Strongly regular graphs from classical groups

Transcription

1 Strongly regular graphs from classical groups Zhe-Xian Wan Chinese Academy of Sciences August 17-22, 2012 SCAC Z. Wan Strongly Regular Graphs 1/ 29

2 Outline Definition Symplectic graphs Orthogonal graphs of odd characteristic Orthogonal graphs of characteristic 2 Unitary graphs Subconstituents of strongly regular graphs from classical groups Z. Wan Strongly Regular Graphs 2/ 29

3 1 Definition Graph = set with a binary non-reflexive symmetric relation (called adjacency). a b, a b. G: graph, V : vertex set, V = n A arc set = {{a, b} a b} For x V, N(x) = {a V a x} G is regular if k (called its valency) s.t. N(x) = k for x V. G is strongly regular if 1 G is regular 2 λ, µ, s.t. N(x) N(y) = λ, N(x) N(y) = µ, x, y V with x y x, y V with x y Parameters of strongly regular graph: {n, k, λ, µ}. Z. Wan Strongly Regular Graphs 3/ 29

4 Distance, diameter, eigenvalues and chromatic number For x, y V, a path of length i joining x and y: a sequence z 0, z 1,, z i V s.t. x = z 0 z 1 z i = y. Distance d(x, y) of x, y = the length of shortest path from x to y in G Diameter d(g) of G = the maximal distance in G Eigenvalues of G = eigenvalues of the adjacency matrix of G Chromatic number χ(g) of G = the least κ s.t. V is partitioned into κ subsets and any two vertices of the same subset are not adjacent Z. Wan Strongly Regular Graphs 4/ 29

5 2 Symplectic graph n = 2v, F q F 2v q = {(a 1, a 2,, a 2v ) a i F q, 1 i 2v}. ( ) 0 I (v) K = I (v) 0 Symplectic group Sp 2v (F q ) = {T GL 2v (F q ) TK t T = K} Symplectic graph Sp(2v, q) α = (a 1, a 2,, a 2v ) F 2v q \{0} [α] = {λα λ F q }, 1-dim subspace generated by α [α] = [kα] for k F q V = {[α] α F 2v q \{0}} α = (a 1, a 2,, a 2v ), β = (β 1, β 2,, β 2v ) F 2v q \ { 0 } [α] [β] αk t β 0 Z. Wan Strongly Regular Graphs 5/ 29

6 Sp(2v, q) Theorem 1. Sp(2v, q) is a strongly regular graph with parameters { q2v 1 q 1, q2v 1, q 2v 2 (q 1), q 2v 2 (q 1) } and eigenvalues q 2v 1, q v 1 and q v 1. Moreover, the chromatic number χ(sp(2v, q)) = q v + 1 and diameter d(sp(2v, q)) = 2. Proof of d = 2: [α], [β] V, [α] [β] or [α] [β]. [α] [β] d([α], [β]) = 1. [α] [β], i.e., αk t β = 0 v 2. T Sp 2v (F q ) s.t. αt = e 1, βt = e 2. e i K t (e v+1 + e v+2 ) = 1, i = 1, 2, d([e 1 ], [e 2 ]) = 2. d([α], [β]) = 2. d = 2. Z. Wan Strongly Regular Graphs 6/ 29

7 Automorphisms of symplectic graph G: graph with vertex set V and adjacent relation. Let τ : V V a τ(a) be bijection satisfying a b τ(a) τ(b). Then τ is called an automorphism of G. All automorphisms of G from a group, denoted by Aut(G). Z. Wan Strongly Regular Graphs 7/ 29

8 Automorphisms of symplectic graph (cont.) Symplectic group Sp(2v, q) = { T GL 2v (q) TK t T = K }. Theorem 2. Sp(2v, q), T Sp 2v (F q ). Let σ T : V V [α] [αt ] Then 1 σ T Aut(Sp(2v, q)) 2 σ T = σ T for T, T Sp 2v (F q ) T = kt for some k F q Proof (1): [α] [β] αk t β = 0 α(tk t T ) t β = 0 (αt )K t (βt ) = 0 [αt ] [βt ] σ T ([α]) σ T ([β]). Z. Wan Strongly Regular Graphs 8/ 29

