Table Of ContentTwin bent functions and Clifford algebras
PaulC.Leopardi
5
1
0
2
v
o
N
1
1 Abstract ThispaperexaminesapairofbentfunctionsonZ2mandtheirrelationship
2
toanecessaryconditionfortheexistenceofanautomorphismofanedge-coloured
]
O graphwhosecoloursaredefinedbythepropertiesofacanonicalbasisforthereal
C representation of the Clifford algebra Rm,m. Some other necessary conditions are
. alsobrieflyexamined.
h
t
a
m
1 Introduction
[
2
v Arecentpaper[11]constructsasequenceofedge-colouredgraphsD m(m>1)with
7 two edge colours, and makes the conjecture that for m>1, there is an automor-
7 phism of D that swaps the two edge colours.This conjecture can be refined into
m
4
thefollowingquestion.
5
0 Question1.1.Considerthesequenceofedge-colouredgraphsD (m>1)asdefined
. m
1 in[11],eachwithredsubgraphD [−1],andbluesubgraphD [1].Forwhichm>1
m m
0 isthereanautomorphismofD thatswapsthesubgraphsD [−1]andD [1]?
m m m
5
1
Notethattheexistenceofsuchanautomorphismautomaticallyimpliesthatthe
:
v subgraphsD [−1]andD [1]areisomorphic.
m m
i Considering that it is known that D [−1] is a strongly regular graph, a more
X m
generalquestioncanbeaskedconcerningsuchgraphs.
r
a
Mathematical Sciences Institute, The Australian National University. e-mail:
[email protected]
1
2 P.C.Leopardi
First,werecalltherelevantdefinition.
Definition1.1.[2, 3, 15]. A simple graph G of order v is strongly regular with
parameters(v,k,l ,m )if
• eachvertexhasdegreek,
• eachadjacentpairofverticeshasl commonneighbours,and
• eachnonadjacentpairofverticeshasm commonneighbours.
Now,themoregeneralquestion.
Question1.2.For which parameters (v,k,l ,m ) is there a an edge-coloured graph
G onv vertices,withtwo edgecolours,red(withsubgraphG [−1])andblue(with
subgraphG [1]),suchthatthesubgraphG [−1]isa stronglyregulargraphwith pa-
rameters (v,k,l ,m ), and such that there exists an automorphism of G that swaps
G [−1]withG [1]?
Remark1.1.Since the existence of such an automorphismimplies thatG [−1] and
G [1]areisomorphic,thisimpliesthatG [1]isalsoastronglyregulargraphwiththe
sameparametersasG [−1].
Questions1.1and1.2wereasked(inaslightlydifferentform)attheworkshop
on“AlgebraicdesigntheorywithHadamardmatrices”inBanffinJuly2014.
Furthergeneralizationgivesthefollowingquestions.
Question1.3.Given a positive integer c>1, for what parameters(v,k,l ,m ) does
thereexistackregulargraphonvverticesthatcanbegivenanedgecolouringwithc
colours,suchthattheedgescorrespondingtoeachcolorforma(v,k,l ,m )strongly
regulargraph?
For what parameters is the c-edge-colouredck regular graph unique up to iso-
morphism?
Remark1.2.ThisquestionappearsonMathOverflow[9],andispartiallyanswered
by Dima Pasechnik and Padraig O´ Catha´in, specifically for the case where the ck
regulargraphisthecompletegraphonv=ck+1vertices.Seetherelevantpapers
by van Dam [6], van Dam and Muzychuk [7], and O´ Catha´in [13]. These partial
answersdonotapplyto the specific case ofQuestion 1.1becausethegraphD is
m
notacompletegraphwhenm>1.
Question1.4.For which parameters (v,k,l ,m ) does the edge-coloured graph G
fromQuestion1.3haveanautomorphismthatpermutesthecorrespondingstrongly
regularsubgraphs?Whichfinitegroupsoccuraspermutationgroupsinthismanner
(i.e. as the group of permutations of strongly regular subgraphs of such an edge-
colouredgraph)?
ThispaperexaminessomeofthenecessaryconditionsforthegraphD tohave
m
anautomorphismasperQuestion1.1.Questions1.2to1.4remainopenforfuture
investigation.
