Lifting Constructions of Strongly Regular Cayley Graphs

Transcription

1 Lftng Constructons of Strongly Rgular Cayly Graphs Koj Momhara Qng Xang Abstract W gv two lftng constructons of strongly rgular Cayly graphs. In th frst constructon w lft a cyclotomc strongly rgular graph by usng a subdffrnc st of th Sngr dffrnc st. Th scond constructon uss quadratc forms ovr fnt flds and t s a common gnralzaton of th constructon of th affn polar graphs [7] and a constructon of strongly rgular Cayly graphs gvn n [15]. Th two constructons ar rlatd n th followng way: Th scond constructon can b vwd as a rcursv constructon, and th strongly rgular Cayly graphs obtand from th frst constructon can srv as startrs for th scond constructon. W also obtan assocaton schms from th scond constructon. Kywords: Cyclotomc strongly rgular graph, Gauss sum, quadratc form, strongly rgular graph. 1 Introducton In ths papr, w assum that th radr s famlar wth th thory of strongly rgular graphs and dffrnc sts. For th thory of strongly rgular graphs, our man rfrncs ar [5] and [18]. For th thory of dffrnc sts, w rfr th radr to Chaptr 6 of [4]. Strongly rgular graphs (srgs) ar closly rlatd to othr combnatoral objcts, such as two-wght cods, two-ntrscton sts n fnt gomtry, and partal dffrnc sts. For ths connctons, w rfr th radr to [5, p. 132] and [7, 22]. Lt Γ b a smpl and undrctd graph and A b ts adjacncy matrx. A vry usful way to chck whthr Γ s strongly rgular s by usng th gnvalus of A (whch ar usually calld gnvalus of Γ). For convnnc, w wll call an gnvalu of Γ rstrctd f t has an gnvctor prpndcular to th all-ons vctor 1. Not that for a k-rgular connctd graph, th rstrctd gnvalus ar smply th gnvalus dffrnt from k. Thorm 1.1. For a smpl k-rgular graph Γ of ordr v, not complt or dglss, wth adjacncy matrx A, th followng ar quvalnt: 1. Γ s strongly rgular wth paramtrs (v, k, λ, µ) for crtan ntgrs λ, µ, 2. A 2 = (λ µ)a + (k µ)i + µj for crtan ral numbrs λ, µ, whr I, J ar th dntty matrx and th all-ons matrx, rspctvly, 3. A has prcsly two dstnct rstrctd gnvalus. On of th most ffctv mthods for constructng srgs s by th Cayly graph constructon. For xampl, th Paly graph P(q) s a class of wll-known Cayly graphs on th fnt fld F q ; that Dpartmnt of Mathmatcs, Faculty of Educaton, Kumamoto Unvrsty, Kurokam, Kumamoto , Japan; Emal: momhara@duc.kumamoto-u.ac.jp Dpartmnt of Mathmatcal Scnc, Unvrsty of Dlawar, Nwark,DE 19716, USA; Emal: xang@math.udl.du 1

2 s, th vrtcs of P(q) ar th lmnts of F q, and two vrtcs ar adjacnt f and only f thr dffrnc s a nonzro squar. Th paramtrs of P(q) ar (v, k, λ, µ) = (4t + 1, 2t, t 1, t), whr q = 4t + 1 s a prm powr. Mor gnrally, lt G b an addtvly wrttn group of ordr v, and lt D b a subst of G such that 0 D and D = D, whr D = { d d D}. Th Cayly graph on G wth conncton st D, dnotd by Cay(G, D), s th graph wth th lmnts of G as vrtcs; two vrtcs ar adjacnt f and only f thr dffrnc blongs to D. In th cas whr Cay(G, D) s strongly rgular, th conncton st D s calld a (rgular) partal dffrnc st. Th survy of Ma [22] contans much of what s known about partal dffrnc sts and about connctons wth strongly rgular Cayly graphs. A classcal mthod for constructng strongly rgular Cayly graphs on th addtv groups of fnt flds s to us cyclotomc classs of fnt flds. Lt p b a prm, f a postv ntgr, and lt q = p f. Lt > 1 b an ntgr such that (q 1), and γ b a prmtv lmnt of F q. Thn th costs C (,q) = γ γ, 0 1, ar calld th cyclotomc classs of ordr of F q. Many authors hav studd th problm of dtrmnng whn a unon D of cyclotomc classs forms a partal dffrnc st. W call Cay(F q, D) a cyclotomc strongly rgular graph f D s a sngl cyclotomc class of F q and Cay(F q, D) s strongly rgular. Extnsv work has bn don on cyclotomc srgs, s [2, 6, 14, 16, 17, 19, 21, 23, 25, 26, 27]. (Som of ths authors usd th languag of cyclc cods n thr nvstgatons nstad of strongly rgular Cayly graphs or partal dffrnc sts. W choos to us th languag of srgs hr.) Th Paly graphs ar prmary xampls of cyclotomc srgs. Also, f D s th multplcatv group of a subfld of F q, thn t s clar that Cay(F q, D) s strongly rgular. Ths cyclotomc srgs ar usually calld subfld xampls. Nxt, f thr xsts a postv ntgr t such that p t 1 (mod ), thn Cay(F q, D) s strongly rgular. S [2]. Ths xampls ar usually calld sm-prmtv. Schmdt and Wht mad th followng conjctur on cyclotomc srgs. Conjctur 1.2. ([26]) Lt F p f b th fnt fld of ordr p f, pf 1 p 1 wth > 1, and C 0 = C (,pf ) 0 wth C 0 = C 0. If Cay(F p f, C 0 ) s strongly rgular, thn on of th followng holds: (1) (subfld cas) C 0 = F p d whr d f, (2) (sm-prmtv cas) 1 p (Z/Z), (3) (xcptonal cas) Cay(F p f, C 0 ) has on of th lvn sts of paramtrs gvn n Tabl 1. Tabl 1: Elvn sporadc xampls No. p f [(Z/Z) : p ] A strongly rgular graph s sad to b of Latn squar typ (rspctvly, ngatv Latn squar typ) f (v, k, λ, µ) = (n 2, r(n ϵ), ϵn+r 2 3ϵr, r 2 ϵr) and ϵ = 1 (rspctvly, ϵ = 1). Typcal xampls 2