9 The case q = 2 Theorem 3. Aut(Sp(2v, F 2 )) = Sp 2v (F 2 ). Proof: By the above example, T σ T is an isomorphism from Sp 2v (F 2 ) into Aut(Sp(2v, F 2 )). Remains to show: any τ Aut(Sp(2v, F 2 )) is of the form σ T where T Sp 2v (F 2 ). α V (Sp(2v, F 2 )) [α] = { α }. τ([α]) contains a unique nonzero vector, which is defined to be τ(α). Define τ(0) = 0. Then τ is a bijective map F 2v 2 F2v 2 with τ(0) = 0. For α, β 1, β 2 F 2v 2 αk t β 1 = τ(α)k t τ(β 1 ) αk t β 2 = τ(α)k t τ(β 2 ) αk t (β 1 + β 2 ) = τ(α)k t (τ(β 1 ) + τ(β 2 )) Z. Wan Strongly Regular Graphs 9/ 29

10 Proof of Theorem 3(cont.) But αk t (β 1 + β 2 ) = τ(α)k t (τ(β 1 + β 2 )) τ(β 1 + β 2 ) = τ(β 1 ) + τ(β 2 ), i.e., τ is linear with τ(0) = 0. τ is nonsingular. τ(1, 0,, 0) τ(0, 1,, 0) Let T =. Then τ(α) = αt for α F2v2.. τ(0, 0,, 1) α, β F 2v 2, α 0, β 0. αk t β = 0 αtk t T t β = 0 The system of linear equations αkx = 0 α(tk t T )x = 0 have the same solutions. TK t T = kk, k 0, k must be 1. T Sp 2v (F 2 ). Z. Wan Strongly Regular Graphs 10/ 29

11 Automorphism of Sp(2v, q) Theorem 4. T Sp 2v (F q ) induces a bijection σ T : V (Sp(2v, q)) V (Sp(2v, q)) [α] [αt ] Then 1 σ T is an automorphism of Sp(2v, q) 2 σ T = σ T for T, T Sp 2v (F q ) σ T = kσ T for some k F q PSp 2v (F q ) = Sp 2v (F q )/ { ±I 2v } can be regarded as a subgroup of Aut(Sp(2v, q)). Z. Wan Strongly Regular Graphs 11/ 29

12 On Aut(Sp(2v, q)) Theorem 5. Let e i = ( i }{{} ), f i = (0 } {{ 0 } ), and let E be the v+i v v subgroup of Aut(Sp(2v, q)) defined as follows Then E = { σ Aut(Sp(2v, q)) : σ([e i ]) = e i, σ([f i ]) = f i, i = 1, 2,, n }. 1 Aut(Sp(2v, q)) = PSp 2v (F q) E 2 If v = 1, E = S q 1 3 If v > 1, E = (F q F q ) ϕ Aut(F q), where ϕ is the natural action of }{{} v Aut(F q) on F q F q defined by }{{} v ϕ(π)((k 1,, k v )) = (π(k 1),, π(k v )), π Aut(F q), k 1,, k v F q. Z. Wan Strongly Regular Graphs 12/ 29

13 3 Orthogonal graphs of odd characteristic F q, q odd. n 2, F n q = { (a 1,, a n ) : a i F q, 1 i n } α F n q \ { 0 }, [α] = { kα : k F q } S n n nonsingular symmetric matrix over F q [α] is isotropic (with respect to S) if αs t α = 0 non-isotropic αs t α 0 Orthogonal graph O(n, q)(w, r, t, S) with vertex set V (O(n, q)) = set of 1-dim isotropic subspaces [α], [β] V (O(n, q)), [α] [β] αs t β 0 Z. Wan Strongly Regular Graphs 13/ 29

