Schrijver, A. (Lex)

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stichting mathematisch centrum AFDELING ZUIVERE WISKUNDE (DEPARTMENT OF PURE MATHEMATICS) A. SCHRIJVER zw 118/78 , A COMPARISON OF BOUNDS OF DELSARTE AND LOVASZ Preprint ~ MC NOVEMBER 2e boerhaavestraat 49 amsterdam PJt,ln;te.d at .t.he. Ma.the.ma.tic.o.t Ce.nbte., 49, 2e. BoeJC.haa.veli.tJr.a.at, Am.o.:teJc.dam. The Ma.thema..t.ic..a.l Ce.nbte., f,ou.n.de.d .the. 11-.th 06 Fe.b1u1.all.y 1946, ,i.,,t, a non-PM f,U J.no:tUu.:ti.on a,.,im,i.ng a.t .the. pJt.omo:ti.on 06 pUll.e. ma.thema:ti.cA and w a.pp..Uc.at,lono. I.t Lo .6ponoo1te.d by .the. Ne..the.Jri.a.ndo GoveJC.nme.n:t .th!c.ough .the. Net:heJli.a.ndo 0Jz.ga.ru.za.tion. f,oJt the. Adva.nc.eme.n:t of, PUAe. Relie.CVtc.h (Z.W.O). AMS (MOS) subject classification scheme (1970): 94Al O, 05C99, (90CXX., 94-XX) • A comparison of bounds of Delsarte and Lovas~*) by A. Schrijver ABSTRACT We aompare two upper bound functions: Delsarte's linear programming bound (an upper bound for the cardinality of cliques in association schemes) and Lovasz's 8-function (an upper bound for the Shannon capacity of a graph). We show that the two bounds can be treated in a unified fashion. Delsarte's linear programming bound can be generalized to a bound 8'(G) for the inde-pendence number a(G) of arbitrary graphs G, such that e'(G) ~ 8(G). On the other hand, if the edge set of G is the union of some classes of a symmetric association scheme, 8(G) may be calculated by means of linear prograrm:ning. We show that for such graphs G the product 8(G)•8(G) is equal to the number of vertices of G. KEY WORDS & PHRASES: linear prog1'amming bound, association scheme, Shannon capacity, positive semi-definite, codes, constant weight codes. *) To appear in IEEE Trans. on Information Theory. 1 1. INTRODUCTION The purpose of this note is to compare two upper bound functions, both being bounds for numbers motivated by more or less information-theoretical problems: Delsarte's linear programming bound, an upper bound for the cardinality of cliques in association schemes, and Lovasz's 8-function, yielding an upper bound for the Shannon capacity of a graph. The first bound may be conceived of as a bound for the independence number a(G) of certain graphs G, whereas Lovasz's bound limits a(Gk), the independence number of the normal product of k copies of G. We first give, in brief, these two bounds and their theoretical background. (A graph is an undirected graph, without loops or multiple edges.) Association schemes and Delsarte's linear programming bound (Delsarte [2], cf. Macwilliams & Sloane [5]). A pair (X,R), where R = (R0 , ••• ,Rn) is a partition of xxx, is called a (symmetric) association scheme, with intersection numbers p~. (i,j,k=O, ••• ,n), if l.J (1) (2) (3) R0 = { (x,x) lxeX}; Rj"/ = { ( y, x) I ( x, ~,) E ¾J = ~, for k = 0, ••• , n; for all i,j,k = O, ••• ,n, and (x,y)e¾: 1 { z I ( x , z ) ER . and ( z , y) ER . } I = p ~ .• l. J l.J k k So pij = pji" We may consider the pairs (X,Ri) as graphs (i=l, ••• ,n). 0 0 (X,R.) is regular of valency v. = p,. (v0 = 1). Therefore p .. = o .. v .• l. l. l.l. l.J l.J l. Let Di be the adjacency matrix of (X,Ri); D0 is the identity matrix. Since, by (3), the symmetric matrices 00 , ••• ,Dn commute there exists a matrix u n O n ) P = (Pk)k,u=O such that Pk, ••• ,Pk are the eigenvalues of Dk (k=O, ••• ,n, 2 and the eigenvalues P~, ••. ,P~ of 0 0 , ••• ,Dn' respectively, have a common eigenvector (u=O, •.• ,n). we may assume that P~ = vk for all k. Set whereµ is the dimension of the common eigenspace of D0 , ••• ,Dn belonging u u u · 1 ( O ) It can be shown that to P0 , ••• ,Pn' respective y u= , .