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 finite graph G are called disconnected if there is no edge between A and B .If V is the set of words of length n over an alphabet { 1 , ,q} and if two words are adjacent whenever their Hamming distance is not equal to afixed δ∈{ 1 , ,n} , then a pair of disconnected sets becomes an equidistant code pair . For disconnected sets A and B we will give a bound for |A|·|B| 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 δ ,andfor q→∞ for any fixed n and δ . In addition, our bound reproves some old results of Ahlswede and others, such as the maximal value of |A|·|B| for equidistant code pairs A ans B in the binary Hamming Scheme. 1 Introduction Throughout G is a finite graph with vertex set V . Two disjoint subsets A and B of V are disconnected ifthereisnoedgebetweenAand B. We define Φ(G)tobethemaximumof | A |·| B | for disconnected sets A and B in G. Suppose V is the set of words of length n over an alphabet { 1, ,q } and define two words adjacent if their Hamming distance (i.e. the number of coordinates in which they differ) is not equal to a fixed δ ∈{ 1, ,n } .Then a pair of disconnected sets becomes an equidistant code pair. the electronic journal of combinatorics 4 (1997), #R7 2 The quantity Φ(G) has an application in information theory and leads to a lower bound for the two-way communication complexity of functions defined 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 Φ(G)forq=2,4and5,foreveryδand n. In this paper we will give a bound for Φ(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 δ,andforq →∞ for any fixed n and δ. 2 Disconnected vertex sets Let V = { 1, ,v } . We define 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 =0if iand j are distinct non-adjacent vertices of G.Letλ 1 (M), ,λ v (M)denote the eigenvalues of a matrix M ∈M (G), such that λ 1 (M) has eigenvector 1 (the all-one vector), so λ 1 (M)=1. Put λ(M)=max i=1 | λ i (M ) | . Lemma 2.1 If A and B are disconnected vertex sets of G and M ∈M (G), then | A |·| B | (v −| A | )(v −| B | ) ≤ λ 2 (M). Proof. See [7] Theorem 2.1, or [11] Lemma 6.1. 2 Theorem 2.2 For any M ∈M (G) Φ(G) ≤ v λ(M) 1+λ(M) . theelectronicjournalofcombinatorics4(1997),#R7 3 Proof.PutΦ=Φ(G)andtakeAandBsuchthatΦ 2 = | A |·| B | .Thenby Lemma2.1 Φ 2 λ 2 (M) ≤ (v 2 − v( | A | + | B | )+Φ 2 ) ≤ (v 2 − 2v | A |·| B | +Φ 2 )=(v − Φ) 2 . Clearlyv ≥ Φ,soΦ ≤ (v − Φ)λ(M),whichyieldstherequiredbound. InordertoinvestigatewhentheboundofTheorem2.2isbestpossible,we define φ(G)=min M∈M(G) v λ(M) 1+λ(M) andwelet M (G)denotethesetofmatricesfrom M (G)forwhichthe aboveminimumisattained.ThusTheorem2.2becomesΦ(G) ≤ φ(G).To determineφ(G)weneedtofindamatrixin M (G).Forthatpurposethe automorphismsofGcanbehelpful. Lemma2.3Let A beanautomorphismgroupofG.Then M (G)contains amatrixwhichisconstantovereachorbitoftheactionof A onV × V. Proof.LetP g denotethepermutationmatrixcorrespondingtog ∈A and takeM ∈M (G).ThenclearlyP g M P g ∈M (G)and,byRayleigh’s principle, | u P g M P g u |≤ λ(M )foreveryunitvectoruorthogonalto1 . Define M= 1 |A| g∈A P g M P g , thenclearly M ∈M (G)andMisconstantover A -orbitsonV × V.Let u(u ⊥ 1)beauniteigenvectorfortheeigenvalue ± λ(M).Thenλ(M)= | u Mu |≤ λ(M ),soλ(M)=λ(M )andhence M ∈M (G). 2 InparticularwemaytakethediagonalconstantifGhasatransitiveauto- morphismgroup.Theorem2.2leadstoamoreexplicitboundintermsofthe LaplacianeigenvaluesofG.