14 On O(2v + δ, q) S is cogredient to one of the following normal forms 0 I (v) 0 I (v) I (v) 0 or I (v) 0 1 z when n = 2v + 1 is odd, z is a fixed element of F q \(F q ) 2 ( 0 I (v) ) or 0 I (v) I (v) 0 I (v) 0 1 z when n = 2v or 2v + 2 respectively. We use S 2v+δ, to cover the cases, where δ = 0, 1 or 2, ( ) 1 =, (1) or z Correspondingly we have O(2v + δ, q) Z. Wan Strongly Regular Graphs 14/ 29

15 On O(2v + δ, q) (cont.) Theorem 6. 1 When v = 1, O(2 1 + δ, q) is complete with q δ + 1 vertices. 2 When v 2, O(2v + δ, q) is strongly regular with parameters where {(q v 1)(q v+δ 1 + 1)/(q 1), q 2v+δ 2, λ, µ}, λ = q 2v+δ 2 q 2v+δ 3 q v 1 + q v+δ 2, µ = q 2v+δ 2 q 2v+δ 3, and eigenvalues q 2v+δ 2, q v+δ 2 and q v 1, χ(o(2v + δ, q)) = q v+δ Z. Wan Strongly Regular Graphs 15/ 29

16 Automorphisms of O(2v + δ, q) Orthogonal group O 2v+δ (F q ) = { T GL 2v+δ (F q ) : TS 2v+δ, t T = S 2v+δ, } Projective orthogonal group PO 2v+δ (F q ) = O 2v+δ (F q )/ { ±I } Theorem 7. Let T O 2v+δ (F q ) and σ T : V (O(2v + δ, q)) V (O(2v + δ, q)) [α] [αt ] Then 1 σ T Aut(O(2v + δ, q)) 2 T 1, T 2 O 2v+δ (F q ), σ T1 = σ T2 T 1 = ±T 2 PO 2v+δ (F q ) can be regarded as a subgroup of Aut(O(2v + δ, q)). Z. Wan Strongly Regular Graphs 16/ 29

17 The case n = 2v Theorem 7. Let v 2 and E be the subgroup of Aut(O(2v, q)) defined as follows: E = { σ Aut(O(2v, q)) : σ([e i ]) = e i, σ([f i ]) = f i, 1 i v }, where e i = ( i }{{} ), f i = (0 } {{ 0 } ). v+i v v Then Aut(O(2v, q)) = PO 2v (F q) E. Moreover, 1 If v = 2, E = Sym(F q) Sym(F q) 2 If v 3, then E = (F q F q ) ϕ Aut(F q) and the isomorphism from }{{} v (F q F q ) ϕ Aut(F q) to E is defined as }{{} v h : (k 1,, k v, π) σ (k1,,k v,π), where σ (k1,,k v,π)([(a 1, a 2, a 2v )]) = [π(a 1 ), k 2 π(a 2 ),, k v π(a v ), k 1 π(a v+1 ), k 1 k 1 2 π(a v+2 ),, k 1 kv 1 π(a 2v )] } n = 2v + 1 details omitted. n = 2v + 2 Z. Wan Strongly Regular Graphs 17/ 29

18 4 Orthogonal graphs of characteristic 2 O(2v + δ, q), q even (Definition omitted) O(2v + 1, q) = S p (2v, q) O(2v + δ, q) (δ = 0 or 2) is strongly regular with parameters where { (q v 1)(q v+δ 1 + 1)/(q 1), q 2v+δ 2, λ, µ }, λ = q 2v+δ 2 q 2v+δ 3 q v 1 + q v+δ 2, µ = q 2v+δ 2 q 2v+δ 3, and eigenvalues q 2v+δ 2, q v+δ 2 and q v 1, χ(o(2v + δ, q)) = q v+δ 1 + 1, except when δ = 0 and v is odd. Aut(O(2v + δ, q)) is determined. (details omitted) Z. Wan Strongly Regular Graphs 18/ 29

