

(Pseudo-)geometric SRGs A SRG(n, k, λ, µ) is pseudo-geometric with characteristic (R, K, T) if n = K(1 + (K − 1)(R − 1) T ), k = R(K − 1), λ = K − 2 + (R − 1)(T − 1), µ = RT. A pseudo-geometric SRG is geometric if it is the point graph of a PG(R, K, T).

The result of Bose Theorem (Bose 1963). Any pseudo-geometric SRG with char- acteristic (R, K, T) is geometric if K > 1 2[R(R − 1) + T(R + 1)(R2 − 2R + 2)]. (1)

• Corollary. Any SRG with parameters of the block graph of a

STS(v) is geometric if v ≥ 69.

The improvement of Neumaier Theorem (Neumaier 1979). Let Γ be a SRG(n, k, λ, µ) with eigenvalues k > r > s. If r > −1 + 1 2s(s + 1)(µ + 1), (2) then Γ is geometric and µ ∈ {s(s + 1), s2} (equivalently, T ∈ {R − 1, R}). Remark If Γ is pseudo-geometric with characteristic (R, K, T), then (1) and (2) are equivalent.

Overview of the proof: basic ideas

• Maximal cliques in (SRGs) have small intersection.
• Large maximal cliques are edge disjoint.
• If the size(= number of leaves) of the largest claw with root

v is small, then there are large maximal cliques containing v.

Overview of the proof: Bose

• If STC1∗ are satisfied and the size of the largest claw with

root v is −s = R, then the neighborhood of v can be decom- posed into −s maximal cliques of size k/(−s).

• Moreover, if the size of the largest claw with root v is −s

for all vertices in Γ, then the above cliques form blocks of a PG(R, K, T).

• If K > [R(R − 1) + T(R + 1)(R2 − 2R + 2)]/2, then STC1

are satisfied and the size of the largest claw with root v is −s = R for all vertices in Γ. (STC1 is a set of technical conditions.)

Overview of the proof: Neumaier

• If we replace (1) by (2), then we can drop the assumption

that the graph is pseudo-geometric.

• If parameters of a PG(R, K, T) satisfy (1), then T ∈ {R, R−1}.

The claw bound The condition r ≤ −1 + 1 2s(s + 1)(µ + 1) is known as the claw bound. From now on we will consider only SRGs for which the claw bound is attained, that is, for which the condition r = −1 + 1 2s(s + 1)(µ + 1) (3) holds.

Graphs attaining the claw bound: parameters

• Lemma. Let Γ be a SRG for which the claw bound is attained.

Then, k = 1 2(µ(−s3 − s2 + 2) − s3 − s2 + 2s), λ = 1 2(µ(s2 + s + 2) + s2 + 3s) − 1,

Graphs attaining the claw bound: s = −2

• (10, 3, 0, 1) – Petersen,
• (16, 6, 2, 2) – Shrikhande + 1× geometric,
• (28, 12, 6, 4) – 3× Chang + T(8),
• (64, 30, 18, 10) – absolute bound.

Graphs attaining the claw bound: s ≤ −3

• Theorem. Let Γ be a SRG(n, k, λ, µ) with eigenvalues k > r > s.

If s ≤ −3 and r = −1 + 1 2s(s + 1)(µ + 1), then Γ is geometric and µ ∈ {s(s + 1), s2}.

• Corollary. Any SRG with parameters of the block graph of a

STS(67) (that is, (737, 96, 35, 9)) is geometric.

Overview of the proof

• Claw of size −s + 1 may appear in unique way.
• If STC2∗ are satisfied and the size of the largest claw with

root v is −s + 1, then the neighborhood of v can be decom- posed into −s + 1 maximal cliques of size k/(−s + 1).

• Moreover, if the size of the largest claw with root v is −s + 1

for all vertices in Γ, then the above cliques form blocks of a PG(R′, K′, T ′) with R′ = −s + 1. (STC2 is another set of technical conditions.)

Overview of the proof (cont.)

• STC1 are satisfied.
• If s ≤ −4, then STC2 are satisfied.
• Either the size of the largest claw with root v is −s for all

vertices in Γ or the size of the largest claw with root v is −s + 1 for all vertices in Γ.

• There are no claws of size −s + 1.

Overview of the proof: s = −3

• Feasible parameters are obtained only for µ ∈ {1, 6, 9}.

If µ = 1, then STC2 are satisfied. For the remaining values of µ we show that STC2 are not necessary.

• For µ = 6 it follows from an elementary counting argument.
• For µ = 9 we use the following.
• Proposition. Let ∆ be a triangle-free 12-valent graph on at most

32 vertices such that the second largest eigenvalue of ∆ is ≤ 2. Then, ∆ is bipartite.

Infinite geometric family: µ = s2 k = 1 2(−s5 − s4 − s3 + s2 + 2s), λ = 1 2(s4 + s3 + 3s2 + 3s) − 1.

• Known for s ≤ −13 (Metsch 1995).
• New for s ≥ −11.

Infinite geometric family: µ = s2 + s k = 1 2(−s5 − 2s4 − 2s3 + s2 + 4s), λ = 1 2(s4 + 2s3 + 4s2 + 5s) − 1.

• Known (Metsch 1991).

Infinite non-geometric family: µ = 1 k = −s3 − s2 + s + 1, λ = s2 + 2s.

• Known for s = −4 (Bose and Dowling 1971).
• Known for s = −4 (Bagchi 2006) (parameters are (1666, 45, 8, 1)).

Infinite non-geometric family: µ = −s − 2 k = 1 2(s4 + 2s3 + s2 − 4), λ = 1 2(−s3 − 2s2 − s − 6).

• Known for s = −3 (Bose and Dowling 1971) (µ = 1).
• Known for s = −4 (Brouwer and Neumaier 1988) (parame-

ters are (1961, 70, 15, 2)).

• New for s ≤ −5.

Infinite non-geometric family: s = −µ5 − 3µ4 − µ3 + 11µ2 + 10µ − 8 8 .

• Known for µ = 2 (Brouwer and Neumaier 1988) (parameters

are (1961, 70, 15, 2)).

• New for µ ≥ 3.

Sporadic examples

• There are 71 other parameter sets with s ≥ −750000,
• none is geometric,
• only five of them have µ ≥ −s.

µ s 36 −30 176 −34 154 −43 1905 −254 850 −390

n k λ µ r s 32180016 8246 442 2 458 −18 165989881 41790 2079 10 2089 −20 560562256 59130 2254 6 2274 −26 830690560 92934 3558 10 3574 −26 622523049 56840 2083 5 2105 −27 1487052121 117720 4039 9 4059 −29 6260912087 482856 16100 36 16094 −30 2206382116 95340 2774 4 2804 −34 62862845606 3376240 99438 176 99296 −34 13027595348 517465 13968 20 13985 −37 229749888277 6018606 140075 154 139964 −43 17344819251 230450 4657 3∗ 4703 −49∗ 95539730521 1430520 28019 21 28049 −51 1453875051457 2969136 31165 6 31254 −95

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