ConsideringthatD [−1]isastronglyregulargraph,thefirstnecessarycondition
m
is that D [1] is also a strongly regular graph, with the same parameters. This is
m
provenasTheorem5.2inSection5.Someothernecessaryconditionsareaddressed
inSection6.
TwinbentfunctionsandCliffordalgebras 3
2 A signedgroup and itsreal monomialrepresentation
ThefollowingdefinitionsandresultsappearinthepaperonHadamardmatricesand
[11], and are presented here for completeness, since they are used below. Further
detailsandproofscanbefoundinthatpaper,unlessotherwisenoted.
ThesignedgroupG of order21+p+q is extensionof Z by Zp+q, definedby
p,q 2 2
thesignedgrouppresentation
G := e (k∈S ) |
p,q {k} p,q
(cid:28)
e2 =−1(k<0), e2 =1(k>0),
{k} {k}
e e =−e e (j6=k) ,
{j} {k} {k} {j}
(cid:29)
whereS :={−q,...,−1,1,...,p}.
p,q
ThefollowingconstructionoftherealmonomialrepresentationP(G )ofthe
m,m
groupG isusedin[11].
m,m
The2×2orthogonalmatrices
. − . 1
E := , E :=
1 1 . 2 1 .
(cid:20) (cid:21) (cid:20) (cid:21)
generate P(G ), the real monomial representation of group G . The cosets of
1,1 1,1
{±I}≡Z inP(G )areorderedusingapairofbits,asfollows.
2 1,1
0↔00↔{±I},
1↔01↔{±E },
1
2↔10↔{±E },
2
3↔11↔{±E E }.
1 2
Form>1,therealmonomialrepresentationP(G )ofthegroupG consists
m,m m,m
ofmatricesoftheformG ⊗G withG inP(G )andG inP(G ).
1 m−1 1 1,1 m−1 m−1,m−1
Thecosetsof{±I}≡Z inP(G )areorderedbyconcatenationofpairsofbits,
2 m,m
whereeachpairofbitsusestheorderingasperP(G ),andthepairsareordered
1,1
asfollows.
0↔00...00↔{±I},
⊗(m−1)
1↔00...01↔{±I ⊗E },
(2) 1
⊗(m−1)
2↔00...10↔{±I ⊗E },
(2) 2
...
22m−1↔11...11↔{±(E E )⊗m}.
1 2
4 P.C.Leopardi
(HereI isusedtodistinguishthis2×2unitmatrixfromthe2m×2m unitmatrix
(2)
I.)Inthispaper,thisorderingiscalledtheKroneckerproductorderingofthecosets
of{±I}inP(G ).
m,m
Werecallhereanumberofwell-knownpropertiesoftherepresentationP(G ).
m,m
Lemma2.1.ThegroupG anditsrealmonomialrepresentationP(G )satisfy
m,m m,m
thefollowingproperties.
1. Pairs of elements of G (and therefore P(G )) either commute or anti-
m,m m,m
commute:forg,h∈G ,eitherhg=ghorhg=−gh.
m,m
2. ThematricesE∈P(G )areorthogonal:EET =ETE=I.
m,m
3. ThematricesE∈P(G )areeithersymmetricandsquaretogiveIorskewand
m,m
squaretogive−I:eitherET =E andE2=I orET =−E andE2=−I.
Taking the positive signed element of each of the 22m cosets listed above de-
finesatransversalof{±I}inP(G )whichisalsoamonomialbasisforthereal
m,m
representationoftheCliffordalgebraR inKroneckerproductorder.Inthispa-
m,m
per,wecallthisorderedmonomialbasisthepositivesignedbasisofP(R ).For
m,m
example, (I,E ,E ,E E ) is the positive signed basis of P(R ). Note: any other
1 2 1 2 1,1
choiceofsignswillgivea differenttransversalof{±I}inP(G ),andhencean
m,m
equivalentorderedmonomialbasis of P(R ), butwe choosepositivesigns here
m,m
fordefiniteness.
Definition2.1.We define the function g :Z →P(G ) to choose the corre-
m 22m m,m
sponding basis matrix from the positive signed basis of P(R ), using the Kro-
m,m
necker product ordering. This ordering also defines a corresponding function on
Z2m,whichwealsocallg .
