Haemers, W.H.

English

Research Papers in Economics

Disconnected vertex sets
and equidistant code pairs
Willem H. Haemers
Department of Econometrics, Tilburg University,
Tilburg, The Netherlands; e-mail: haemers@kub.nl
Submitted: October 28, 1996; Accepted: January 29, 1997.
Abstract
Two disjoint subsets A and B of a vertex set V of a ¯nite graph
G are called disconnected if there is no edge between A and B.I fV
is the set of words of length n over an alphabet f1;:::;qgand if two
words are adjacent whenever their Hamming distance is not equal to
a¯ x e d±2f 1 ;:::;ng, then a pair of disconnected sets becomes an
equidistant code pair.
For disconnected sets A and B we will give a bound for jAj¢j B j
in terms of the eigenvalues of a matrix associated with G. In case
the complement of G is given by a relation of an association scheme
the bound takes an easy form, which applied to the Hamming scheme
leads to a bound for equidistant code pairs. The bound turns out to
be sharp for some values of q, n and ±,a n df o rq!1for any ¯xed n
and ±. In addition, our bound reproves some old results of Ahlswede
and others, such as the maximal value of jAj¢jBjfor equidistant code
pairs A ans B in the binary Hamming Scheme.
1 Introduction
Throughout G is a ¯nite graph with vertex set V . Two disjoint subsets A
and B of V are disconnected i ft h e r ei sn oe d g eb e t w e e nAand B. We de¯ne
©(G)t ob et h em a x i m u mo f
q
j A j¢j Bjfor disconnected sets A and B in G.
Suppose V is the set of words of length n over an alphabet f1;:::;qg and
de¯ne two words adjacent if their Hamming distance (i.e. the number of
coordinates in which they di®er) is not equal to a ¯xed ± 2f 1 ;:::;ng.T h e n
a pair of disconnected sets becomes an equidistant code pair.the electronic journal of combinatorics 4 (1997), #R7 2
The quantity ©(G) has an application in information theory and leads
to a lower bound for the two-way communication complexity of functions
de¯ned on V £V that are constant over the non-edges of G. About ten years
ago this application caused some activity in the study of equidistant code
pairs. The best result is due to Ahlswede [1], who gives the exact value of
©(G)f o rq=2 ,4a n d5 ,f o re v e r y±and n.
In this paper we will give a bound for ©(G) in terms of eigenvalues of
a matrices associated with G. In case the complement of G is given by
a relation of an association scheme the bound takes an easy form, which
applied to the Hamming scheme leads to a bound for equidistant code pairs.
This bound is not as accurate as Ahlswede's result, but it is more general
and it turns out to be sharp for some values of q, n and ±,a n df o rq!1
for any ¯xed n and ±.
2 Disconnected vertex sets
Let V = f1;:::;vg. We de¯ne M(G) to be the collection of symmetric v£v
matrices M with all row and column sums equal to 1, such that (M)i;j =0i f
iand j are distinct non-adjacent vertices of G.L e t¸ 1 ( M ) ;:::;¸ v(M)d e n o t e
the eigenvalues of a matrix M 2M ( G ), such that ¸1(M) has eigenvector 1
(the all-one vector), so ¸1(M)=1 .P u t
¸ ( M)=m a x
i 6 =1
j¸i(M)j:
Lemma 2.1 If A and B are disconnected vertex sets of G and M 2M ( G ) ,
then
jAj¢j Bj
( v¡j Aj )(v ¡j Bj )
·¸
2( M) :
Proof. See [7] Theorem 2.1, or [11] Lemma 6.1. 2
Theorem 2.2 For any M 2M ( G )
©(G) · v
¸(M)
1+¸ ( M)
:theelectronicjournalofcombinatorics4(1997),#R7 3
Proof.Put©=©(G)andtakeAandBsuchthat©2 =jAj¢jBj.Thenby
Lemma2.1
©2
¸2(M)
·(v
2 ¡v(jAj+jBj)+©
2)·(v
2¡2v
q
jAj¢jBj+©
2)=(v¡©)
2:
Clearlyv¸©,so©·(v¡©)¸(M),whichyieldstherequiredbound. 