3 of srgs of Latn squar typ or ngatv Latn squar typ com from nonsngular quadrcs n th projctv spac PG(m 1, q), whr m s vn. It sms that w know mor xampls of srgs of Latn squar typ than srgs of ngatv Latn squar typ, s [10]. Our frst man rsult n ths papr s a constructon of ngatv Latn squar typ strongly rgular Cayly graphs Cay(F q 2, D) by lftng a cyclotomc strongly rgular graph on F q. Th proof rls on th Davnport-Hass lftng formula on Gauss sums. In our scond man thorm, w wll gv a rcursv constructon of strongly rgular Cayly graphs by usng quadratc forms ovr fnt flds undr th assumpton that crtan strongly rgular Cayly graphs xst on a ground fld. Ths constructon gnralzs th followng two constructons. Thorm 1.3. ([7]) Lt Q : V = F n q F q b a nonsngular quadratc form, whr n s vn and q s an odd prm powr, and lt D = {x F n q Q(x) s a nonzro squar n F q }. Thn, Cay(V, D) s a strongly rgular graph (whch s th so-calld affn polar graph). Fng t.al [15] gav th followng constructon usng unform cyclotomy. Thorm 1.4. ([15]) Lt p b a prm, > 2, q = p 2jr, whr r 1, (p j + 1), and j s th smallst such postv ntgr. Lt Q : V = F n q F q b a nonsngular quadratc form, whr n s vn, and lt D (,q) C = {x F n q Q(x) C (,q) } for 0 1. Thn, Cay(V, D (,q) C ) s strongly rgular for all 0 1. Th strongly rgular Cayly graphs obtand n Scton 3 can b usd as startrs for th scond constructon. In ths way, w obtan a fw nfnt famls of strongly rgualr Cayly graphs wth Latn squar typ or ngatv Latn squar typ paramtrs. Furthrmor, w dscuss assocaton schms rlatd to th scond constructon and obtan svral nw assocaton schms. 2 Background on Gauss sums and strongly rgular Cayly graphs Lt p b a prm, f a postv ntgr, and q = p f. Th canoncal addtv charactr ψ of F q s dfnd by ψ : F q C, ψ(x) = ζ Tr q/p(x) p, whr ζ p = xp( 2π p ) s a complx prmtv p-th root of unty and Tr q/p s th trac from F q to F p. All complx charactrs of (F q, +) ar gvn by ψ a, whr a F q. Hr ψ a s dfnd by ψ a (x) = ψ(ax), x F q. (2.1) For a multplcatv charactr χ of ordr of F q, w dfn th Gauss sum G f (χ ) = x F q χ (x)ψ(x). From th dfnton w s clarly that G f (χ ) Z[ζ p ], th rng of algbrac ntgrs n th cyclotomc fld Q(ζ p ). Lt σ a,b b th automorphsm of Q(ζ p ) dfnd by σ a,b (ζ ) = ζ a, σ a,b (ζ p ) = ζ b p, whr gcd (a, ) = gcd (b, p) = 1. Blow w lst svral basc proprts of Gauss sums [3]: () G f (χ )G f (χ ) = q f χ s nontrval; 3

4 () G f (χ p ) = G f (χ ), whr p s th charactrstc of F q ; () G f (χ 1 ) = χ ( 1)G f (χ ); (v) G f (χ ) = 1 f χ s trval; (v) σ a,b (G f (χ )) = χ a (b)g f (χ a ). In gnral, xplct valuatons of Gauss sums ar vry dffcult. Thr ar only a fw cass whr th Gauss sums hav bn valuatd. Th most wll-known cas s th quadratc cas,.., th ordr of χ s two. Th nxt smpl cas s th so-calld sm-prmtv cas (also known as unform cyclotomy or pur Gauss sum), whr thr xsts an ntgr j such that p j 1 (mod ), whr s th ordr of th multplcatv charactr nvolvd. Th xplct valuatons of Gauss sums n ths cass ar gvn n [3]. Th nxt ntrstng cas s th ndx 2 cas whr th subgroup p gnratd by p (Z/Z) s of ndx 2 n (Z/Z) and 1 p. In ths cas, t s known that can hav at most two odd prm dvsors. Many authors hav nvstgatd ths cas, s [29] for a complt soluton to th problm of valuatng ndx 2 Gauss sums. Rcntly, ths ndx 2 Gauss sums wr usd n th constructon of nw nfnt famls of strongly rgular graphs. S [14, 16]. Now w rcall th followng wll-known lmma n algbrac graph thory (s.g., [5]). Lmma 2.1. Lt (G, +) b an ablan group and D a subst of G such that 0 D and D = D. Thn, th rstrctd gnvalus of Cay(G, D) ar gvn by ψ(d), ψ Ĝ \ {ψ 0}, whr Ĝ s th charactr group of G and ψ 0 s th trval charactr. Lt q b a prm powr and lt C (,q) = γ γ, 0 1, b th cyclotomc classs of ordr of F q, whr γ s a fxd prmtv root of F q. In ordr to chck whthr a canddat subst D = I C(,q) s a conncton st of a strongly rgular Cayly graph, by Thorm 1.1 and Lmma 2.1, t s nough to show that ψ(ad) = x D ψ(ax), a F q, tak xactly two valus, whr ψ s th canoncal addtv charactr of F q. Not that th sum ψ(ad) can b xprssd as a lnar combnaton of Gauss sums (cf. [16]) by usng th orthogonalty of charactrs: ψ(ad) = 1 G f (χ 1 ) χ(aγ ), (2.2) I χ C 0 s th subgroup of F q consstng of all χ whch ar trval on C (,q) 0. Thus, th computa- whr C0 tons ndd to show whthr a canddat subst D = I C(,q) s a conncton st of a strongly rgular Cayly graph ar ssntally rducd to valuatng Gauss sums. Howvr, as prvously sad, valuatng Gauss sums xplctly s vry dffcult. In Scton 3 of ths papr, w wll gv a contructon of strongly rgular graphs by lftng a cyclotomc srg. To prov that our constructon ndd gvs rs to srgs, w do not valuat th Gauss sums nvolvd; nstad, w us th Davnport-Hass lftng formula statd blow. Thorm 2.2. ([3]) Lt χ b a nontrval multplcatv charactr of F q = F p f and lt χ b th lft of χ to F q = F p fs,.., χ (α) = χ(norm q /q(α)) for α F q, whr s 2 s an ntgr. Thn G fs (χ ) = ( 1) s 1 (G f (χ)) s. 3 Lftng cyclotomc strongly rgular graphs va subdffrnc sts In ths scton, w gv a constructon of strongly rgular Cayly graphs by lftng a cyclotomc strongly rgular graph va a subdffrnc st of th Sngr dffrnc sts. W start by rvwng a constructon of th Sngr dffrnc sts. 4