2 m
Forexample,
g (0)=g (00)=I, g (1)=g (01)=E ,
1 1 1 1 1
g (2)=g (10)=E , g (3)=g (11)=E E .
1 1 2 1 1 1 2
3 Two bent functions
We nowdefine two functions,s and t on Z2m, andshow thatbothofthese are
m m 2
bent.First,recalltherelevantdefinition.
Definition3.1.[8,p.74].
ABooleanfunction f :Zm→Z isbentifitsHadamardtransformhasconstant
2 2
magnitude.Specifically:
1. TheSylvesterHadamardmatrixH ,oforder2m,isdefinedby
m
1 1
H := ,
1 1 −
(cid:20) (cid:21)
H :=H ⊗H , for m>1.
m m−1 1
TwinbentfunctionsandCliffordalgebras 5
2. ForaBooleanfunction f :Zm→Z ,definethevector f by
2 2
f :=[(−1)f[0],(−1)f[1],...,(−1)f[2m−1]]T,
wherethevalueof f[i],i∈Z2m isgivenbythevalueof f onthebinarydigitsof
i.
3. Intermsofthesetwodefinitions,theBooleanfunction f :Zm→Z isbentif
2 2
H f =C[1,...,1]T.
m
(cid:12) (cid:12)
forsomeconstantC. (cid:12) (cid:12)
The first function, s is defined and shown to be bent in [11]. We repeat the
m
definitionhere.
Definition3.2.Weusethebasiselementselectionfunctiong ofDefinition2.1to
m
definethesign-of-squarefunctions :Z2m→Z as
m 2 2
1↔g (i)2=−I
s (i):= m
m (0↔gm(i)2=I,
foralliinZ2m.
2
Remark3.1.Property3fromLemma2.1ensuresthats iswelldefined.Also,since
m
eachg (i)isorthogonal,s (i)=1ifandonlyifg (i)isskew.
m m m
From the property of Kronecker products that (A⊗B)T =AT ⊗BT, it can be
shown thats can also be calculated fromi∈Z2m as the parity of the numberof
m 2
occurrencesofthebitpair01ini,i.e.s (i)=1ifandonlyifthenumberof01pairs
m
isodd.Alternatively,fori∈Z ,s (i)=1ifandonlyifthenumberof1digitsin
22m m
thebase4representationofiisodd.
Thefollowinglemmaisprovenin[11].
Lemma3.1.Thefunctions isabentfunctiononZ2m.
m 2
Thebasiselementselectionfunctiong alsogivesrisetoasecondfunction,t
m m
onZ .
22m
6 P.C.Leopardi
Definition3.3.Wedefinethenon-diagonal-symmetryfunctiont onZ andZ2m
m 22m 2
asfollows.
ForiinZ2:
2
1 ifi=10, sothatg (i)=±E ,
t (i):= 1 2
1
(0 otherwise.
ForiinZ2m−2:
2
t (00⊙i):=t (i),
m m−1
t (01⊙i):=s (i),
m m−1
t (10⊙i):=s (i)+1,
m m−1
t (11⊙i):=t (i).
m m−1
where⊙denotesconcatenationofbitvectors,ands isthesign-of-squarefunction,
asabove.
Itiseasytoverifythatt (i)=1ifandonlyifg (i)issymmetricbutnotdiagonal.
m m
This can be checked directly for t . For m >1 it results from properties of the
1
Kronecker product of square matrices, specifically that (A⊗B)T =AT ⊗BT, and
thatA⊗BisdiagonalifandonlyifbothAandBarediagonal.
Thefirstmainresultofthispaperisthefollowing.
Theorem3.1.Thefunctiont isabentfunctiononZ2m.
m 2
The proof of Theorem3.1 uses the followingresult, due to Tokareva[16], and
stemmingfromtheworkofCanteaut,Charpinandothers[4,TheoremV.4][5,The-
orem2].Theresultreliesonthefollowingdefinition.