InordertoinvestigatewhentheboundofTheorem2.2isbestpossible,we
de¯ne
Á(G)=min
M2M(G)
v
¸(M)
1+¸(M)
andweletM0(G)denotethesetofmatricesfromM(G)forwhichthe
aboveminimumisattained.ThusTheorem2.2becomes©(G)·Á(G).To
determineÁ(G)weneedto¯ndamatrixinM0(G).Forthatpurposethe
automorphismsofGcanbehelpful.
Lemma2.3LetAbeanautomorphismgroupofG.ThenM0(G)contains
amatrixwhichisconstantovereachorbitoftheactionofAonV£V.
Proof.LetPg denotethepermutationmatrixcorrespondingtog2Aand
takeM0 2M 0 (G).ThenclearlyPgM0P>
g 2M 0 (G)and,byRayleigh's
principle,ju>PgM0P >
g uj·¸(M 0 )foreveryunitvectoruorthogonalto1 .
De¯ne
M=
1
jAj
X
g2A
PgM
0P
>
g ;
thenclearly M2M(G)andMisconstantoverA-orbitsonV£V.Let
u(u?1 )beauniteigenvectorfortheeigenvalue§¸(M).Then¸(M)=
ju > Muj·¸(M 0),so¸(M)=¸(M 0)andhence M2M 0(G). 2
InparticularwemaytakethediagonalconstantifGhasatransitiveauto-
morphismgroup.Theorem2.2leadstoamoreexplicitboundintermsofthe
LaplacianeigenvaluesofG.(IfAisthestandardadjacencymatrixofGand
Disthediagonalmatrixcontainingthevertexdegrees,thenF=D¡Ais
theLaplacianmatrixofG.IteasilyfollowsthatFispositivesemi-de¯nite
andsingular;seeforexampleBrualdiandRyser[6].)theelectronicjournalofcombinatorics4(1997),#R7 4
Theorem2.4SupposeFistheLaplacianmatrixofGandlet0=¹ 1·
¹ 2·:::·¹ v betheeigenvaluesofF,then
Á(G)·
v
2
Ã
1¡
¹ 2
¹ v
!
withequalityifGhasanautomorphismgroupthatactstransitivelyonthe
edges.
Proof.De¯ne
M=
¡2
¹2 +¹v
F+I:
ThenM2M(G)and¸(M)=(¹ v¡¹ 2)=(¹ v+¹ 2),whichyieldstheinequal-
ity.
SupposeGhasanautomorphismgroupwhichactstransitivelyonthe
edges.Then,byLemma2.3thereexistsamatrixM0 2M 0 (G)suchthat
M 0=xF+DforsomeconstantxanddiagonalmatrixD.NowM 0 1 =1
givesD=Iandso
¸(M
0)=maxfjx¹2 +1j;jx¹v +1jg:
Itfollowsthat¸(M0)isminimalifx¹2 +1=¡x¹v ¡1,thatis,ifx=
¡2=(¹2 +¹v).ThusM2M 0(G). 
Example.SupposeGisthetriangulargraphT(2m)(thatis,thelinegraph
ofK2m).Thenv=m(2m¡1),¹2 =2mand¹v =4m¡2.Theorem2.4
givesÁ(G)=
³
m
2
´
.Weeasilyhavethat©(G)¸
³
m
2
´
,so©(G)=
³
m
2
´
.
Nextweconsiderassociationschemes.Fortheoryandnotationsee[4],[5]
or[9].LetSbeann-classassociationschemede¯nedonthesetVwith
relationsR0;:::;R n.For±2f1;:::;ngwedenotebyG± thegraph(V;R±)
andbyG± thecomplementofG±.
Theorem2.5IfQisthematrixofdualeigenvaluesofS,then
Á(G±)·
v
Pn
j=0 jQ±;jj
:the electronic journal of combinatorics 4 (1997), #R7 5
Equality holds if the automorphism group of S acts transitively on each rela-
tion.