•. ,n. n (5) I u=O n 1 U V and l PkQk = k=O m· o uv' I I -1 T . t . where m = X. So P and m •Q represent inverse ma rices. Coding theorists are interested in two families of association schemes: the families of Hamming schemes and Johnson schemes, respectively. Let n and q be natural numbers and let X be the set of vectors of length n, with entries in {O, ••• ,q-1}. Moreover let, fork= O, ••• ,n: (6) ¾ = {(x,y)EXXX where dH(x,y) denotes the Hamming distance between the vectors x and y, i.e. the number of coordinate places in which x and y differ. Let R = (R0 , ••. ,Rn). As can be checked easily (X,R) is a symmetric association scheme; schemes obtained in this way are called Hamming schemes. For Hamming schemes the values of vk, µu and P~ are given by: (7) (r') k (n). (q-l) u, vk = k . (q-1 l , µu = u k (-1)j(q-1)k-j(~) (n-~) k Pu = ¾: (u) = I = I (-q)j(q-l)k-j (~-~)(~)I k j=O J k-J j=O -J J for k,u = 0 (K() · h , • • • ,n K u 1.s t e Krawtchouk polynomial of degree k in the 3 variable u) • The second family is obtained as follows. Let v and n be natural numbers and let X be the set of 0,1-vectors of length v with exactly nones (n~½v). Moreover, let, fork= O, .•. ,n, ( 8) { (x,y) EXXX I dJ (x,y) = k}, where dJ(x,y) = ½dH(x,y) is the Johnson distance between x and y. Let R = (R0 , ••• ,Rn). Then (X,R) is a symmetric association scheme; schemes constructed in this manner are called Johnson schemes. Their parameters are: (9) vk = (~) (~=~) ' µ = (V) - (u~ 1) = v-2u+1. (v) , u u v-u+l u k _ k-j(n-j) (n-u) (v-n+j-u) k (-l) j (~) (n-~) (v-n~u) , Pu = Ek(u) = I ( 1) k . . . = I k j=O -J J J j=O J k-J k-J for k,u = O, ••. ,n (Ek(u) is the Eberlein polynomial of degree 2k in the variable u). (A third family of symmetric association schemes is formed by strongly regular graphs. These are exactly those graphs (X,R1) such that (X,R) is a symmetric association scheme, where R = (R0 ,R1 ,R2 ), R2 = (XXX)\(R0 uR1). It follows that the complementary graph of a strongly regular graph is strongly regular.) The main problem in combinatorial coding theory is to estimate the maximum size of any subset C (a "code") of (the set X in) Hamming and Johnson schemes such that no two elements in C have (Hamming or Johnson) distance less than a given valued. A generalized translation of this problem into the language of association schemes needs the notion of an M-clique; given OEMc{O, •.. ,n} a subset Y of Xis an M-clique if (x,y)EUkEMRk for all x,yEY. 4 So the coding problem is to determine the maximum cardinality of {0,d,d+1, •.. ,n}-cliques·in Hamming and Johnson schemes. To obtain an upper bound for the size of cliques in a symmetric association scheme (X,R) define, for Ycx, the inner distribution (a0 , .... ,an) of Y by (10) fork= 0, ... ,n; so a 0 = 1 and I~=O ¾ = IYI. Moreover, if Y is an M-clique then ak = 0 if k4M. Delsarte showed that, for the inner distribution of any sybset Y of X, one has n o 1 ) I ¾ Q~ ~ o , k=O for u = 0, ... ,n. Therefore, for M-cliques Y one has (12) IYI :5 max{l~=O ¾ I a0 , ... ,an~0; a 0=1; ¾=0 for k4M; I~=O ¾Q~ ~ O} = = min{I:=o bu I bo, ... ,bn~0; bo=1; r:=o buP~:50 for ke:M\{0}}. The equality in (12) follows from the duality theorem of linear programming. This bound on the size of cliques is called Delsarte's linear programming bound. One may apply linear programming techniques to calculate its value -see [1] for applications in coding theory. The following result of Delsarte shows that the linear programming bound is a sharpening of the Hamming bound in coding theory. Let (X,R) be a symmetric association scheme, with R = (R0 , ••• ,Rn)' and let 0EMc{0, ••• ,n} and M = {0}u({0, ••• ,n}\M). Then (13) the product of the linear programming bound for M-cliques and the linear programming bound for ~cliques is at most lxl. 5 Hence IYI • IZl$1XI for M-cliques Y and M-cliques Z. Taking. M = {O,d,d+l, ••• ,n} in a Hamming scheme the Hamming bound follows. The Shannon capacity and Lovasz's bound. Lovasz [4] introduced, for any graph G, the number 8(G), which is an upper bound for the "Shannon capacity" 0(G). Let a.