Definition3.4.Forabentfunction f onZmthedualfunction f isgivenby
2
(H [f]) =:2m/2(−1)f(i). e
m i
e
Lemma3.2.[16,Theorem1]Ifabinaryfunction f onZ2mcanbedecomposedinto
2
fourfunctions f ,f ,f ,f onZ2m−2 as
0 1 2 3 2
f(00⊙i)=: f (i), f(01⊙i)=: f (i),
0 1
f(10⊙i)=: f (i), f(11⊙i)=: f (i),
2 3
whereallfourfunctionsarebent,withdualfunctionssuchthat f +f +f +f =1,
0 1 2 3
then f isbent.
e e e e
ProofofTheorem3.1.InLemma3.2,set f = f :=t ,f =s ,f =s +
0 3 m−1 1 m−1 2 m−1
1.Clearly, f = f .Also, f = f +1,sinceH [f ]=−H [f ].Therefore f +
0 3 2 1 m−1 2 m−1 1 0
f + f + f =1. Thus, these four functionssatisfy the premise of Lemma 3.2, as
1 2 3
longasbothes e andt e areebent. e
m−1 m−1
e e e
TwinbentfunctionsandCliffordalgebras 7
Itis knownthats is bentforall m. Itis easy to show thatt is bent,directly
m 1
fromitsdefinition.Thereforet isbent. ⊓⊔
m
4 Bent functions and Hadamarddifference sets
The following well known properties of Hadamard difference sets and bent func-
tionsarenotedin[11].
Definition4.1.[8,pp.10and13].
Thek-elementsetDisa(v,k,l ,n)differencesetinanabeliangroupGoforderv
ifforeverynon-zeroelementginG,theequationg=d −d hasexactlyl solutions
i j
(d,d )withd,d inD.Theparametern:=k−l .A(v,k,l ,n)differencesetwith
i j i j
v=4niscalledaHadamarddifferenceset.
Lemma4.1.[8,Remark2.2.7][12,14].AHadamarddifferencesethasparameters
oftheform
(v,k,l ,n)=(4N2,2N2−N,N2−N,N2)
or (4N2,2N2+N,N2+N,N2).
Lemma4.2.[8, Theorem6.2.2]The Booleanfunction f :Zm →Z is bentif and
2 2
onlyifD:= f−1(1)isaHadamarddifferenceset.
Together,theseproperties,alongwithLemma3.1andTheorem3.1,areusedhere
toprovethefollowingresult.
Theorem4.1.Thesetss −1(1)andt −1(1)arebothHadamarddifferencesets,with
m m
thesameparameters
(v ,k ,l ,n )=(4m,22m−1−2m−1,22m−2−2m−1,22m−2).
m m m m
Proof. Both s and t are bent functions, as per Lemma 3.1 and Theorem 3.1
m m
respectively.Therefore,byLemma4.2,boths −1(1)andt −1(1)areHadamarddif-
m m
ferencesets. Inbothcases, therelevantabeliangroupisZ2m, with order4m. Thus
2
inLemma4.1wemustsetN=2m−1toobtainthateither
(v ,k ,l ,n )=(4m,22m−1−2m−1,22m−2−2m−1,22m−2)or
m m m m
(v ,k ,l ,n )=(4m,22m−1+2m−1,22m−2+2m−1,22m−2).
m m m m
Since s (i)=1 if and only if g (i) is skew, and t (i)=1 if and only if g (i) is
m m m m
symmetric but not diagonal,not only are these conditionsmutually exclusive,but
also,forallm>1,thenumberofiforwhichs (i)=t (i)=0ispositive.Theseare
m m
theiforwhichg (i)isdiagonal.Thusk =22m−1−2m−1ratherthan22m−1+2m−1.
m m
Theresultfollowsimmediately. ⊓⊔
As a check, the parametersk can also be calculated directly, using the recursive
m
definitionsofeachofs andt .
m m
8 P.C.Leopardi
5 Bent functions and strongly regulargraphs
This section examines the relationship between the bent functionss and t and
m m
thesubgraphsD [−1]andD [1]fromQuestion1.1.
m m
First we revise some known properties of Cayley graphs and strongly regular
graphs, as noted in the previous paper on Hadamard matrices and Clifford alge-
bras[11], includingthe resultof BernasconiandCodenotti[1] onthe relationship
betweenbentfunctionsandstronglyregulargraphs.
FirstwerecallaspecialcaseofthedefinitionofaCayleygraph.
