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

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

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

Introduction Regulariy Classical DRGs Strongly Regular Graphs Tight DRGs Distance-regular graphs Tight & classical 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 ) or SRG( n , k , λ, µ ), in which case the complement graph Γ is SRG( n , n − k − 1 , n − 2 k + µ − 2 , n − 2 k + λ ) .

Introduction Regulariy Classical DRGs Strongly Regular Graphs Tight DRGs Distance-regular graphs Tight & classical Intersection array Examples trivial: K n , t · K n , and the complement K t × 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 C 5 is SRG(5 , 2 , 0 , 1), the Petersen graph is SRG(10 , 3 , 0 , 1).

Introduction Regulariy Classical DRGs Strongly Regular Graphs Tight DRGs Distance-regular graphs Tight & classical Intersection array More examples of strongly regular graphs L ( K v ) is strongly regular with parameters � v � n = , k = 2( v − 1) , λ = v − 2 , µ = 4 . 2 For v � = 8 this is the unique srg with these parameters. Similarly, L ( K v , v ) = K v × K v is strongly regular, with parameters n = v 2 , k = 2( v − 2) , λ = v − 2 , µ = 2 . For v � = 4 this is the unique srg with these parameters.

Introduction Regulariy Classical DRGs Strongly Regular Graphs Tight DRGs Distance-regular graphs Tight & classical 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 � v � � v − 1 � 2 n = � , k = s s − 1 − 1 , � s 2 λ = v − 1 s − 1 − 2 + ( s − 1) 2 , µ = s 2 . 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 ( K v ). (If D is a square design, i.e., v − 1 = s ( s − 1), then its line graph is the complete graph K v .)

Introduction Regulariy Classical DRGs Strongly Regular Graphs Tight DRGs Distance-regular graphs Tight & classical 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 ( v 2 log v / 6 − cv 2 ) . 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 , 3 s , s + 3 , 9), for n = ( s + 1)(2 s + 3) / 3 and s ≡ 0 or 2 (mod 3).

Introduction Regulariy Classical DRGs Strongly Regular Graphs Tight DRGs Distance-regular graphs Tight & classical 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 = v 2 , 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 = 2 s − 1. For s = 2 we obtain the lattice graph K v × K v . The number of Latin squares of order k is asymptotically equal to exp ( k 2 log k − 2 k 2 ) .

Introduction Regulariy Classical DRGs Strongly Regular Graphs Tight DRGs Distance-regular graphs Tight & classical 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 n ≤ 1 - Absolute bounds 2 m σ ( m σ + 3) , and if q 1 n ≤ 1 11 � = 0 also 2 m σ ( m σ + 1) .

Introduction Regulariy Classical DRGs Strongly Regular Graphs Tight DRGs Distance-regular graphs Tight & classical Intersection array Paley graph P (13) 5 1 −3 6 2 6 2 −2 −6 3 −1 −5 4 −6 3 −5 4 0 −4 5 −4 5 1 −3 6 imbedded on a torus −3 6 2 −2 −6 The Shrikhande graph and P (13) are the only distance-regular graphs which are locally C 6 (one has µ = 2 and the other µ = 3).

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

Introduction Regulariy Classical DRGs Strongly Regular Graphs Tight DRGs Distance-regular graphs Tight & classical Intersection array Clebsch graph Two drawings of the complement of the Clebsch graph.

Introduction Regulariy Classical DRGs Strongly Regular Graphs Tight DRGs Distance-regular graphs Tight & classical 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 K 2 and some are 2 . K 1 .

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

Introduction Regulariy Classical DRGs Strongly Regular Graphs Tight DRGs Distance-regular graphs Tight & classical Intersection array Moore graphs Let Γ be a graph of diameter d . Then Γ has girth at most 2 d + 1 , while in the bipartite case the girth is at most 2 d . Graphs with diameter d and girth 2 d + 1 are called Moore graphs ( Hoffman and Singleton ). Bipartite graphs with diameter d and girth 2 d are known as generalized polygons ( Tits ).

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

Introduction Regulariy Classical DRGs Strongly Regular Graphs Tight DRGs Distance-regular graphs Tight & classical Intersection array Distance-regular graphs ◮ A graph Γ, diameter d , vertex set V Γ, and Γ i ( 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 p h ij ( u , v ) := | Γ i ( u ) ∩ Γ j ( v ) | . ◮ Γ is distance-regular if the values of p h ij := p h (a cubic number of them – O ( d 3 )) ij ( u , v ) only depend on the choice of distances h , i , j and not on the particular vertices u , v .

Introduction Regulariy Classical DRGs Strongly Regular Graphs Tight DRGs Distance-regular graphs Tight & classical Intersection array Intersection array ◮ Distance-regular graphs are regular with valency k := p 0 11 and have subconstituents Γ i ( u ) (0 ≤ i ≤ d ) of size k i := p 0 ii and valency a i := p i 1 , i . b 1 b 2 b 3 k k 2 k 3 k d k · · · c 2 c 3 c d u 1 a 1 a 2 a 3 a d ◮ Set b i := p i 1 , i +1 , c i := p i 1 , i − 1 ; note a i + b i + c i = k (0 ≤ i ≤ d ). ◮ All p h ij can be determined from the intersection array { k , b 1 , . . . , b d − 1 ; 1 , c 2 , . . . , c d } (2 d − 1 parameters) . ◮ Eigenvalues and their multiplicities can be computed directly from the intersection array.

Introduction Classical parameters Classical DRGs Known families Tight DRGs Open cases Tight & classical 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 H q ( d , e ) (e ≥ d) and Grassmann graphs J q ( e , d ) (e ≥ 2 d) , 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, b ′ := b 2 / ( θ − k + b 1 + c 2 − b ) , d , k , c d , b := b 1 / (1+ θ ) and where θ = θ 1 is the second eigenvalue in the Q -polynomial ordering, cf. [BCN89, Prop. 8.1.5].