Proof. Let A0;:::;A n (with A0 = I) be the adjacency matices of S and let
E0;:::;E n (with vE0 = J, the all-one matrix) be the minimal idempotents.
Then the matrix Q of dual eigenvalues is given by
vEj =
n X
i=0
Qi;jAi; for j =0 ;:::;n:
So v(Ej)k;l = Q±;j whenever fk;lg2R ±. De¯ne P± = fjj1 · j · n; Q±;j >
0g, m =
P
j2P± Q±;j + 1
2 and
M = 1
m(1
2I ¡
X
j2P±
(Ej ¡ Q±;jE0)):
Then, since Ej1 =0for j 6= 0 one readily veri¯es that M 2M ( G ± ).
Moreover
P
j2P± Ej has only 0 and 1 as eigenvalues. This implies that
¸i(M)=§
1
2 m for i 6=1 ,s o¸ ( M)=
1
2 m, and thus we ¯nd Á(G±) ·
v
2m+1.B y
use of
Pn
j=0 Q±;j =0a n dQ ±;0 =1 ,w eo b t a i n
m+
1
2 =1+
X
j 2P±
Q±;j =
1
2
n X
j=0
jQ±;jj;
and the required inequality follows.
Next assume that S admits an automorphism group which is transitive on
each relation. Then by Lemma 2.3 there exists a matrix M0 2M 0( G ±)w h i c h
is a linear combination of A0;:::;A n,t h a ti s ,M 0is in the Bose-Messner
algebra of S.L e t ¸ j 0 ( M 0 ) denote the eigenvalue of M0 whose eigenspace
is given by Ej. We claim that we may assume that ¸j0(M0)=¸ ( M 0 )i f
Q ±;j · 0. Indeed, suppose this is not the case, then de¯ne d = ¸(M0) ¡
¸j0(M0), m =1¡dQ±;j and M00 =
1
m(M0 + d(Ej ¡ Q±;jE0)). It follows that
M00 2M ( G ±), m ¸ 1a n d¸ j 0( M 00)=¸ ( M 00)=
1
m¸ ( M 0)·¸ ( M 0). So we can
rede¯ne M0 = M00, which proves the claim. Similarly, we may assume that
¸j0(M0)=¡ ¸ ( M 0)i fQ ±;j ¸ 0a n dj6 = 0. It now follows that
E =
1
2¸(M0)
(¸(M
0)I ¡ M
0 +( ¸ ( M
0)¡1)E0)theelectronicjournalofcombinatorics4(1997),#R7 6
haseigenvalue0and1only,andhenceEisanidempotentofS.Therefore
EisthesumofthoseEj thatcorrespondtotheeigenvalue1ofE,that
isE=
P
j2P± Ej.Inaddition,since(M0)k;l =0forfk;lg2R ± ,wehave
P
j2P± Q±;j =
1
2¸(M0) ¡ 1
2.ThisimpliesthatM0 =M,soM2M 0(G ±). 
AgraphG ±ina2-classassociationschemeisthesameasastronglyregular
graph.AnexampleofsuchagraphisthetriangulargraphT(m),described
intheexampleabove.Itisnotdi±culttoseethatforstronglyregulargraphs
theboundsofTheorem2.4andTheorem2.5coincide.
3 Equidistantcodepairs
SupposeV=f1;:::;qg n,thesetofwordsoflengthnoveranalphabetofsize
q,andde¯netwowordstobeinrelationR± iftheirHammingdistance(the
numberofcoordinateplacesinwhichtheydi®er)equals±.Thisde¯nesthe
wellknownHammingassociationschemeH(n;q).ForagraphG± inH(n;q)
twodisconnectedsetsinG± arecalledequidistantcodepairs(atdistance±)
andwewrite©± andÁ± insteadof©(G±)andÁ(G ±)respectively.
Lemma3.1
©
2
± ¸max
0·±0·±
Ã
n¡±0
±¡±0
!