(IfAisthestandardadjacencymatrixofGand Disthediagonalmatrixcontainingthevertexdegrees,thenF=D − Ais theLaplacianmatrixofG.IteasilyfollowsthatFispositivesemi-definite andsingular;seeforexampleBrualdiandRyser[6].) theelectronicjournalofcombinatorics4(1997),#R7 4 Theorem2.4SupposeFistheLaplacianmatrixofGandlet0=µ 1 ≤ µ 2 ≤ ≤ µ v betheeigenvaluesofF,then φ(G) ≤ v 2 1 − µ 2 µ v withequalityifGhasanautomorphismgroupthatactstransitivelyonthe edges. Proof.Define M= − 2 µ 2 +µ v F+I. ThenM ∈M (G)andλ(M)=(µ v − µ 2 )/(µ v +µ 2 ),whichyieldstheinequal- ity. SupposeGhasanautomorphismgroupwhichactstransitivelyonthe edges.Then,byLemma2.3thereexistsamatrixM ∈M (G)suchthat M =xF+DforsomeconstantxanddiagonalmatrixD.NowM 1=1 givesD=Iandso λ(M )=max {| xµ 2 +1 | , | xµ v +1 |} . Itfollowsthatλ(M )isminimalifxµ 2 +1= − xµ v − 1,thatis,ifx= − 2/(µ 2 +µ v ).ThusM ∈M (G). Example.SupposeGisthetriangulargraphT(2m)(thatis,thelinegraph ofK 2m ).Thenv=m(2m − 1),µ 2 =2mandµ v =4m − 2.Theorem2.4 givesφ(G)= m 2 .WeeasilyhavethatΦ(G) ≥ m 2 ,soΦ(G)= m 2 . Nextweconsiderassociationschemes.Fortheoryandnotationsee[4],[5] or[9].LetSbeann-classassociationschemedefinedonthesetVwith relationsR 0 , ,R n .Forδ ∈{ 1, ,n } wedenotebyG δ thegraph(V,R δ ) andbyG δ thecomplementofG δ . Theorem2.5IfQisthematrixofdualeigenvaluesofS,then φ(G δ ) ≤ v n j=0 | Q δ,j | . 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 A 0 , ,A n (with A 0 = I) be the adjacency matices of S and let E 0 , ,E n (with vE 0 = J, the all-one matrix) be the minimal idempotents. Then the matrix Q of dual eigenvalues is given by vE j = n i=0 Q i,j A i , for j =0, ,n. So v(E j ) k,l = Q δ,j whenever { k, l }∈ R δ . Define P δ = { j | 1 ≤ j ≤ n, Q δ,j > 0 } , m = j∈P δ Q δ,j + 1 2 and M = 1 m ( 1 2 I − j∈P δ (E j − Q δ,j E 0 )). Then, since E j 1 =0for j = 0 one readily verifies that M ∈M (G δ ). Moreover j∈P δ E j has only 0 and 1 as eigenvalues. This implies that λ i (M)= ± 1 2m for i =1,soλ(M)= 1 2m , and thus we find φ(G δ ) ≤ v 2m+1 .By use of n j=0 Q δ,j =0andQ δ,0 =1,weobtain m+ 1 2 =1+ j∈P δ Q δ,j = 1 2 n j=0 | Q δ,j | , 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 M ∈M (G δ )which is a linear combination of A 0 , ,A n ,thatis,M is in the Bose-Messner algebra of S.Letλ j (M ) denote the eigenvalue of M whose eigenspace is given by E j . We claim that we may assume that λ j (M )=λ(M )if Q δ,j ≤ 0. Indeed, suppose this is not the case, then define d = λ(M ) − λ j (M ), m =1 − dQ δ,j and M = 1 m (M + d(E j − Q δ,j E 0 )). It follows that M ∈M (G δ ), m ≥ 1andλ j (M )=λ(M )= 1 m λ(M ) ≤ λ(M ). So we can redefine M = M , which proves the claim. Similarly, we may assume that λ j (M )= − λ(M )ifQ δ,j ≥ 0andj = 0. It now follows that E = 1 2λ(M ) (λ(M )I − M +(λ(M ) − 1)E 0 ) theelectronicjournalofcombinatorics4(1997),#R7 6 haseigenvalue0and1only,andhenceEisanidempotentofS.Therefore EisthesumofthoseE j thatcorrespondtotheeigenvalue1ofE,that isE= j∈P δ E j .Inaddition,since(M ) k,l =0for { k,l }∈ R δ ,wehave j∈P δ Q δ,j = 1 2λ(M ) − 1 2 .ThisimpliesthatM =M,soM ∈M (G δ ). AgraphG δ ina2-classassociationschemeisthesameasastronglyregular graph.AnexampleofsuchagraphisthetriangulargraphT(m),described intheexampleabove.Itisnotdifficulttoseethatforstronglyregulargraphs theboundsofTheorem2.4andTheorem2.5coincide. 