Researchers in the past have studied eigenvalues of Cayley digraphs or
graphs. We are interested in characterizing Cayley digraphs on a finite Abelian
group G whose eigenvalues are algebraic integers in a given number field K. And
we succeed in finding a method to do so by proving Theorem 1. Also, the number
of such Cayley digraphs is computed.Comment: 9 page

Li, Fei

English

arXiv

Researchers in the past have studied eigenvalues of Cayley digraphs or graphs. We are interested in characterizing Cayley digraphs on a finite commutative group $G$ whose eigenvalues are algebraic integers in a given number field $K.$ We succeed in finding a method to do so. The number of such Cayley digraphs are computed

Contributions to Discrete Mathematics (E-Journal, University of Calgary)

Volume 15, Number 2, Pages 148–152ISSN 1715-0868A METHOD TO DETERMINE ALGEBRAICALLYINTEGRAL CAYLEY DIGRAPHS ON FINITE ABELIANGROUPFEI LIAbstract. Researchers in the past have studied eigenvalues of Cayleydigraphs or graphs. We are interested in characterizing Cayley digraphson a finite commutative group G whose eigenvalues are algebraic integersin a given number field K. We succeed in finding a method to do so.The number of such Cayley digraphs are computed.1. Introduction and Main ResultsIn this paper, the finite digraphs without loops and multiple edges areconsidered. Let G be a group and let S be a subset of G that does notcontain the identity. Then the Cayley digraph D(G,S) is the digraph withvertex set G and edge set E(D(G,S)) = {gh | hg−1 ∈ S} (see [7]). If S isinverse-closed, then the definition reduces to that of a Cayley graph. WhenG is cyclic, D(G,S) is called a circulant digraph. For the prerequisites forCayley digraphs, we refer to [4, 7].A digraph is called algebraically integral over a number field K if itsspectrum consists of some algebraic integers of K. Since 1974, Harary andSchwenk (see [8]) began to research integral graphs, which in fact are al-gebraically integral over the rational number field Q. For some literaturestudying integral Cayley graphs and the spectrums of Cayley digraphs, seee.g., [1, 2, 3, 5, 12, 14].We call D(G,S) Gaussian integral if it is algebraically integral over theGaussian field Q(i). Several authors have studied digraphs with Gaussianspectra (see [6, 9, 15]). In our previous results, algebraically integral circu-lant digraphs were determined completely(see [11]).In this paper, we obtain the necessary and sufficient conditions for Cayleydigraphs on finite abelian groups to be algebraically integral. Also, thenumber of such algebraically integral Cayley digraphs are calculated.Received by the editors December 13, 2015, and in revised form June 14, 2018.2010 Mathematics Subject Classification. 05C20.Key words and phrases. Cayley digraph, integral graph, algebraic integer, abeliangroup.This research was partially supported by the NSSFC (No.13CTJ006).©2020 University of Calgary148ALGEBRAICALLY INTEGRAL CAYLEY DIGRAPHS 149Let n be the order of G. By the theorem of finite abelian groups, G isisomorphic to a direct sum of some cyclic groups. In the following, we assumethat G = C1⊕C2⊕ · · ·⊕Cm, where Ci = Z/niZ (ni not necessary beinga power of a prime number) for some integer ni, 1 ≤ i ≤ m. Here Z is the setof integers. We assume without loss of generality the fixed number field Kis contained in the nth cyclotomic field Q(ζn). The main results are statedas follows:• (Theorem 2.1) The Cayley digraph D(G,S) is algebraically integralover K if and only if S is a union of some orbits Hgi’s.