Definition5.1.The Cayley graph of a binary function f :Zm →Z is the undi-
2 2
rectedgraphwithadjacencymatrixF givenbyF = f(g +g ),forsomeordering
i,j i j
(g ,g ,...)ofZm.
1 2 2
TheresultofBernasconiandCodenotti[1]ontherelationshipbetweenbentfunc-
tionsandstronglyregulargraphsisthefollowing.
Lemma5.1.[1,Lemma12].TheCayleygraphofabentfunctiononZmisastrongly
2
regulargraphwithl =m .
WeusethisresulttoexaminethegraphD .Thefollowingtwodefinitionsappear
m
inthepreviouspaper[11]andarerepeatedhereforcompleteness.
Definition5.2.LetD bethegraphwhoseverticesarethen2=4m canonicalbasis
m
matrices of the real representation of the Clifford algebra R , with each edge
m,m
havingoneoftwocolours,−1(red)and1(blue):
• MatricesA andA areconnectedbyarededgeiftheyhavedisjointsupportand
j k
areanti-amicable,i.e.A A−1isskew.
j k
• Matrices A and A are connected by a blue edge if they have disjoint support
j k
andareamicable,i.e.A A−1issymmetric.
j k
• OtherwisethereisnoedgebetweenA andA .
j k
Wecallthisgraphtherestrictedamicability/anti-amicabilitygraphoftheClifford
algebraR ,therestrictionbeingtherequirementthatanedgeonlyexistsforpairs
m,m
ofmatriceswithdisjointsupport.
Definition5.3.ForagraphG withedgescolouredby-1(red)and1(blue),G [−1]
denotestheredsubgraphofG ,thegraphcontainingalloftheverticesofG ,andall
ofthered(-1)colourededges.Similarly,G [1]denotesthebluesubgraphofG .
Thefollowingtheoremispresentedin[11].
Theorem5.1.Forallm>1,thegraphD [−1]isstronglyregular,withparameters
m
v =4m,k =22m−1−2m−1,l =m =22m−2−2m−1.
m m m m
Unfortunately, the proof given there is incomplete, proving only that D [−1] is
m
stronglyregular,withoutshowingwhyk =22m−1−2m−1andl =m =22m−2−
m m m
2m−1.Inthissection,werectifythisbyprovingthefollowing.
TwinbentfunctionsandCliffordalgebras 9
Theorem5.2.Forallm>1,bothgraphsD [−1]andD [1]isstronglyregular,with
m m
parametersv =4m,k =22m−1−2m−1,l =m =22m−2−2m−1.
m m m m
Proof. SinceeachvertexofD isacanonicalbasiselementoftheCliffordalgebra
m
R ,wecanimposetheKroneckerproductorderingonthevertices,labellingeach
m,m
vertexAbyg −1(A)∈Z2m.Thelabelk (a,b)ofeachedge(g (a),g (b))ofD
m 2 m m m m
dependsona+binthefollowingway:
k (a,b):=t (a+b)−s (a+b), thatis,
m m m
−1, s (a+b)=1 (⇔g (a+b)isskew),
m m
k m(a,b)=0, s m(a+b)=tm(a+b)=0 (⇔gm(a+b)isdiagonal),
1, tm(a+b)=1 (⇔gm(a+b)issymmetricbutnotdiagonal).
Thus D [−1] is isomorphic to the Cayley graph of s on Z2m, and D [1] is iso-
m m 2 m
morphictotheCayleygraphoft onZ2m.Since,byLemma3.1andTheorem3.1,
m 2
boths andt arebentfunctionsonZ2m,Lemma5.1impliesthatbothD [−1]and
m m 2 m
D [1]arestronglyregulargraphs.
m
It remains to determine the graph parameters. Firstly, v is the number of ver-
m
tices,whichis4m.