3 Equidistantcodepairs SupposeV= { 1, ,q } n ,thesetofwordsoflengthnoveranalphabetofsize q,anddefinetwowordstobeinrelationR δ iftheirHammingdistance(the numberofcoordinateplacesinwhichtheydiffer)equalsδ.Thisdefinesthe wellknownHammingassociationschemeH(n,q).ForagraphG δ inH(n,q) twodisconnectedsetsinG δ arecalledequidistantcodepairs(atdistanceδ) andwewriteΦ δ andφ δ insteadofΦ(G δ )andφ(G δ )respectively. Lemma3.1 Φ 2 δ ≥ max 0≤δ ≤δ n − δ δ − δ (q − 1) δ−δ q 2 q 2 δ . Proof.TakeforAthesetofwords(x 1 , ,x n )with1 ≤ x i ≤ q 2 ifi ≤ δ and x i =1ifi>δ ,andletBconsistofthewords(x 1 , ,x n )with q 2 <x i ≤ qif i ≤ δ andx i =1forpreciselyδ − δ valuesofi>δ .ThenAandBforman equidistantcodepairatdistanceδwithsizes q 2 δ and q 2 δ n−δ δ−δ (q − 1) δ−δ , 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 δ,j = n k=0 ( − 1) k (q − 1) j−k δ k n − δ j − k forδ,j ∈{ 0, ,n } . TheautomorphismgroupofH(n,q)istransitiveoneachrelation,soTheo- rem2.5givestheexactvalueofφ δ foralln,qandδ. ExampleIfn=6andq=6,thenforj=0, ,6therespectivevaluesof Q 4,j are1,6, − 9, − 44,111, − 90and25.Theorem2.5givesφ 4 =46656/286 ≈ 163.13.WithLemma3.1(takeδ =2)wefind45 √ 6 ≤ Φ 4 ≤ 23328/143. ThisexampleshowsthatourboundwillnotproveAhlswede’sconjecture. Butitcangiveinterestingresultsinsomecases. Theorem3.2Ifq>2then φ δ ≤ q n (q − 2) n−δ 2 δ . Equalityholdsifandonlyifδ=n. Proof.TheinequalityfollowsfromTheorem2.5and n j=0 | Q δ,j |≥| n j=0 ( − 1) j Q δ,j | = n k=0 δ k n j=0 ( − 1) j−k (q − 1) j−k n − δ j − k =2 δ (q − 2) n−δ . Ifjrunsfrom0ton,Q n,j alternatesinsign,sowehaveequalityifδ=n. Thedualeigenvaluesofanyassociationschemesatisfy n j=0 1 µ j Q δ,j Q n,j =0 ifδ =n(µ j =rkE j ).ThereforeQ δ,j cannotalternateinsignifδ =n,so thenwehavestrictinequality. Corollary3.3Ifq>2then Φ δ ≤ q n (q − 2) n−δ 2 δ . Equalityholdsifandonlyifδ=nandqiseven. theelectronicjournalofcombinatorics4(1997),#R7 8 Proof.Ifδ=nandqiseven,Lemma3.1(withδ =δ)givesΦ n ≥ ( q 2 ) n , whichequalsφ n .Ifqisodd,( q 2 ) 2n isnotaninteger,soΦ n =φ n . WeseethatthelowerboundofLemma3.1andtheupperboundofCorol- lary3.3tendtothesamevalue( q 2 ) δ ifq →∞ .Moreprecisely: Corollary3.4 Φ δ = q 2 δ +O(q δ−1 )(q →∞ ). Forq ≥ 4AhlswedeandM¨ors[3]showedthatΦ δ <Φ n ifδ<n.Thisresult nowfollowsdirectlyfromCorollary3.3whenqisevenand,byLemma3.1, alsowhenqisoddandnisnottoobig. Forq=3notmuchisknownaboutΦ δ .Ahlswede[1]hasaconstruction forequidistantcodepairsandconjecturesthatitisbestpossible.Ifthisis truethenΦ δ attainsitsmaximalvalue 3 2 n 3 2 n 2 ifδ= 2n 3 .Theorem3.2givesφ n =( 3 2 ) n ,thuswehavethatΦ n <Φ δ if δ= 2n 3 (n>2).ByuseofTheorem2.5,strongerresultsarepossible,butit turnsoutthattheboundφ δ isnotgoodenoughtoprovethatΦ δ ismaximal ifδ= 2n 3 . Forq=2thevalueΦ δ isknownforallδ;see[1].Itattainsthemaxi- malvalue2 n 2 ifδ= n 2 andifδ= n 2 .Thisresultwasfirstprovedby Ahlswede,ElGamalandPang[2]andhasseveraldifferentproofsnow(see [1]).Weshallseethatourboundprovidesyetanotherproof.Theconstruc- tionisasfollows.TakeforAthesetofwords(x 1 , ,x n )withx 2i−1 =x 2i for1 ≤ i ≤ n 2 andx n =1ifnisodd.TakeforBthesetofwordswith x 2i−1 =x 2i for1 ≤ i ≤ n 2 andx n fixedifnisodd.ThenAandBare equidistantcodepairsatdistance n 2 or n 2 and | A | = | B | =2 n 2 . Theorem3.5Ifq=2then φ δ ≤ 2 n 2 forδ ∈{ 1, ,n } . Equalityholdsifδ ∈{ n 2 , n 2 } . theelectronicjournalofcombinatorics4(1997),#R7 9 Proof.Withi= √ − 1wehave n j=0 i j Q δ,j = n k=0 i −k δ k n j=0 i j−k n − δ j − k =(1 − i) δ (1+i) n−δ = 2 n 2 ω, whereinω=e πi 4 (n−2δ) .Hence n j=0 | Q δ,j | = jeven | Q δ,j | + jodd | Q δ,j |≥ 2 n 2 ( | Reω | + | Imω | )= 2 n 2 , andtheinequalityfollowsbyuseofTheorem2.5.Theconstructionabove showsthatΦ δ =φ δ =2 n 2 if | n 2 − δ |≤ 1 2 . WithasimilarargumentandabitmoreworkasintheproofofTheorem3.2 itcanbeseenthatif | n 2 − δ |≥ 1theboundφ δ isstrictlylessthan2 n 2 . 