5 Lt p b a prm, f 1, m 3 b ntgrs and q = p f. Lt L b a complt systm of cost rprsntatvs of F q n F q m. W may assum that L s chosn n such a way that Tr q m /q(x) = 0 or 1 for any x L. Lt Thn, s a Sngr dffrnc st. L 0 = {x L Tr qm /q(x) = 0} and L 1 = {x L Tr qm /q(x) = 1}. H 0 = {x F q m/f q x L 0 } (3.1) Not that any nontrval multplcatv charactr χ of xponnt (q m 1)/(q 1) of F q nducs a m charactr of th quotnt group F q m/f q, whch wll b dnotd by χ also. Morovr, vry charactr of F q m/f q arss n ths way. By a rsult of Yamamoto [28], for any nontrval multplcatv charactr χ of xponnt (q m 1)/(q 1) of F qm, w hav χ(h 0 ) = G fm (χ)/q. Now, lt F q C 0 (:= C (,qm ) 0 ) F q m b a subgroup such that [F q m : C 0] =. Thn C 0 = C 0 /F q F q m/f q. Lt S b a complt systm of cost rprsntatvs of C 0 n F q m/f q, and G = {s s S} F q m/c 0. Thn, by assumpton, [F q m : C 0] dvds (q m 1)/(q 1),.., = G (q m 1)/(q 1). From now on, w assum that Cay(F q m, C 0 ) s strongly rgular. Thn H 0 sc 0, s S, tak xactly two valus. (S [7] or [26].) It follows that H 0 sc 0 H 0 C 0 = 0 or δ, whr δ s a nonzro ntgr. For any nontrval multplcatv charactr χ of F q m of xponnt, w hav χ(h 0 ) = s S H 0 sc 0 χ(s) = s S( H 0 sc 0 H 0 C 0 )χ(s) = δ s S χ(s), whr Thus S = {s S : H 0 sc 0 H 0 C 0 = δ}. (3.2) χ(s) = χ(h 0) δ s S = G fm(χ). (3.3) δq It follows that δ s a powr of p, and S := {s s S } G s a (, S, λ )-dffrnc st, whch s usually calld a subdffrnc st of H 0. S Scton 6 of [26]. Th trm subdffrnc st was frst ntroducd by McFarland [24]. Lt γ b a prmtv lmnt of F q 2m and lt ω = Norm q 2m /q m(γ) = +1, whch s a prmtv γqm lmnt of th subfld F q m of F q 2m. Lt C (,q2m ) j = γ j γ and C (,qm ) j = ω j ω = ω j C 0. Thorm 3.1. Assum that F q C 0 F q m b a subgroup such that [F q m : C 0] =, C 0 = C 0, and Cay(F q m, C 0 ) s strongly rgular. Lt I = {0 1 ω S }, whr S s dfnd n (3.2) and ω stands for ωf q. Lt Thn Cay(F q 2m, D) s also strongly rgular. D = I C (,q2m ). 5

