arXiv:1001.0046v1 [math.CO] 31 Dec 2009THE CAUCHY-SCHWARZ INEQUALITY IN CAYLEY GRAPH AND TOURNAMENT STRUCTURES ON FINITE FIELDS STEPHAN FOLDES AND L ´ASZL´O MAJOR Abstract. The Cayley graph construction provides a natural grid struc ture on a finite vector space over a field of prime or prime square car dinality, where the characteristic is congruent to 3 modulo 4, in addition to the quadratic residue tournament structure on the prime subfield. Distanc e from the null vector in the grid graph defines a Manhattan norm. The Hermiti an inner prod- uct on these spaces over finite fields behaves in some respects similarly to the real and complex case. An analogue of the Cauchy-Schwarz ine quality is valid with respect to the Manhattan norm. With respect to the non-t ransitive order provided by the quadratic residue tournament, an analogue o f the Cauchy- Schwarz inequality holds in arbitrarily large neighborhoo ds of the null vector, when the characteristic is an appropriate large prime. 1.Manhattan norms and grid graphs We consider the finite fields FpandFp2of prime and prime square cardinality, wherep≡3 mod 4. The field Fp2has a natural graph structure with the field elements as vertices, two distinct vertices u,zbeing adjacent if ( z−u)4= 1. The subfieldFpofFp2then induces a subgraph in which xandyare adjacent if and only if (y−x)2= 1.The graph Fp2is isomorphic to the Cartesian square C2 p=Cp/squareCp, whereCpis ap-cycle and within Fp2the induced subgraph Fpis itself a p-cycle. Clearly the graph Fp2is not planar, but can be drawn as a grid on the torus. For any connected graph whose vertex set is a group, the distanc e of any vertex zfrom the identity element of the group is called the normofz, denoted N(z). In general, distances and norms measured in connected subgraph s induced by sub- groups can be larger than distances and norms measured with refe rence to the whole graph. However, with respect to the distance-preserving s ubgraph induced byFpinFp2, the norm of any z∈Fpis the same as its norm with respect to the whole graph Fp2: this is simply the length of the shortest path from 0 to zin the cycle induced by Fp. Forq=porq=p2, then-dimensional vector space Fn qis also endowed with the Cartesian product graph structure Fq/square···/squareFqisomorphic to Cn porC2n p. The norm of a vector v= (v1,...,v n) inFn qis then equal to the sum N(v1)+···+N(vn) and we also write N(v) for this vector norm. The Gaussian integers Z[i] also constitute a graph in which uandzare adjacent if and only if ( z−u)4= 1. Date: Dec 24, 2009. 1991Mathematics Subject Classification. Primary 05C12, 05C20, 05C25; Secondary 06F99, 11T99. Key words and phrases. Cauchy-Schwarz inequality, triangle inequality, submult iplicativity, finite field, quadratic field extension, quadratic residue to urnament, grid graph, Manhattan dis- tance, discrete norm, Gaussian integers, graph product, gr aph quotient, Cayley graph. 12 STEPHAN FOLDES AND L ´ASZL´O MAJOR It iseasytoseethatthenorminthis infinite Manhattan grid satisfiesthetriangle and submultiplicative inequalities N(u+z)≤N(u)+N(z) N(uz)≤N(u)N(z) To emphasize that the norms on Fp2,Fn p2andZ[i] are understood with reference to the specific grid graphs defined above, we call these norms Manhattan norms . Throughout this paper we think of Fp2as the ring quotient Z[i]/(p). 2.Graph quotients and Cayley graphs Given a graph G(undirected, with possible loops) on vertex set Vand an equi- valence relation ≡onV, thequotient graph G/≡is defined as follows: the vertices ofG/≡are the equivalence classes of ≡, and classes A,Bare adjacent if for some a∈A,b∈B, the elements a,bare adjacent in G. Note that the distance of Ato Bin the quotient graph is at most equal to, but possibly less than the m inimum of the distances atobfor alla∈A,b∈B. Note also that G/≡can have loops even ifGhas not. Given a group Gwith identity element eand a set Γ of group elements that generates G, the(left) Cayley graph C(G,Γ) ofGwith respect to Γ has vertex set G, elements a,b∈Gbeing considered adjacent if ab−1orba−1belongs to Γ. For each congruence ≡of the group G, corresponding to some normal subgroup H, Γ yields a generating set Γ ≡ofG/≡consisting with those classes of ≡that intersect Γ. The graph quotient of C(G,Γ) by the equivalence ≡coincides with the Cayley graph of the quotient graph G/≡with respect to Γ ≡. ForR⊆Ginducing a connected subgraph [ R] inC(G,Γ), denote by dR(x,y) the distance function of the subgraph [ R]. Denoting by xHtheH-coset of any x∈G, this relates to norms in C(G,Γ) andC(G,Γ)/≡as follows: for all x∈R, dR(x,e)≥N(x)≥N(xH) Both inequalities can be strict. However, we have: Cayley Graph Quotient Lemma. Let a group Gwith identity ebe generated byΓ⊆G, and consider any normal subgroup Hwith corresponding congruence ≡. There is a set R⊆Ghaving exactly one element in common with each congruence class modulo H, and such that for every x∈R dR(x,e) =N(x) =N(xH) Proof.We can define the unique (representative) element r(A)∈R∩Afor each cosetAby induction on the distance d(H,A) ofAfromHinC(G,Γ)/≡. Let r(H) =e. Assuming r(A) defined for all Awithd(H,A)≤m, let a coset Bhave distance m+1 from H. Choose any coset Aadjacent to Bwithd(H,A) =mand elements a∈A,b∈Bthat are adjacent in C(G,Γ). Letr(B) =ba−1r(A)./square We can apply the above lemma in the case where G=Z[i], Γ ={1,i}and H=pZ[i] ={pa+pbi:a,b∈Z}foraprimeinteger p≡3 mod 4. Now C(G,Γ)and C(G,Γ)/≡are the Manhattan grid graphs on Z[i] andZ[i]/H=Fp2, respectively. Referringtothe set Rofrepresentativesinthe lemma, forany H-cosetsX,Yletx,y be the unique elements in X∩R,Y∩R. Asxy∈XY, we have N(XY)≤N(xy). By the submultiplicative inequality in Z[i] we have N(xy)≤N(x)N(y). Using the3 lemmawehave N(x)N(y) =N(X)N(Y). Thisyieldsasubmultiplicativeinequality inFp2and a similar reasoning on the coset X+Yyields a triangle inequality: Triangle and Submultiplicative Inequalities in Fp2.For allu,zinFp2 N(u+z)≤N(u)+N(z) N(uz)≤N(u)N(z) /square This indicates that Manhattan distance provides a well-behaved not ion of neigh- borhood of 0 in the finite fields Fp2. 3.Squares in Fpand non-transitive order For each prime p≡3 mod 4 the quadratic residue tournament onFpis the directed graph with vertex set Fpin which there is an arrowfrom vertex xto vertex yify−xis a non-zero square in Fp, in which case we write x