Tight distance-regular graphs with classical parameters Jano s - PowerPoint PPT Presentation

Tight distance-regular graphs with classical parameters Janoˇ s Vidali Joint work with Aleksandar Juriˇ si´ c University of Ljubljana Faculty of Mathematics and Physics Faculty of Computer and Information Science University of Primorska Andrej Maruˇ siˇ c Institute May 27, 2015

Introduction Classical DRGs Distance-regular graphs Tight DRGs Intersection array Tight & classical Distance-regular graphs ◮ Let Γ be a graph of diameter d with vertex set V Γ, and Γ i ( u ) be the set of vertices of Γ at distance i from u ∈ V Γ. ◮ For u , v ∈ V Γ with ∂ ( u , v ) = h , let the intersection numbers be p h ij := | Γ i ( u ) ∩ Γ j ( v ) | . ◮ The graph Γ is distance-regular if the values of p h ij ( u , v ) only depend on the choice of distances h , i , j and not on the particular vertices u , v .

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

Introduction Classical parameters Classical DRGs Known families Tight DRGs Open cases Tight & classical Distance-regular graphs with classical parameters ◮ A. Neumaier [BCN89] observed that the intersection arrays of many known distance-regular graphs can be expressed in terms of just four parameters. ◮ A distance-regular graph of diameter d has classical parameters ( d , b , α, β ) if its intersection array satisfies b i = ([ d ] − [ i ])( β − α [ i ]) (0 ≤ i ≤ d − 1) and c i = [ i ](1 + α [ i − 1]) (1 ≤ i ≤ d ) , i =0 b i is the b -analogue of n . where [ n ] := [ n ] b := � n − 1 ◮ The parameter b is an integer distinct from 0 and − 1.

Introduction Classical parameters Classical DRGs Known families Tight DRGs Open cases Tight & classical name α + 1 β + 1 d b Johnson graphs J ( e , d ), e ≥ 2 d d 1 2 e − d + 1 Grassmann graphs J q ( e , d ), e ≥ 2 d d q q + 1 [ e − d + 1] Twisted Grassmann graphs ˆ J q (2 d + 1 , d ) q + 1 [ d + 2] d q Hamming graphs H ( d , e ) 1 1 d e H i ( d , 4), 1 ≤ i ≤ d / 2 Doob graphs ˆ 1 1 4 d Halved cubes 1 2 H ( n , 2) d 1 3 m + 1 Dual polar graphs B d ( q ) d q 1 q + 1 Dual polar graphs C d ( q ) d q 1 q + 1 Dual polar graphs D d ( q ) d q 1 2 Hemmeter graphs ˆ D d ( q ) d q 1 2 q 2 Halved dual polar graphs D n , n ( q ) d [3] q [ m + 1] q q 2 Ustimenko graphs ˆ D n , n ( q ) d [3] q [ m + 1] q q 2 + 1 Dual polar graphs 2 D d +1 ( q ) d q 1 q 3 + 1 Dual polar graphs 2 A 2 d ( q ) q 2 d 1 Dual polar graphs 2 A 2 d − 1 ( q ) q 2 d 1 q + 1 1+ q 2 1 − ( − q ) d +1 or d − q 1 − q 1 − q q e Bilinear forms graphs H q ( d , e ), e ≥ d d q q q 2 q 2 q m Alternating forms graphs Alt n ( q ) d q 2 q 2 q m Quadratic forms graphs Q n − 1 ( q ) d − ( − q ) d Hermitean forms graphs Her d ( q ) d − q − q Triality graphs 3 D 4 , 2 ( q ) 1 3 − q [3] q 1 − q q 4 q 4 q 9 Affine E 6 ( q ) graphs 3 q 4 Exceptional Lie graphs E 7 , 7 ( q ) 3 [5] q [10] q Gosset graph E 7 (1) 3 1 5 10 Witt graph M 23 3 − 2 − 1 6 Witt graph M 24 3 − 2 − 3 11 Coset graph of the extended ternary Golay code 3 − 2 − 2 9 q is a prime power; m = n = 2 d + 1 or m + 1 = n = 2 d

Introduction Classical parameters Classical DRGs Known families Tight DRGs Open cases Tight & classical 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 have either α = b − 1 or α = b [BCN89, Bro11]. ◮ We have proven nonexistence 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.

Introduction Tight distance-regular graphs Classical DRGs Local graphs Tight DRGs Equitable partitions Tight & classical Tight distance-regular graphs A. Juriˇ si´ c, J. H. Koolen and P. Terwilliger [JKT00] established the fundamental bound for distance-regular graphs: � k � � k � ka 1 b 1 θ 1 + θ d + ≥ − ( a 1 + 1) 2 . a 1 + 1 a 1 + 1 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].