6 Proof: Lt ψ 1 b th canoncal addtv charactr of F q 2m and lt χ b a multplcatv charactr of ordr of F q 2m. Th rstrctd gnvalus of Cay(F q 2m, D) ar ψ 1 (γ a D), 0 a 1. By (2.2), n ordr to show that Cay(F q 2m, D) s strongly rgular, w comput th sums 1 T a = ψ 1 (γ a D) + I = G 2fm (χ x ) χ x (γ a+ ), x=1 I whr a = 0, 1,..., 1. Snc (q m 1), χ must b th lft of a charactr, say χ, of F q m. By th Davnport-Hass lftng formula, w hav 1 T a = x=1 By th dfnton of I, w hav Hnc I χ x (ω a )G fm (χ x )G fm (χ x ) χ x (ω ) I χ x (ω ) = χ x (s) = G fm(χ x ). δq s S T a = 1 1 χ x δq (ω a )G fm (χ x )G fm (χ x )G fm (χ x ) x=1 = qm 1 δ 1 x=1 χ x (ω a )G fm (χ x ), (3.4) whr n th last stp w usd th fact that G fm (χ x )G fm (χ x ) = χ x ( 1)q m. By th assumpton that Cay(F q m, C (,qm ) 0 ) s strongly rgular, w hav 1 x=1 χx (ω a )G fm (χ x ), a = 0, 1,..., 1, tak xactly two valus. W conclud that T a, 0 a 1, tak xactly two valus too. Thrfor Cay(F q 2m, D) s also strongly rgular. Not that th st D has sz D = (qm 1) I (q m + 1). By applyng Thorm 3.1 to th known cyclotomc srgs n th statmnt of Conjctur 1.2, w obtan a lot of strongly rgular Cayly graphs. W frst apply Thorm 3.1 to th sm-prmtv xampls. In ths cas, w hav I = S = 1 by [11, p. 23]. Corollary 3.2. Lt p b a prm, 2, q m = p 2jr, whr m = 2jr, r 2, (p j + 1), and j s th smallst such postv ntgr. Thn thr xsts an (n 2, r(n + 1), n + r 2 + 3r, r 2 + r) strongly rgular Cayly graph wth n = q m and r = (q m 1)/. Th proof s straghtforward. W omt t. Nxt w apply Thorm 3.1 to th subfld xampls. Corollary 3.3. Lt q b a prm powr, m 3 a postv ntgr and a any postv dvsor of m. Thn thr xsts an (n 2, r(n + 1), n + r 2 + 3r, r 2 + r) strongly rgular Cayly graph wth n = q m and r = q m a 1. Proof: W apply Thorm 3.1 to th subfld xampls of cyclotomc srgs. W us th sam notaton as n th statmnt and proof of Thorm 3.1. Thn, by [11, p. 23], w hav C 0 = F q a, = qm 1 q a 1, I = S = qm a 1 q a 1, δ = qa 1 and th rstrctd gnvalus of Cay(F q m, C 0 ) ar 1 and q a 1. Th corollary now follows by straghtforward computatons usng (3.4). Rmark 3.4. In th cas whr q = 2 and a = 1, th paramtrs of th strongly rgular Cayly graphs obtand n Corollary 3.3 ar (2 2m, (2 m + 1)(2 m 1 1), 2 m 1 (2 m 1 1) 2, 2 m 1 (2 m 1 1)). 6

7 Thn, th st D {0} clarly forms a dffrnc st wth paramtrs (2 2m, 2 m 1 (2 m 1), 2 m 1 (2 m 1 1)), whch s a Hadamard dffrnc st n th lmntarty ablan 2-group of ordr 2 2m. Ths dffrnc st was frst dscovrd n [12, p. 105]. Th corrspondng bnt functon s a monomal quadratc bnt functon. Fnally, w apply Thorm 3.1 to th lvn sporadc xampls of cyclotomc srgs. In ths cas, th valus of k := S ar gvn n [26, Tabl II]. Corollary 3.5. Thr xsts a (q 2, r(q+1), λ, µ) ngatv Latn squar typ strongly rgular Cayly graph, whr r = k(q 1)/, n ach of th followng cass: (q,, k) = (3 5, 11, 5), (5 9, 19, 9), (3 12, 35, 17), (7 9, 37, 9), (11 7, 43, 21), (17 33, 67, 33) (3 53, 107, 53), (5 18, 133, 33), (41 81, 163, 81), (3 144, 323, 161), (5 249, 499, 249). 4 Strongly rgular Cayly graphs from quadratc forms Lt V b an n-dmnsonal vctor spac ovr F q. A functon Q : V F q s calld a quadratc form f () Q(αv) = α 2 Q(v) for all α F q and v V, () th functon B : V V F q dfnd by B(u, v) = Q(u + v) Q(u) Q(v) s blnar. W say that Q s nonsngular f th subspac W of V wth th proprty that Q vanshs on W and B(v, w) = 0 for all v V and w W s th zro subspac (quvalntly, w say that Q s nonsngular f t can not b wrttn as a form n fwr than n varabls by any nvrtbl lnar chang of varabls). If F q has odd charactrstc or V s vn-dmnsonal ovr an vncharactrstc fld F q, thn Q s nonsngular f and only f B s nondgnrat [8, p. 14]. But ths s not ncssarly tru n gnral. Now assum that n s vn f q s vn and n s arbtrary othrws. Thn, Q : V = F n q F q s a nonsngular quadratc form f and only f th assocatd polar form B(x, y) = Q(x + y) Q(x) Q(y) s nondgnrat; th charactrs ϕ b, b V, of (V, +) dfnd by ϕ b (x) = ψ 1 (B(b, x)), x V, (4.1) whr ψ 1 s th canoncal addtv charactr of F q, ar all th complx charactrs of (V, +). W can lnarly xtnd th charactrs ϕ b to th whol group rng C[V ]: for A = g V a gg C[V ], w dfn ϕ b (A) = g V a gϕ b (g). It s wll known that a nonsngular quadratc form on V = F n q, whr n s vn, s quvalnt to thr x 1 x 2 + x 3 x x n 1 x n, (4.2) or x 1 x 2 + x 3 x x n 3 x n 2 + (ax 2 n 1 + bx n 1 x n + cx 2 n), (4.3) whr ax 2 n 1 + bx n 1 x n + cx 2 n s rrducbl ovr F q. A nonsngular quadratc form quvalnt to (4.2) (rsp. (4.3)) s calld hyprbolc (rsp. llptc). Lmma 4.1. ([20, Thorm 3.2]) Lt q = p f, whr p a prm and f 1 s an ntgr, and lt Q b a nonsngular quadratc form on V = F n q wth n = 2m vn. Thn ψ 1 (Q(x)) = ϵq m, x V whr ϵ = 1 or 1 accordng as Q s hyprbolc or llptc. 7