SinceD [−1]is regular,we can determinek by examiningonevertex,g (0).
m m m
The edges (g (0),g (b)) of D [−1] are those for which s (b)= 1, that is, the
m m m m
edges where b is in the Hadamard difference set s −1(1). Thus, by Theorem 4.1,
m
k =2N2−N=22m−1−2m−1,whereN=2m−1.
m
SinceD [−1]isastronglyregulargraph,itholdsthat
m
(v −k −1)m =k (k −1−l )
m m m m m m
[15,p.158]andhence,sincel =m ,wemusthave(v −1)l =k (k −1).We
m m m m m m
nownotethat
k (k −1)=(2N2−N)(2N2−N−1)=(v −1)(22m−2−2m−1),
m m m
sothatl =m =22m−2−2m−1.
m m
Runningthroughtheseargumentsagain,withD [1]substitutedforD [−1]and
m m
t substitutedfors ,yieldsthesameparametersforD [1]. ⊓⊔
m m m
Remark5.1.Amoreelementaryderivationofthevalueofl forD [−1]follows.
m m
There are k (k −1) ordered pairs (a,b) with a6=b and s (a)=s (b)=1.
m m m m
Sincek (k −1)=(N2−N)(4N2−1),thisgivesexactlyN2−N=22m−2−2m−1
m m
orderedpairsforeachofother4m−1verticesofD [−1].
m
Also, considering that s −1(1) is a Hadamard difference set, and for c∈Z2m,
m 2
c6=0,consideroneofthepairs(a,b)suchthats (a)=s (b)=1andc=a+b.
m m
Thusb=a+cands (a)=s (a+c)=1.Therefore,thegraphD [−1]contains
m m m
theedges(g (0),g (a)),(g (0),g (b)),(g (c),g (a)),and(g (c),g (b)).
m m m m m m m m
10 P.C.Leopardi
Thus, in the graph D [−1], the vertices g (0) and g (c) have the two vertices
m m m
g (a)and g (b) in common.Thisis truewhetheror notthereis an edgebetween
m m
g (0) and g (c). The pair (b,a) yields the same four edges. Running through all
m m
suchpairs(a,b)andusingTheorem4.1again,wesee thatl =m =2N2−N=
m m
22m−2−2m−1.
6 Other necessary conditions
Thissectionexaminestwoothernecessaryconditionsfortheexistenceofanauto-
morphismofD thatswapsD [−1]withD [1].Thefirstconditionfollows.
m m m
Theorem6.1.If an automorphism q : D → D exists that swaps D [−1] with
m m m
D [1], then there is an automorphismQ :D →D that also swaps D [−1] with
m m m m
D [1],leavingg (0)fixed.
m m
Proof. Forthepurposesofthisproof,assumetheKroneckerproductorderingofthe
canonicalbasiselementsofR anddefinetheone-to-onemappingf :Z2m→Z2m
m,m 2 2
suchthatq (g (a))=g (f (a))foralla∈Z2m.Theconditionthatq swapsD [−1]
m m 2 m
withD [1]isequivalenttothecondition
m
k (f (a)+f (b))=−k (a+b),
m m
wherek isasdefinedintheproofofTheorem5.2above.
m
LetF (a):=f (a)+f (0)foralla∈Z2m.ThenF (a)+F (b)=f (a)+f (b)for
2
alla,b∈Z2m,andtherefore
2
k (F (a)+F (b))=k (f (a)+f (b))=−k (a+b).
m m m
NowdefineQ :D →D suchthatQ (g (a))=g (F (a))foralla∈Z2m. ⊓⊔
m m m m 2
Thesecondconditionissimplytonotethatifq swapsD [−1]withD [1],then
m m
for any induced subgraph G ⊂D and its image q (G ), the corresponding edges
m
(A,B)and(q (A),q (B))willalsohaveswappedcolours.
These two conditions were used to design a backtracking search algorithm to
findanautomorphismthatsatisfiesQuestion1.1orruleoutitsexistence.Twoim-
plementationsof the search algorithmwere coded:one using Python,and a faster
implementationusing Cython.The sourcecode is available on GitHub [10]. Run-
ningthesearchconfirmstheexistenceofanautomorphismform=1,2,and3,but
rulesitoutform=4.OnanIntel(cid:13)R CoreTM [email protected],theCython
implementationofsearchform=4takesabout15hourstorun.
Sincethispaperwassubmitted,theauthorhasfoundasimpleproofthatanauto-
morphismsatisfyingQuestion1.1doesnotexistform>4:SeearXiv:1504.02827
[math.CO].