(G) be the maximum number of independent (i.e. pairwise non-adjacent) points in a graph G, and let G·H denote the (normal) product of graphs G and H, i.e. the point set of G·H is the cartesian product of the point sets of G and H, whereas two distinct points of G•H are adjacent iff in both coordinate places the elements are adjacent or equal. Gk denotes the product of k copies of G. Shannon [9] introduced the following number for graphs G: (14) 0(G) = sup l¼(Gk) k =lim~ k-+00 which is called the Shannon capacity of G. If one considers the points of Gas letters in an alphabet, two points being adjacent iff they are "confoundable", then a.(Gk) may be interpreted as the maximum number of k-letter messages such that any two of them are inconfoundable in at least one coordinate place. Since a.(G)ksa.(Gk), it follows that a.,(G)$0(G). Equality does not hold in 2 2 general; e.g. a.(c5 ) = 2, whereas a.(c5 ) = 5 $ 0(c5) • Lovasz showed that, in fact, 0(G) = /s. Actually, he gave a general upper bound for 0(G) as follows. Let G = (V,E) be a graph, with vertex set V = {1, •.. ,n}, and define 6 (15) 0(G) •- min{le.vA I A= (a .. ) is a symmetric nxn-matrix such that l.J a .. = 1 if {i,j}ti;E}, l.J where le.VA denotes the largest eigenvalue of A. Now, if ~(G) = k, each matrix A satisfying the conditions mentioned in (15) has a kXk all-one principal submatrix (with largest eigenvalue k), hence levA~k. Therefore a(G) ~ 0(G). Since, as Lovasz proved, 0(G•H) = 0(G) ·0(H) for all graphs k k G and H, one has a(G )~0(G) , which yields the stronger inequality 0(G)~0(G) (Haemers [3] showed the existence of graphs G with 0(G)<8(G)). Moreover Lovasz showed ( 16) 9 (G) = maxn: .. b,. I B=(b .. ) J.,J J.J J.J is an nxn positive semi-definite matrix, with TrB=l, and b .. =O whenever {i,j}EE}. l.J So 9(G) may be considered as both a maximum and a minimum, which makes the function e easier to handle. Lovasz found, inter alia, for graphs G (with n points): (17) 0(G)·0(G) ~ n (where G is the complementary graph), with equality and (18) if G is vertex-transitive; -nA n 9(G) ~ ~---:::;:- if G is regular (A 1 and An being the largest and 1 n smallest eigenvalues of the adjacency matrix of G), with equality if G is edge-transitive. A consequence of (18) is: let v~2n and let K(v,n) be the graph whose vertices are then-subsets of some fixed v-set, two vertices being adjacent iff they are disjoint; such graphs are called Kneser-graphs. Then (19) 8 (K(v,n)) = (v-1) n-1 7 (by (18) it is sufficient to calculate the eigenvalues of K(v,n)), generalizing (v-1) . the Erdos-Ko-Rado theorem, which says that a(K(v,n)) = n-l The theories of Delsarte and Lovasz appear to have certain common characteristics, such as bounding cliques or independent sets in graphs, using eigenvalue-techniques on matrices determined by graphs, yielding relations between a graph and its complement, and being applicable to allied structures such as "constant weight codes" and Kneser-graphs. The purpose of this note is to go further into this relationship. Clearly, Delsarte's linear programming bound may be conceived of as an upper bound for a(G) for graphs G whose edge set is the union of some classes R. l. of a symmetric association scheme (X,R). We show that Delsarte's bound can be extended to a bound 8' (G) for a(G) for arbitrary graphs G; the description of 8 1 (G) has many features in common with Lovasz's 8(G). It will follow that e' (G) ~ e (G) (in general e' (G) -/ e (G)). On the other hand, if the edge set of G is the union of some classes of a symmetric association scheme (X,R) the number 8(G) may be calculated by means of a linear program obtained from (12) by dropping the nonnegativity constraints for a 0 , ••• ,an. It follows that also for such graphs G ~ne has 8(G)·8(G) = !xi (cf. (13) and (17)). 