8 For ach u F q, dfn D u = {x V Q(x) = u}, and w us th sam D u to dnot th corrspondng group rng lmnt z D u z C[V ]. For a subst X of F q, w wrt D X = x X D x, whch s vwd as an lmnt of C[V ]. Now, w gv th followng ky lmma. Lmma 4.2. Lt q = p f b a prm powr and n = 2m b an vn postv ntgr. Lt Q : V = F n q F q b a nonsngular quadratc form. For any (q 1), lt C (,q) = ω ω and C (,q2 ) = γ γ, 0 1, dnot th cyclotomc classs of ordr of F q and F q 2, rspctvly, whr γ s a fxd prmtv lmnt of F q 2 and ω = Norm q2 /q(γ). Thn, for any 0 b V, and ϕ b (D (,q) C ) = { ϵq m 1 q 1, f Q(b) = 0, ϵq m 1 ψ 1(γ +s C (,q2 ) 0 ), f Q(b) C (,q) ϕ b (D 0 ) = { ϵq m 1 (q 1), f Q(b) = 0, ϵq m 1, f Q(b) 0, s for 0 s 1, whr ϵ = 1 or 1 accordng as Q s hyprbolc or llptc, ϕ b s dfnd n (4.1), and ψ 1 s th canoncal addtv charactr of F q 2. Proof: W comput th valus of ϕ b (D (,q) C ). For b V \ {0}, w hav q ϕ b (D (,q) C ) = χ b (x) ψ 1 (u(q(x) y)) u F q y C (,q) x V = ψ 1 (B(b, x) + uq(x))ψ u ( C (,q) ) x V u F q = ψ 1 (B(b, x) + uq(x))ψ u ( C (,q) ) + q 1 ψ 1 (B(b, x)). x V x V u F q Snc x V ψ 1(B(b, x)) = 0 and B(b, x) + uq(x) = u 1 Q(b) + uq(x + u 1 b) for u F q, w hav q ϕ b (D (,q) C ) = x V By Lmma 4.1, w obtan q ϕ b (D C (,q) ) = ϵq m u F q 1 = ϵq m a=0 a=0 u F q ψ 1 ( u 1 Q(b) + uq(x + u 1 b))ψ u ( C (,q) ). (4.4) ψ 1 ( u 1 Q(b))ψ u ( C (,q) ) (q 1)/ 1 c=0 ψ 1 (ω a c Q(b))ψ 1 (ω a+c C (,q) ) 1 (q 1)/ 1 = ϵq m ψ 1 (ω a+ C (,q) 0 ) ψ 1 (ω a c Q(b)) = Blow w furthr prov that c=0 { ϵq m q 1, f Q(b) = 0, ϵq m 1 a=0 ψ 1(ω a+ C (,q) 0 )ψ 1 (ω a+s C (,q) 0 ), f Q(b) C s (,q) for 0 s 1. 1 ψ 1 (ω a+ C (,q) 0 )ψ 1 (ω a+s C (,q) 0 ) = ψ 1(γ +s C (,q2 ) 0 ). a=0 Lt χ b a multplcatv charactr of ordr of F q 2. Snc (q 1), χ must b th lft of a charactr, say χ, of F q. Thn, by th orthogonalty of charactrs and th Davnport-Hass lftng 8

9 formula on Gauss sums, w hav 1 ψ 1 (ω a+ C (,q) 0 )ψ 1 (ω a+s C (,q) 0 ) a=0 = G f (χ x )G f (χ y )χ x (ω )χ y (ω s ) = 1 x=0 y=0 1 G f (χ x ) 2 χ x (ω +s ) x=0 = 1 1 G 2f (χ x )χ x (γ +s ) x=0 = ψ 1(γ +s C (,q2 ) 0 ). Smlarly, for b V \ {0}, w hav q ϕ b (D 0 ) = ( 1 a=0 { ϵq m (q 1), f Q(b) = 0, ϵq m, f Q(b) 0. χ x+y (ω a ) ) Th proof s now complt. Now w gv th man thorm of ths scton. Thorm 4.3. Lt q b a prm powr, > 1 b an ntgr dvdng q 1, and I b a subst of {0, 1,..., 1}. Lt γ b a fxd prmtv lmnt of F q 2 and ω = Norm q2 /q(γ). Lt C (,q2 ) = γ γ and E = I C(,q). Assum that Cay(F q 2, ) I C(,q2 ) s a ngatv Latn squar typ srg. Thn, for any nonsngular quadratc form Q : V = F n q F q, whr n = 2m, th Cayly graph Cay(V, D E ) s strongly rgular. Proof: By assumpton, Cay(F q 2, ) I C(,q2 ) s an srg of ngatv Latn squar typ. Its paramtrs ar (q 2, r(q +1), q +r 2 +3r, r 2 +r), whr r = I (q 1)/. Th rstrctd gnvalus of ths srg ar r and r q. It follows from Lmma 4.2 that ϕ b (D E ) = I ϕ b(d C (,q) ), b V \ {0}, tak xactly two valus, namly ϵq m 1 I (q 1)/ and ϵq m 1 ( I (q 1)/ q). Thus Cay(V, D E ) s strongly rgular. Rmark 4.4. () Not that th srg Cay(V, D E ) obtand n th abov thorm s of Latn squar typ or ngatv Latn squar typ accordng as ϵ = 1 or 1. On can s that undr th sam assumptons as n Thorm 4.3, Cay(V, D E (D 0 \ {0})) s also strongly rgular snc Cay(F q 2, ) I C(,q2 ) s a ngatv Latn squar typ srg, whr I = {0, 1,..., 1} \ I. Furthrmor, t s wll known that Cay(V, D 0 ) s also strongly rgular wth Latn squar typ or ngatv Latn squar typ paramtrs accordng to ϵ = 1 or 1[7]. () Th most mportant condton n Thorm 4.3 s that Cay(F q 2, ) I C(,q2 ) s strongly rgular. Ths condton s trvally satsfd n th followng cas. Lt = 2 and I = {0}. Thn, Cay(F q 2, C (2,q2 ) 0 ) s obvously strongly rgular wth ngatv Latn squar paramtrs. Thus, th aformntond condton s trvally satsfd. In ths cas, th srg Cay(V, D E ) obtand from Thorm 4.3 s xactly th affn polar graph. W thus hav rcovrd Thorm 1.3. Th strongly rgular Cayly graphs obtand n Scton 3 satsfy th assumptons of Thorm 4.3, namly, dvds q 1 and Cay(F q 2, ) I C(,q2 ) s a ngatv Latn squar typ srg. Thus, w can us th srgs obtand n Scton 3 as startrs to obtan nw ons. W frst consdr th smprmtv cas. Lt p b a prm, > 2, q = p 2jr, whr r 1, (p j +1), and j s th smallst such postv ntgr. In ths cas, E s chosn as th dual of th sm-prmtv cyclotomc strongly 9