19 5 Unitary graphs F q 2 any characteristic : F q 2 F q 2 a a = a q involutive automorphism of F q 2 with fixed field F q n 2 F n q = { (a 2 1,, a n ) : a i F q 2, 1 i n } α F n q, α 0, [α] = { kα : k F 2 q } 1-dim subspace generated 2 by α. k F q, [α] = [kα]. 2 Z. Wan Strongly Regular Graphs 19/ 29

20 Unitary graph U(n, q 2 ) H: n n/f q 2 t H = H, H is called Hermitian H is assumed to be nonsingular Hermitian α F n q 2, α 0, [α] is isotropic (w.r.t. H), if αh t α = 0, otherwise, non-isotropic, if αh t α 0 U(n, q 2 ) unitary graph (w.r.t. H) with vertex set V (U(n, q 2 )) = set of 1-dim isotropic subspaces adjacency: [α] [β] αh t β = 0 Z. Wan Strongly Regular Graphs 20/ 29

21 On U(n, q 2 ) Theorem 9. 1 n = 2 or 3, U(2, q 2 ) and U(3, q 2 ) are complete graphs with q + 1 and q vertices, respectively. 2 n 4, U(n, q 2 ) is a strongly regular graph with parameters { (qn ( 1) n )(q n 1 ( 1) n 1 ), q 2n 3, λ, µ }, q 2 1 where λ = q 2n 3 q 2n 5 + ( 1) n 3 q n 2 + ( 1) n 2 q n 3, µ = q 2n 3 q 2n 5 and eigenvalues q 2n 3, ( 1) n q n 3 and ( 1) n 1 q n 2 when n is even, q 2n 3, ( 1) n 1 q n 2 and ( 1) n q n 3 when n is odd. χ(u(n, q 2 )) remains to be determined. Z. Wan Strongly Regular Graphs 21/ 29

22 Automorphisms of U(n, q 2 ) Unitary group U n (F q 2) = { T GL n (F q 2) : TH t T = H } S = { a F q 2 : aa = 1 } Z n = { ai n : a S } PU n (F q 2) = U n (F q 2)/Z n projective unitary group Theorem 10. Let T U n (F q 2) and σ T : V (U n (F q 2)) V (U n (F q 2)) [α] [αt ] Then 1 σ T Aut(U(n, q 2 )). 2 T 1, T 2 U n (F q 2), σ T1 = σ T2 T 1 = kt 2, k S. Thus PU n (F q 2) can be regarded as a subgroup of Aut(U(n, q 2 )). Z. Wan Strongly Regular Graphs 22/ 29

23 On Aut(U(n, q 2 )) Theorem 11. Let E be the subgroup of Aut(U(n, q 2 )) defined as where Then E = { σ Aut(U(n, q 2 )) : σ([e i ]) = e i, σ([f i ]) = f i, 1 i v }, e i = ( i }{{} ), f i = (0 } {{ 0 } ). v+i n v v 1 Aut(U(n, q 2 )) = PU n (F q 2) E 2 If n = 2, E = S q 1 3 If n = 3, E = S q If n 6, (details omitted) The cases n = 4 and 5 remain to be determined. Z. Wan Strongly Regular Graphs 23/ 29

24 6 Subconstituents of strongly regular graphs from classical groups Take U(n, q 2 ) as an example. Assume v 2. d(u(n, q 2 )) = 2. For [α] V (U(n, q 2 )), Γ i ([α]) = the induced subgroup of U(n, q 2 ) with vertices at distance i from [α], i = 1, 2. It suffices to consider Γ i (e 1 )(= Γ i ), e 1 = (1, 0,, 0). Z. Wan Strongly Regular Graphs 24/ 29