(q¡1)
±¡±0
µ¹q
2
º»q
2
¼¶±0
:
Proof.TakeforAthesetofwords(x1;:::;x n)with1·x i·
q
2 ifi·±0 and
xi =1ifi>± 0,andletBconsistofthewords(x1;:::;x n)with
q
2 <x i·qif
i·±0 andxi 6=1forprecisely±¡± 0valuesofi>± 0.ThenAandBforman
equidistantcodepairatdistance±withsizesb
q
2c±0 andd
q
2e±0³
n¡±0
±¡±0
´
(q¡1)±¡±0,
respectively. 2
TheaboveconstructionwasgivenbyAhlswede[1].Heprovesthatequality
holdsforq=4andq=5andconjecturesequalityforallq¸4.Forq=2
andq=3thereexistbetterconstructionsofequidistantcodepairs(see
below).theelectronicjournalofcombinatorics4(1997),#R7 7
TheHammingschemeisself-dual,whichmeansthatthedualeigenvalues
coincidewiththeeigenvalues.Theyaregivenby(see[8]):
Q±;j =
n X
k=0
(¡1)
k(q¡1)
j¡k
Ã
±
k
!Ã
n¡±
j¡k
!
for±;j2f0;:::;ng:
TheautomorphismgroupofH(n;q)istransitiveoneachrelation,soTheo-
rem2.5givestheexactvalueofÁ± foralln,qand±.
ExampleIfn=6andq=6,thenforj=0;:::;6therespectivevaluesof
Q4;j are1,6,¡9,¡44,111,¡90and25.Theorem2.5givesÁ4 =46656=286¼
163:13.WithLemma3.1(take±0 =2)we¯nd45
p
6·©4 ·23328=143.
ThisexampleshowsthatourboundwillnotproveAhlswede'sconjecture.
Butitcangiveinterestingresultsinsomecases.
Theorem3.2Ifq>2then
Á± ·
qn
(q¡2)n¡±2±:
Equalityholdsifandonlyif±=n.
Proof.TheinequalityfollowsfromTheorem2.5and
n X
j=0
jQ±;jj¸j
n X
j=0
(¡1)
jQ±;jj=
¯ ¯ ¯ ¯ ¯ ¯
n X
k=0
Ã
±
k
! n X
j=0
(¡1)
j¡k(q¡1)
j¡k
Ã
n¡±
j¡k
!¯ ¯ ¯ ¯ ¯ ¯
=2
±(q¡2)
n¡±:
Ifjrunsfrom0ton,Qn;j alternatesinsign,sowehaveequalityif±=n.
Thedualeigenvaluesofanyassociationschemesatisfy
Pn
j=0
1
¹jQ±;jQn;j =0
if±6=n(¹j =rkE j).ThereforeQ±;j cannotalternateinsignif±6=n,so
thenwehavestrictinequality. 
Corollary3.3Ifq>2then
©± ·
qn
(q¡2)n¡±2±:
Equalityholdsifandonlyif±=nandqiseven.theelectronicjournalofcombinatorics4(1997),#R7 8
Proof.If±=nandqiseven,Lemma3.1(with±0 =±)gives© n¸(
q
2) n ,
whichequalsÁn.Ifqisodd,(
q
2)2n isnotaninteger,so©n 6=Án. 
WeseethatthelowerboundofLemma3.1andtheupperboundofCorol-
lary3.3tendtothesamevalue(
q
2) ±ifq!1.Moreprecisely:
Corollary3.4
©± =
µq
2
¶±
+O(q
±¡1)(q!1):
Forq¸4AhlswedeandMÄors[3]showedthat©± <©n if±<n.Thisresult
nowfollowsdirectlyfromCorollary3.3whenqisevenand,byLemma3.1,
alsowhenqisoddandnisnottoobig.