Introduction Tight distance-regular graphs Classical DRGs Local graphs Tight DRGs Equitable partitions Tight & classical Local graphs of tight graphs Theorem [JKT00, BCN89]: For any vertex u of a tight distance-regular graph Γ, the local graph Γ( u ) is strongly regular with nontrivial eigenvalues b 1 b 1 τ = − 1 − and σ = − 1 − 1 + θ d 1 + θ 1 and multiplicities m τ = a 1 ( a 1 − σ )( σ + 1) m σ = a 1 ( a 1 − τ )( τ + 1) and ( a 1 + στ )( τ − σ ) . ( a 1 + στ )( σ − τ )

Introduction Tight distance-regular graphs Classical DRGs Local graphs Tight DRGs Equitable partitions Tight & classical 1-homogeneity A partition { C i } t i =1 of V Γ is equitable if there exist parameters n ij such that every vertex in C i has precisely n ij neighbours in C j . 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 any partition of the graph corresponding to the distances from two adjacent vertices is equitable with the same parameters. u p 1 . . . p 1 p 1 . . . p 1 b 1 2 , d i − 1 , i i , i + 1 − d 1 , d − d 1 − p 1 p 1 p 1 . . . p 1 . . . p 1 a 1 i − 1 , i + 1 , d 1 , − ii dd i − 1 i + 1 d 1 − p 1 . . . . . . p 1 p 1 p 1 b 1 d 1 , 1 i + 1 , i − 1 i , i − d , d v d 2 − −

Introduction Tight distance-regular graphs Classical DRGs Local graphs Tight DRGs Equitable partitions Tight & classical The CAB property A graph Γ has the CAB property [JK00] if for any two vertices u , v ∈ V Γ, the partition of the local graph Γ( u ) corresponding to the distances from v is equitable with parameters only depending on the distance ∂ ( u , v ). u · · · · · · v c h a h b h Γ( v ) Γ h − 1 ( v ) Γ h ( v ) Γ h +1 ( v )

Introduction Tight distance-regular graphs Classical DRGs Local graphs Tight DRGs Equitable partitions Tight & classical Characterization Theorem [JKT00, JK00]: Let Γ be a distance-regular graph with a 1 � = 0 and a d = 0. The following are equivalent: ◮ Γ is 1-homogeneous, ◮ Γ has the CAB property, ◮ Γ is tight.

Introduction Condition Classical DRGs Parameters of partitions Tight DRGs Feasible family Tight & classical Local graphs Tight distance-regular graphs with classical parameters Proposition : A distance regular graph with classical parameters ( d , b , α, β ) and d ≥ 3 is tight iff β = 1 + α [ d − 1] and b , α > 0 . All known examples have b = 1: ◮ halved cubes 1 2 H (2 d , 2), ( d , b , α, β ) = ( d , 1 , 1 , d ), ◮ Johnson graphs J (2 d , d ), ( d , b , α, β ) = ( d , 1 , 2 , 2 d − 1), and ◮ the Gosset graph E 7 (1), ( d , b , α, β ) = (3 , 1 , 4 , 9). These graphs are uniquely determined by their parameters.

Introduction Condition Classical DRGs Parameters of partitions Tight DRGs Feasible family Tight & classical Local graphs Parameters of partitions The parameters of the CAB and 1-homogeneous partitions of a tight distance-regular graphs with classical parameters can be computed explicitly: a 1 − γ h α h β h δ h c h a h b h γ h a 1 − α h − β h a 1 − δ h β h = b (1+ α b h [ d − h − 1]) , α h = 1+ α [ h − 1] , γ h = δ h − 1 = α ( b +1)[ h − 1] , a i − 1 − ρ i a d 2 − ρ d a 1 − ρ 2 a i − ρ i + 1 0 1 − − c i − b i + c d b d 1 c d u b 1 b 2 b i c i p 1 1 1 1 2 1 . . . p 1 p 1 . . . − − − b 1 2 , k d d a 1 i − 1 , i i , i + 1 − d 1 ρ 2 ρ i 1 1 ρ d 1 ρ i + − − ρ d 1 b d 1 σ i − τ i − σ i + τ i + σ d τ 1 σ i τ i − 1 p 1 1 1 p 1 1 1 p 1 − a 1 . . . p 1 . . . i − 1 , i + 1 , d 1 , − ii i − 1 i + 1 d 1 ρ d − b d 1 ρ i + 1 ρ 2 ρ i ρ d − 1 − 1 1 p 1 a 1 . . . p 1 p 1 . . . b 1 d 1 , k d i , i − 1 i + 1 , i − c i − c i c d 1 c d v b 1 1 b 2 b i b i + 2 b d 1 2 d 1 1 − − − − a 1 − ρ 2 1 − ρ i a i − ρ i + 0 a i − a d 2 − ρ d 1 1 − − ρ i = α b i − 2 ( b +1)[ i − 1] , σ i = [ i − 1](1+ α [ i − 1]) , τ i = b i +1 [ d − i − 1](1+ α b i [ d − i − 1]) .