10 rgular Cayly graph Cay(F q, C (,q) 0 ). Thn, E = (q 1)/ by [11, p. 23]. (Hr, rplac q m of Corollary 3.2 wth q.) By applyng Thorm 4.3 to srgs of Corollary 3.2, w hav th followng corollary. Corollary 4.5. Lt p b a prm, > 2, q = p 2jr, r 2, (p j + 1), and j s th smallst such postv ntgr. Thn thr xsts a (q 2m, r(q m ϵ), ϵq + r 2 3ϵr, r 2 ϵr) strongly rgular Cayly graph wth r = q m 1 (q 1)/. W hav thus rcovrd Thorm 1.4. Nxt w consdr th subfld cas. Lt q = p st and = pst 1 p s 1. Hr, E s chosn as th dual of th subfld cyclotomc strongly rgular Cayly graph Cay(F q, C (,q) 0 ). Thn, E = p s(t 1) 1 by [11, p. 23]. (W rplacd n = q m and r = q m a 1 n Corollary 3.3 wth n = p st and r = p s(t 1) 1 rspctvly.) Thn, w hav th followng corollary. Corollary 4.6. Thr xsts a (p 2stm, (p stm ϵ)r, ϵp stm + r 2 3ϵr, r 2 ϵr) strongly rgular Cayly graphs for any prm p and postv ntgrs s, t, and m, whr ϵ = ±1 and r = p st(m 1) (p s(t 1) 1). Fnally, w consdr th sporadc cass. Lt q, and k b as thos n Corollary 3.5. In ths cas, E s chosn as th dual of sporadc cyclotomc strongly rgular Cayly graphs Cay(F q, C (,q) 0 ). Thn, E = (q 1) k. (Not that th numbr k s th sz of th subdffrnc st corrspondng to th cyclotomc srg Cay(F q, C (,q) 0 ), s [26, Tabl II].) Hnc, w obtan th followng corollary. Corollary 4.7. Thr xsts a (q 2m, r(q m ϵ), ϵq m + r 2 3ϵr, r 2 ϵr) strongly rgular Cayly graph for any m 1, whr ϵ = ±1 and r = q m 1 (q 1) k, n ach of th followng cass: (q,, k) = (3 5, 11, 5), (5 9, 19, 9), (3 12, 35, 17), (7 9, 37, 9), (11 7, 43, 21), (17 33, 67, 33) (3 53, 107, 53), (5 18, 133, 33), (41 81, 163, 81), (3 144, 323, 161), (5 249, 499, 249). 5 Rmarks on assocaton schms Th rsults on srgs obtand n Scton 4 hav mplcatons on assocaton schms. Lt X b a fnt st. A (symmtrc) assocaton schm wth d classs on X conssts of sts (bnary rlatons) R 0, R 1,..., R d whch partton X X and satsfy (1) R 0 = {(x, x) x X}; (2) R s symmtrc for all ; (3) for any, j, k {0, 1,..., d} thr s an ntgr p k,j such that gvn any par (x, y) R k {z X (x, z) R, (z, y) R j } = p k,j. Not that ach of th symmtrc rlatons R can b vwd as an undrctd graph G = (X, R ). Thn, th graphs G, 1 d, dcompos th complt graph wth vrtx st X. An srg wth vrtx st X and ts complmnt form an assocaton schm on X wth two classs. If p k,j = pk j, for all, j, k, thn th assocaton schm s sad to b commutatv. Lt (X, {R } d =0 ) b a commutatv assocaton schm. For ach, 0 d, lt A dnot th adjacncy matrx of G = (X, R ). Thn A A j = d k=0 pk,j A k and A A j = A j A, for all 0, j d. It follows that A 0, A 1,..., A d gnrat a commutatv algbra (ovr th rals) of dmnson d + 1, whch s calld th Bos-Msnr algbra of th schm (X, {R } d =0 ). Th Bos- Msnr algbra has a unqu st of prmtv dmpotnts E 0 = (1/ X )J, E 1,..., E d, whr J s 10