8 ~ 2. A COMPARISON OF THE BOUNDS OF DELSARTE AND LOVASZ First recall the following strong form of the duality theorem of linear programming. Let c and D be closed convex cones in lRk and IRm, respectively, * * * with dual cones C and D, respectively (that is, C consists of all vectors in IRk having a nonnegative inner product with each element of C). Let M be a real-valued mxk-matrix, and let CE IRk and dE IRm. Then I I * * (20) max{cx XEC; d-MXED} = min{yd yED; yM-cEC }, provided that the object sets are nonempty and closed. Furthermore, notice that the closed convex cone of all real-valued symmetric positive semi-definite nxn-matrices, conceived of as n2 -vectors, has as dual cone the set of real-valued nxn-matrices u such that yTUy~O for all real n-vectors y. (So symmetric matrices in the dual cone are positive semi-definite.) For convenience, we use the following inner product notation for nxn-matrices (21) (b .. ) : 1.J n A*B = I i, j=l a .. ·b .. , 1.J 1.J that is, A*B T = Tr(A B). So A*I TrA and A*J = I~,j=l a ..• 1.J Let G be a graph, with point set {1, ..• ,n}. Lovasz defined (22) 8(G) = max{I, .b .. I B=(b .. ) is a symmetric positive semi-definite 1.,J 1.J 1.J nxn-matrix with TrB=l, and b .. =O if {i,j}EE} = 1.J min {.te.vA I A= (a .. ) is a symmetric nxn-matrix with a .. =1 if 1.J 1.] {i,j}4E}. 9 Now define 8 1 (G) as follows. (23) 8'(G) = max(L, .b .. j B=(b .. ) is a nonnegative symmetric positive l,J l] l] semi-definite nxn-matrix with TrB=1, and b .. =O if {i,j}EE}, lJ so the difference with (22) is the restriction of the range for the maximum to nonnegative matrices B. THEOREM 1. a(G) ~ 8' (G) ~ 0(G). PROOF. Clearly, e• (G) ~ 0(G). Suppose yc{l, ... ,n} is an independent set with a(G)=k elements. Define b .. = 1/k if i,jEY, and b .. = 0, otherwise. l] l] Then B=(b .. ) is nonnegative and positive semi-definite with trace 1, and lJ b .. =0 if {i,j}EE. Furthermore, I .. b .. = k. Hence a(G) = k ~ 8' (G). D lJ l,J l] THEOREM 2. 0'(G) = min{le.vA / A=(a .. ) is a symmetric nxn-matrix with lJ a .. d if {i,j}E/:E}. lJ PROOF. By definition (24) 8' (G) = max{B*J I B=(b .. ) is a symmetric positive semi-definite lJ nxn-matrix such that: B*I=1, B*F .. =O for lJ {i,j}EE, and B*F .. ?:0 for {i,jHE}, 1. J where F .. is the nxn-(0,1)-matrix with only ones in the positions (i,j) and l] (j,i). From the above-mentioned form of the linear programming duality theorem it follows that this maximum equals 10 {25) minP. e:R I M•(m ) is a s"mmetric nxn-matrix; m .. ~O if {i,j}4E; ij ~-- l.J AI+M-J is positive semi-definite}. Putting A= J-M, one has, since, for symmetric A, the largest eigenvalue of A is equal to the minimum value of A such that AI-A is positive semi-definite, (26) 8' (G) = min{levA j A= (a .. ) is a symmetric nxn-matrix such that l.J a .. ~1 if {i,jHE}. 0 l. J Since the largest eigenvalue of a matrix is not increased by decreasing diagonal elements we may suppose that the minimum is attained by some A with ones on the diagonal. We secondly prove that for graphs derived from symmetric association schemes e• (G) coincides with Delsarte's linear programming bound. Let (X,R) be a symmetric association scheme, with R = (R0 , ... ,Rn), and let Oe:Mc{O, ... ,n}. Let G=(X,E) be the graph with E = UitMRi. Clearly, M-cliques in the association scheme coincide with independent sets in G. THEOREM 3. 6'(G) is equal to the linear programming bound for M-cliques in (X,R). PROOF. The linear programming bound is, by definition (cf. (12)) (27) max{L~=O ak l a0 , ... ,an~O; a0=1; ak=O for ktM; l~=O ~Q~ ~ 0 for u=O, ... ,n}. Let a0 , ..• ,an attain this maximum, and put ( 28) (b .. ) = B l.J 11 n ak \' --D l m·v k' k=O k where m, vk and Dk are as in section 1. Then B satisfies the conditions mentioned in (23); Bis positive semi-definite since, by the commutativity of n0 , ••• ,Dn, the matrix B has eigenvalues (29) n a l _k_ Pu= k=O m·vk k for u=O, ... ,n. Since Dk*J = vk·m, it follows that B*J = I .. b .. = Ikak. l.,J l.J Therefore, the linear programming bound is at most 8 1 (G). To prove the converse, let b 0 , .