Forq=3notmuchisknownabout©±.Ahlswede[1]hasaconstruction
forequidistantcodepairsandconjecturesthatitisbestpossible.Ifthisis
truethen©± attainsitsmaximalvalue
µ3
2
¶b n
3 c
2
n
2
if±=d2n
3 e.Theorem3.2givesÁn =(
3
2) n ,thuswehavethat© n<© ± if
±=d2n
3 e(n>2).ByuseofTheorem2.5,strongerresultsarepossible,butit
turnsoutthattheboundÁ± isnotgoodenoughtoprovethat©± ismaximal
if±=d2n
3 e.
Forq=2thevalue© ± isknownforall±;see[1].Itattainsthemaxi-
malvalue2d n
2 e if±=bn
2candif±=dn
2e.Thisresultwas¯rstprovedby
Ahlswede,ElGamalandPang[2]andhasseveraldi®erentproofsnow(see
[1]).Weshallseethatourboundprovidesyetanotherproof.Theconstruc-
tionisasfollows.TakeforAthesetofwords(x1;:::;x n)withx 2i¡1=x 2i
for1·i·
n
2 andxn =1ifnisodd.TakeforBthesetofwordswith
x2i¡1 6=x2i for1·i· n
2 andxn ¯xedifnisodd.ThenAandBare
equidistantcodepairsatdistancebn
2cordn
2eandjAj=jBj=2 bn
2c.
Theorem3.5Ifq=2then
Á± ·2
bn
2c for±2f1;:::;ng:
Equalityholdsif±2fb
n
2c;d n
2eg.theelectronicjournalofcombinatorics4(1997),#R7 9
Proof.Withi=
p
¡1wehave
n X
j=0
i
jQ±;j =
n X
k=0
i
¡k
Ã
±
k
! n X
j=0
i
j¡k
Ã
n¡±
j¡k
!
=(1¡i)
±(1+i)
n¡± =2
n
2!;
wherein!=e
¼i
4 (n¡2±).Hence
n X
j=0
jQ±;jj=
X
jeven
jQ±;jj+
X
jodd
jQ±;jj¸2
n
2(jRe!j+jIm!j)=2
dn
2e;
andtheinequalityfollowsbyuseofTheorem2.5.Theconstructionabove
showsthat©± =Á± =2 bn
2cifjn
2 ¡±j·1
2. 
WithasimilarargumentandabitmoreworkasintheproofofTheorem3.2
itcanbeseenthatifjn
2 ¡±j¸1theboundÁ± isstrictlylessthan2b n
2 c.
4 Concludingremarks
ThereissomesimilaritybetweenourfunctionÁ(G)andLov¶asz'sfunction
µ(G)(see[12]).Thelatterfunctiongivesanupperboundforacoclique
(independentsetofvertices)inG,whichis,inasense,avertexsetdiscon-
nectedtoitself.Lovasz'sµ(G)isknowntobecomputableinpolynomialtime
(see[10]),butwedonotknowthecomplexityofthecomputationofÁ(G)
and©(G).
IfweapplyTheorem2.5totheJohnsonassociationschemeJ(m;n),we
obtainboundsforequidistantcodepairsin(binary)constantweightcodes.
Itseemscertainlyworthwhiletoworkthisoutandweintendtodoso.The
problemisthattheformulasforthedualeigenvaluesarerathercomplicated.
Forthespecialcasethat±=nwebelievedthat
P
j jQ±;jj=
³ m
2
n
´
,buthad
noproof,untilVolkerStrehl(privatecommunication)provideduswitha
computer-generatedproofbyuseofZeilberger'salgorithm.
Acknowledgement.Theresearchforthispaperwasdonewhentheau-
thorwasaguestoftheSonderforschungsbereich\DiskreteStructureninder
Mathematik"attheUniversityofBielefeld.SpecialthanksgoestoProfessor
RudiAhlswedefortheinvitation,andseveralfruitfuldiscussionsconcerning
thesubject.the electronic journal of combinatorics 4 (1997), #R7 10
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Disconnected vertex sets and equidistant code pairs.

Disconnected vertex sets and equidistant code pairs.

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