11 th all-ons matrx. Thus, th algbra has two bass, {A 0 d} and {E 0 d}. W dnot by P th bas-chang matrx such that (A 0, A 1,..., A d ) = (E 0, E 1,..., E d ) P. Th ntrs n th th column of P ar th gnvalus of A, 0 d. Th matrx P s calld th frst gnmatrx (or charactr tabl) of th assocaton schm. Gvn a d-class commutatv assocaton schm (X, {R } 0 d ), w can tak unon of classs to form graphs wth largr dg sts (ths procss s calld a fuson). It s not ncssarly guarantd that th fusd collcton of graphs wll agan form an assocaton schm on X. If an assocaton schm has th proprty that any of ts fusons s also an assocaton schm, thn w call th assocaton schm amorphc. A wll-known and mportant xampl of amorphc assocaton schms s gvn by th cyclotomc assocaton schms on F q whn th cyclotomy s unform [2]. For a partton Λ 0 = {0}, Λ 1,..., Λ d {1, 2,..., d}, lt R Λ = k Λ R k. Th followng smpl crtron, calld th Banna-Muzychuk crtron, s vry usful for dcdng whthr (X, {R Λ } d forms an assocaton schm or not. Lt P b th frst gnmatrx of th assocaton schm (X, {R } d =0 ). Thn, (X, {R Λ } d =0 ) forms an assocaton schm f and only f thr xsts a partton, 0 d, of {0, 1,..., d}, wth 0 = {0} such that ach (, Λ j )-block of P has a constant row sum. Morovr, th constant row sum of th (, Λ j )-block s th (, j) ntry of th frst gnmatrx of th fuson schm. (For a proof, s [1].) From now on, w us th sam notaton and assumptons as thos n Lmma 4.2. Lt G = Cay(V, T ), whr T = {, } V /{1, 1}, and R 0 = {(x, x) x V }, R = E(Cay(V, T )). Thn, t s obvous that (V, {R } V /{1, 1} =0 ) forms a commutatv assocaton schm. Lt P b th frst gnmatrx of ths schm. Th (, j) ntry of th prncpal part of P (th matrx obtand by rmovng th frst row and column from P ) of th schm s gvn by χ (T j ), whr both th rows and columns ar labld by th lmnts of V /{1, 1}. Wrt E 1 = D 0 \ {0} and E s+2 = D (,q) C, s = {x V Q(x) whr D 0 and D C (,q) s C (,q) s ar dfnd by D 0 = {x V Q(x) = 0} and D (,q) C s }. Snc E = E for 1 + 1, th substs (Λ = :=)E /{1, 1} V /{1, 1}, 1 + 1, ar wll dfnd. Thn, th row sums of th (, Λ j )-block ar gvn by χ b (E j ), b E. On th othr hand, Lmma 4.2 mpls that for ach par, j, th sum χ b (E j ) ar constant for all b E. Thus, by th Banna-Muzychuk crtron, th partton gvs a fuson schm of (V, {R } V /{1, 1} =0 ). In summary, w hav th followng rsult. Thorm 5.1. Lt q = p f b a prm powr and n = 2m b an vn postv ntgr. Lt Q : V = F n q F q b a nonsngular quadratc form. For any (q 1), lt C (,q) = ω ω, 0 1, dnot th cyclotomc classs of ordr of F q. Thn th dcomposton of th complt graph on V by Cay(V, D 0 \ {0}) and Cay(V, D (,q) C ), 0 1, gvs a ( + 1)-class assocaton schm. Nxt, w gv a gnral suffcnt condton for a fuson of th assocaton schm n Thorm 5.1 to b an assocaton schm. Thorm 5.2. Lt q = p f b a prm powr and n = 2m b an vn postv ntgr. Lt Q : V = F n q F q b a nonsngular quadratc form. For any (q 1), lt C (,q) = ω ω and C (,q2 ) = γ γ, 0 1, dnot th cyclotomc classs of ordr of F q. Assum that thr xsts a partton A, 1 d, of { 0 1} such that th dcomposton of th complt graph of F q 2 by Cay(F q, l A C (,q2 ) l ), 1 d, s a fuson schm of th -class cyclotomc schm on F q 2. Thn, th dcomposton of th complt graph on V by Cay(V, D 0 \ {0}) and Cay(V, ), 1 d, gvs a (d + 1)-class assocaton schm. D l A C (,q) l Th abov thorm follows mmdatly from Lmma 4.2. Ths can b sn as follows. By th assumpton that th graph dcomposton by Cay(F q 2, l A C (,q2 ) l ), 1 d, gvs a fuson 11 =0 )