25 On Γ 1 Theorem 12. Γ 1 is quasi-strongly regular with parameters {q 2n 3, k, b 1, b 2, b 3; q 2n 3 2q 2n 5 + q 2n 7 + ( 1) n 1 q n 2 + ( 1) n q n 3 }, where k = q 2n 3 q 2n 5 + ( 1) n 1 q n 2 + ( 1) n q n 3, b 1 = q 2n 3 2(q 2n 5 ( 1) n 1 q n 2 ( 1) n q n 3 ), b 2 = q 2n 3 + q 2n 7 2(q 2n 5 ( 1) n q n 2 ( 1) n q n 3 ), b 3 = q 2n 3 + q 2n 7 2(q 2n 5 ( 1) n 1 q n 2 ( 1) n q n 3 ) ( 1) n 1 q n 4. Z. Wan Strongly Regular Graphs 25/ 29

26 On Γ 2 Theorem 13. When n = 4 or 5, i.e., v = 2, Γ 2 is strongly regular with parameters {ν, q 2n 5, q 2n 5 q 2n 7 + ( 1) n 1 q n 2 + ( 1) n q n 3, q 2n 5 }, and when n 6, i.e., v 3, Γ 2 is quasi-strongly regular with parameters {ν, q 2n 5, q 2n 5 q 2n 7 + ( 1) n 1 q n 2 + ( 1) n q n 3 ; q 2n 5 q 2n 7, q 2n 5 }, where ν = (q 2n 3 ( 1) n 1 q n ( 1) n q n 1 q 2 )(q 2 1) 1. Z. Wan Strongly Regular Graphs 26/ 29

27 Reference [1] X.L. Hubaut, Strongly regular graphs, Discrete math. 13(1975), pp [2] J.J. Seidel, Strongly regular graphs, in: Surveys in Combinatories, Proc. 7th Brit. Comb. conf., B.bllobás(ed), London Math. Soc. LNS 38, Cambridge, 1979, pp [3] A.E. Brouwer, A.M. Cohn, A. Neumaier, Distance Regular Graphs, Springer-Verlag, Berlin, Heidelberg, [4] W. Golightly, W. Haynworth and D.G. Sarvate, A family of connected quasi-strongly regular graphs, Congress Numerantium, 124(1997), pp [5] C. Godsil, G. Royle, Algebraic Graph Theory, Graduate Texts in Mathematics, Vol.207, Springer-Verlage, [6] Z. Wan, Geometry of Classical Groups over Finite Fields, 2nd edition, Scinece Press, Beijing/New York, [7] Z. Tang, Z. Wan, Symplectic graphs and their automorphisms, European J. Combin. 27(2006), pp [8] Z. Gu, Z. Wan, Orthogonal graphs of odd characteristic and their automorphisms, Finite Fields and Their Applications, 14(2008), pp Z. Wan Strongly Regular Graphs 27/ 29

28 Reference (cont.) [9] F. Li, Y. Wang, Subconstituents of symplectic graphs, European J. Combin. 29(2008), pp [10] Z. Wan and K. Zhou, Unitary graphs and their automorphisms, Annals of Combinatorics: Volume 14, Issue 3(2010), pp [11] Z. Gu, Z. Wan, Subconstituents of orthogonal graphs of odd characteristic, Linear Algebra and Its Applications, 434(2011), pp [12] Z. Gu, Z. Wan, Automorphisms of subconstituents of symplectic graphs, accepted for publications in Algebra Colloquium. [13] Z. Gu, Z. Wan, K. Zhou, Subconstituents of unitary groups over finite feilds, in preparation. [14] Z. Gu, Z. Wan, K. Zhou, Automorphisms of subconstitutents of unitary groups over finite fields, in preparation. [15] Z. Gu, Z. Wan, K. Zhou, Subconstituents of orthogonal graphs of odd characteristic, continued, in preparation. Z. Wan Strongly Regular Graphs 28/ 29

29 Thanks!!! Z. Wan Strongly Regular Graphs 29/ 29