•. ,bn attain the minimum in (12), and let ;\=lb. Define u u (30) A Since ;\I-A has eigenvalues n (31) I k=O n b \' ~ Qu _ l) Pv l µ k • k u=O u n b = l ~-m-6 - okO ~ 0 u=O µu uv (v=O, ... ,n) the matrix A has largest eigenvalue at most A. Furthermore, by (4) and (12), a .. ;:c: 1 if {i,j} ,l E. Therefore, the minimum in (12) is at l.J least the minimum of theorem 2, or the linear programming bound is at least 8' (G) . D If (X,R) is a Johnson scheme with n classes (cf. Delsarte [2]) and { } (v-1) M= O, ... ,n-1 , then G=K(v,n). As Lovasz showed that 8(K(v,n))= n-l , also Delsarte's linear programming bound yields the Erdos-Ko-Rado theorem. 12 Using techniques similar to those used in the proof of theorem 3 one proves for symmetric association schemes (X,R) and graphs G related as mentioned before theorem 3: THEOREM 4. 8(G) = max{I~=O ¾ I a 0=1; ¾=O for k4M; I~=O akQ~~O for u=O, •.. ,n} = min{I:=0 bu I bo,·••1bn~O; bo=l; r:=O buP~ = 0 for keM\{0}}. PROOF. Similar to the proof of theorem 3. D So for graphs derived from symmetric association schemes there is an easier way to calculate the 8-value. As a generalization of Delsarte's result (13) one has THEOREM 5. Let the edge set E of the graph G = (V,E) be the union of some classes of a symmetric association scheme. Then 8(G) ·8(G) lxl. PROOF. Lovasz proved that for all graphs G:8(G) ·8(G)~lxl. Now suppose Eis the union of some classes of an association scheme, as described before theorem 3. Then by theorem 4, 8(G) = Ikak, for some a 0 , ..• ,an, where a 0=1, ak=O for k4M and lk¾Q~~O for u=O, ... ,n. set (32) b = u Then b 0 , ... ,bn~O and b 0=1; furthermore, for kiM (cf. (5)): (33) n I b Pu= 0 u k u= ak-m = 8(G) = O, so b 0 , ... ,bn satisfy the conditions mentioned in the minimum-side of theorem 4, with G instead of G. Also n 13 !xi (34) I b u=O u = 8 (G) • Since, by theorem 4, I b ;,: 8 (G) we have shown that 8 (G) -8 (G) ~ Ix!. D u u Because there are (many) strongly regular graphs that are not vertex-transitive (cf. Seidel [8]) theorem 5 is not included in (17). M.R. Best found the following example of a graph G with 8 1 (G) < 8(G). The points of Gare all vectors in {0,1}6 , two vectors being adjacent iff their Hamming distance is at most 3 (so the edge set is the union of some classes of a Hamming scheme). Then 8 1 (G) = 4 whereas 8(G) = 16 3 • After completing this research I learnt that partially similar results have been obtained, independently, by McEliece, Rodemich & Rumsey [7] (cf. [6]). Their functions aL(G) and 8L(G) are equal to 8 1 (G) and 8(G), respectively. REFERENCES [1] M.R. Best, A.E. Brouwer, F.J. Macwilliams, A.M. Odlyzko and N.J.A. Sloane, "Bounds for binary codes of length less than 25", IEEE Trans. Inform. Theory, vol. IT-24, pp. 81-93, 1978. [2] P. Delsarte, An algebraic approach to the association schemes in coding theory, Philips Res. Reports Suppl. 10, 1973. 14 [3] w. Haemers, "On the problems of Lovasz concerning the Shannon capacity of a graph", IEEE Trans. Inform. Theory (to appear). [4] L. Lovasz, "On the Shannon capacity of a graph", IEEE Trans. Inform. Theory (to appear). [SJ F.J. Macwilliams and N.J.A. Sloane, The theory of error-correcting codes, Amsterdam: North-Holland, 1977. [6] R.J. McEliece, "The bounds of Delsarte and Lovasz, and their applications to coding theory", presented at the International Centre for Mechanical Sciences, Udine, Italy, Summer 1978. [7] R.J. McEliece, E.R. Rodemich and H.C. Rumsey, Jr., "The Lovasz bound and some generalizations", J. Comhinatorics, Information and System Sciences, vol. 3, 1978 (to appear). [SJ J.J. Seidel, "Graphs and two-graphs", in Proc. Fifth Southeastern Conf. on Comh., Graph Theory and Computing, Winnipeg: Utilitas, 1974, pp. 125-143. [9] C.E. Shannon, "The zero-error capacity of a noisy channel", IRE Trans. Inform. Theory, vol. IT-3, pp. 3-15, 1956. 

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