12 schm of th -class cyclotomc schm on F q 2, thr xsts a partton Λ h, 1 h d, of { 0 1} such that for ach 1 h, d, ψ 1(γ s l A C (,q2 ) l ) ar constant for all s Λ h,.., ϕ b ) ar constant for all b V such that Q(b) (D l A C (,q) l Λ h C (,q) l by Lmma 4.2. Smlarly, ϕ b ) ar constant for all b V such that Q(b) = 0. Furthrmor, ϕ (D l A C (,q) b (D 0 ) s dtrmnd accordng to Q(b) = 0 or not. Thus, by th Banna-Muzychuk crtron, th concluson of Thorm 5.2 follows. W also rmark that f th assumd assocaton schm of F q 2 s amorphc, thn so s th rsultng schm on V. Th condton of th abov thorm s trvally satsfd n th followng cas. Lt p b a prm, > 2, q = p 2jr, r 2, (p j +1), and j s th smallst such postv ntgr. In ths cas, snc th -class cyclotomc assocaton schm on F q 2 s amorphc, any fuson of th schm n Thorm 5.1 forms an assocaton schm. Ths rcovrs Corollary 2.4 of [15]. Also, qut rcntly, an nfnt famly of (prmtv and non-amorphc) thr-class assocaton schms on F 2 6s satsfyng th assumpton of Thorm 5.2 was found [13, Thorm 7 ()]. Fnally, th followng thorm of Van Dam [9] allows us to put th rsult of Thorm 4.3 n Scton 4 n th contxt of assocaton schms. Thorm 5.3. Lt {G 1, G 2,..., G d } b a dcomposton of th complt graph on a st X, whr ach G s strongly rgular. If G ar all of Latn squar typ or all of ngatv Latn squar typ, thn th dcomposton s a d-class amorphc assocaton schm on X. By usng Thorm 4.3 and part () of Rmark 4.4 n conjuncton wth Thorm 5.3, w hav th followng: Corollary 5.4. Undr th sam assumptons as n Thorm 4.3, th strongly rgular dcomposton Cay(V, D E ), Cay(V, D F q \E), Cay(V, D 0 \ {0}) ylds a 3-class amorphc assocaton schm. Acknowldgmnts Th work of K. Momhara was supportd by JSPS undr Grant-n-Ad for Rsarch Actvty Startup Th work of Q. Xang was don whl h s a Program Offcr at NSF. Th vws xprssd hr ar not ncssarly thos of th NSF. Rfrncs [1] E. Banna, Subschms of som assocaton schms, J. Algbra, 144 (1991), [2] L. D. Baumrt, W. H. Mlls, R. L. Ward, Unform cyclotomy, J. Numbr Thory, 14 (1982), [3] B. Brndt, R. Evans, K. S. Wllams, Gauss and Jacob Sums, Wly, [4] T. Bth, D. Jungnckl, H. Lnz, Dsgn Thory, Vol. I, 2nd dt., Cambrdg Unvrsty Prss, [5] A. E. Brouwr, W. H. Hamrs, Spctra of Graphs, Sprngr, Unvrstxt, [6] A. E. Brouwr, R. M. Wlson, Q. Xang, Cyclotomy and strongly rgular graphs, J. Alg. Combn., 10 (1999), [7] R. Caldrbank, W. M. Kantor, Th gomtry of two-wght cods, Bull. London Math. Soc., 18 (1986),

13 [8] P. J. Camron, Fnt gomtry and codng thory, Lctur Nots for Socrats Intnsv Programm, Fnt Gomtrs and Thr Automorphsms, Potnza, Italy, Jun [9] E. R. van Dam, Strongly rgular dcompostons of th complt graphs, J. Alg. Combn., 17 (2003), [10] J. A. Davs, Q. Xang, Ngatv Latn squar typ partal dffrnc sts n nonlmntary ablan 2-groups, J. London Math. Soc., 70 (2004), [11] P. Dlsart, An algbrac approach to th assocaton schms of codng thory, Phlps Rs. Rpts Suppl., No. 10, [12] J. F. Dllon, Elmntary Hadamard dffrnc sts, Ph.D. thss, Unvrsty of Maryland, [13] T. Fng, K. Momhara, Thr-class assocaton schms from cyclotomy, ArXv: [14] T. Fng, K. Momhara, Q. Xang, Constructons of strongly rgular Cayly graphs and skw Hadamard dffrnc sts from cyclotomc classs, ArXv: [15] T. Fng, B. Wn, Q. Xang, J. Yn, Partal dffrnc sts from quadratc forms and p-ary wakly rgular bnt functons, to appar n th procdngs of th confrnc n honor of Kqn Fng. [16] T. Fng, Q. Xang, Strongly rgular graphs from unons of cyclotomc classs, J. Combn. Thory, Sr. B, 102 (2012), [17] G. G, Q. Xang, T. Yuan, Constructon of strongly rgular Cayly graphs usng ndx four Gauss sums, J. Alg. Combn., DOI /s y. [18] C. Godsl, G. Royl, Algbrac Graph Thory, GTM 207, Sprngr-Vrlag, [19] C. L. M. d Lang, Som nw cyclotomc strongly rgular graphs, J. Alg. Combn., 4 (1995), [20] D. B. Lp, L. M. Schullr, Zros of a par of quadratc forms dfnd ovr a fnt fld, Fnt Flds Appl., 5 (1999), [21] J. H. van Lnt, A. Schrjvr, Constructon of strongly rgular graphs, two-wght cods and partal gomtrs by fnt flds, Combnatorca, 1 (1981), [22] S. L. Ma, A survy of partal dffrnc sts, Ds. Cods Cryptogr., 4 (1994), [23] R. J. McElc, Irrducbl cyclc cods and Gauss sums, n Combnatorcs, pp (Proc. NATO Advancd Study Inst., Brukln, 1974; M. Hall, Jr. and J. H. van Lnt (Eds.)), Part 1, Math. Cntr Tracts, Vol. 55, Math. Cntrum, Amstrdam, Rpublshd by Rdl, Dordrcht, 1975 (pp ). [24] R. L. McFarland, Sub-dffrnc sts of Hadamard dffrnc sts, J. Combn. Thory, Sr. A, 54 (1990), [25] K. Momhara, Cyclotomc strongly rgular graphs, skw Hadamard dffrnc sts, and ratonalty of rlatv Gauss sums, Europ. J. Combn., to appar. [26] B. Schmdt, C. Wht, All two-wght rrducbl cyclc cods?, Fnt Flds Appl., 8 (2002), [27] T. Storr, Cyclotomy and Dffrnc Sts, Markham Publshng Company, [28] K. Yamamoto, On Jacob sums and dffrnc sts, J. Combn. Thory, Sr. A, 3 (1967),

14 [29] J. Yang, L. Xa, Complt solvng of xplct valuaton of Gauss sums n th ndx 2 cas, Sc. Chna Sr. A, 53 (2010),