4 Concludingremarks Thereissomesimilaritybetweenourfunctionφ(G)andLov´asz’sfunction θ(G)(see[12]).Thelatterfunctiongivesanupperboundforacoclique (independentsetofvertices)inG,whichis,inasense,avertexsetdiscon- nectedtoitself.Lovasz’sθ(G)isknowntobecomputableinpolynomialtime (see[10]),butwedonotknowthecomplexityofthecomputationofφ(G) andΦ(G). IfweapplyTheorem2.5totheJohnsonassociationschemeJ(m,n),we obtainboundsforequidistantcodepairsin(binary)constantweightcodes. Itseemscertainlyworthwhiletoworkthisoutandweintendtodoso.The problemisthattheformulasforthedualeigenvaluesarerathercomplicated. Forthespecialcasethatδ=nwebelievedthat j | Q δ,j | = 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 References [1] R. Ahlswede, On code pairs with specified Hamming distances, Colloquia Mathematica Societas J´anos Bolyai 52, Combinatorics, Eger (Hungary), 1987, pp. 9-47. [2] R. Ahlswede, A. El Gamal and K.F. Pang, A two-family extremal prob- lem in Hamming space, Discrete Math. 49 (1984), 1-5. [3] R. Ahlswede and M. M¨ors, Inequalities for code pairs, Europ. J. Com- binatorics 9 (1988), 175-181. [4] A.E.Brouwer,A.M.CohenandA.Neumaier,Distance-regular graphs, Springer-Verlag, Berlin, 1989. [5] A.E. Brouwer and W.H. Haemers, Association schemes, chapter 15 in: Handbook of Combinatorics (R. Graham, M. Gr¨otschel and L. Lov´asz eds.), Elsevier Science, Amsterdam, 1995, pp. 747-771. [6] R.A. Brualdi and H.J. Ryser, Combinatorial matrix theory, Cambridge University Press, Cambridge, 1991. [7] E.R. van Dam and W.H. Haemers, Eigenvalues and the diameter of graphs, Linear Multilinear Algebra 39 (1995), 33-44. [8] Ph. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Research Rep. Suppl. 10, 1973. [9] C.D. Godsil, Algebraic combinatorics, Chapman and Hall, New York - London, 1993. [10] M. Gr¨otschel, L. Lov´asz and A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization, Combinatorica 1 (1981), 169-197. [11] W.H. Haemers, Interlacing eigenvalues and graphs, Linear Algebra Appl. 226-228 (1995), 593-616. [12] L. Lov´asz, On the Shannon capacity of a graph, IEEE Trans. Inform. Theory 25 (1979), 1-7. . Disconnected vertex sets and equidistant code pairs Willem H. Haemers Department of Econometrics, Tilburg University, Tilburg, The Netherlands; e-mail: haemers@kub.nl Submitted:. distance is not equal to afixed δ∈{ 1 , ,n} , then a pair of disconnected sets becomes an equidistant code pair . For disconnected sets A and B we will give a bound for |A|·|B| in terms of the eigenvalues. ,q } n ,thesetofwordsoflengthnoveranalphabetofsize q,anddefinetwowordstobeinrelationR δ iftheirHammingdistance(the numberofcoordinateplacesinwhichtheydiffer)equalsδ.Thisdefinesthe wellknownHammingassociationschemeH(n,q).ForagraphG δ inH(n,q) twodisconnectedsetsinG δ arecalledequidistantcodepairs(atdistanceδ) andwewriteΦ δ and δ insteadofΦ(G δ )and (G δ )respectively. Lemma3.1 Φ 2 δ ≥ max 0≤δ ≤δ n − δ δ − δ (q − 1) δ−δ q 2 q 2 δ . Proof.TakeforAthesetofwords(x 1 ,