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 � V � that | V | = n , B ⊆ , and each pair of vertices lies exactly in 3 one 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, I R ) 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) ) , k = R ( K − 1) , T 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) k = 3 v − 9 , , 6 2 λ = v + 3 , µ = 9 . 2

(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) ) , k = R ( K − 1) , T 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)( R 2 − 2 R + 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 2 s ( s + 1)( µ + 1) , (2) then Γ is geometric and µ ∈ { s ( s + 1) , s 2 } (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 STC 1 ∗ 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)( R 2 − 2 R + 2)] / 2, then STC 1 are satisfied and the size of the largest claw with root v is − s = R for all vertices in Γ. ( STC 1 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 2 s ( 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 2 s ( 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, 1 2( µ ( − s 3 − s 2 + 2) − s 3 − s 2 + 2 s ) , k = 1 2( µ ( s 2 + s + 2) + s 2 + 3 s ) − 1 , = λ

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 2 s ( s + 1)( µ + 1) , then Γ is geometric and µ ∈ { s ( s + 1) , s 2 } . 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 STC 2 ∗ 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. ( STC 2 is another set of technical conditions.)

Overview of the proof (cont.) • STC 1 are satisfied. • If s ≤ − 4, then STC 2 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 STC 2 are satisfied. For the remaining values of µ we show that STC 2 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 = s 2 1 2( − s 5 − s 4 − s 3 + s 2 + 2 s ) , k = 1 2( s 4 + s 3 + 3 s 2 + 3 s ) − 1 . λ = • Known for s ≤ − 13 (Metsch 1995). • New for s ≥ − 11.

Infinite geometric family: µ = s 2 + s 1 2( − s 5 − 2 s 4 − 2 s 3 + s 2 + 4 s ) , k = 1 2( s 4 + 2 s 3 + 4 s 2 + 5 s ) − 1 . λ = • Known (Metsch 1991).

Infinite non-geometric family: µ = 1 − s 3 − s 2 + s + 1 , k = s 2 + 2 s. λ = • 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 1 2( s 4 + 2 s 3 + s 2 − 4) , k = 1 2( − s 3 − 2 s 2 − s − 6) . λ = • 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 = − µ 5 − 3 µ 4 − µ 3 + 11 µ 2 + 10 µ − 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 3 ∗ − 49 ∗ 17344819251 230450 4657 4703 95539730521 1430520 28019 21 28049 − 51 1453875051457 2969136 31165 6 31254 − 95

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