• (Proposition 3.2) For G = (Z/n1Z)⊕(Z/n2Z)⊕ · · ·⊕(Z/nmZ),there are at most 2r(G,K) Cayley digraphs algebraically integral overK.Remark. Based on Theorem 2.1, we can get a method to determine integraland Gaussian integral Cayley digraphs on a finite abelian group completely.2. Proof of Main TheoremLet H = Gal(Q(ζn)/K) be the Galois group of Q(ζn) over K. Since H ⊆Gal(Q(ζn)/Q) ∼= (Z/nZ)∗, for each σ ∈ H, there is an element a ∈ (Z/nZ)∗such that σ(ζn) = ζan. Notice that n = n1n2 · · ·nm. For each ni, we definea group operation pii of H on Z/niZ by pii(σ, x) = σ(x) = ax (mod ni) forx ∈ Z/niZ. Then we have a operation of H on G by σ(x1, x2, . . . , xm) =(pi1(σ, x1), pi2(σ, x2), . . . , pim(σ, xm)) for (x1, x2, . . . , xm) ∈ G. Using the orbitdecomposition formula, G \ {0} is the disjoint union of the distinct orbits,and we can write G \ {0} = ⊔i∈I Hgi, where I is some index set, and the giare elements of distinct orbits.We obtain a necessary and sufficient condition for the Cayley digraphD(G,S) to be algebraically integral over K as in the following theorem.Theorem 2.1. The Cayley digraph D(G,S) is algebraically integral over Kif and only if S is a union of some orbits Hgi’s.In the following, we give a proof of Theorem 2.1. Let Ĝ denote thegroup of multiplicative homomorphisms from G to C∗ (see [13]), where Cis the complex field. Then we have an isomorphism G ∼= Ĝ. Also we setG = {hu|hu = (hu1, hu2, · · · , hum), 1 ≤ u ≤ n}. For later use, we supposehn = (0, 0, · · · , 0). For each hu ∈ G, denote by χu ∈ Ĝ the homomorphismsatisfying χu(hv) =∏mi=1 ζhuihvini , hv ∈ G. In fact, χu is also called a char-acter of G. The spectrum of the Cayley digraph D(G,S) is given by [2],spec(D(G,S)) = {λ1, λ2, . . . , λn}, where λu =∑hv∈S χu(hv).Proposition 2.2. If S is a union of some orbits Hgi’s, then D(G,S) isalgebraically integral over K.Proof. For every element σ ∈ H and each orbit Hgi, we have σHgi ={σ(hgi) | h ∈ H} = Hgi. So σS = S, since S is a union of some orbits Hgi’s.150 FEI LIThusσ(λu) =∑hv∈Sσ(χu(hv)) =∑hv∈Sσ(m∏i=1ζhuihvini ) =∑hv∈Sm∏i=1σ(ζhuihvini )=∑hv∈Sm∏i=1ζhuiσ(hvi)ni =∑hv∈Sχu(σ(hv)) = λu.By the Galois theory (see [10]) for finite Galois extensions, we get λu ∈ K.Notice that λu are algebraic integers, hence D(G,S) is algebraically integralover K. In the other hand, we have the following proposition.Proposition 2.3. If D(G,S) is algebraically integral over K, then S is aunion of some orbits Hgi’s .Before the proof, we require some preparation. Let Γ be an (n− 1)-ordersquare matrix. Namely, we suppose Γ = (γuv), where γuv is the entry of theuth row and vth column, with γuv = χu(hv). For the matrix Γ, we have thefollowing lemma.Lemma 2.4. The matrix Γ is nonsingular.Proof. Let χ be a nontrivial character of finite abelian group G, we knowthat∑g∈G χ(g) = 0. Then the sum of all entries in each row of Γ is −1.Therefore, to prove this lemma, it suffices to show the n× n block matrixΓ′ =(Γ 11 1)is invertible.Let F be the space of complex valued class functions on G. All the ele-ments of Ĝ form an orthonormal basis of F (see [13]). Suppose there existn complex numbers ku(1 ≤ u ≤ n) such that∑nu=1 kuRu = (0, 0, . . . , 0),where Ru is the uth row of the matrix Γ′. Then we have the class function∑nu=1 kuχu = 0. So ku = 0 for all u, 1 ≤ u ≤ n, which shows that the rowvectors of Γ′ are linearly independent. Hence Γ′ is invertible. Let τ be an (n− 1)-dimension column vector,τ = (v1, v2, · · · , vn−1)T ,with vi = 1 for hi ∈ S and 0, otherwise. It is easy to see thatΓτ = (λ1, λ2, · · · , λn−1)T .Let τi be the (n − 1)-dimension column vector for the orbit Hgi just as τfor S. We denote by W the vector space {ω ∈ Kn−1 | Γω ∈ Kn−1} and byV ⊂ Kn−1 the vector space spanned by the vectors {τi|i ∈ I}. We obtainthe following lemma for W and V.Lemma 2.5. W and V are the same vector space.ALGEBRAICALLY INTEGRAL CAYLEY DIGRAPHS 151Proof. By Proposition 2.2, Γτi ∈ Kn−1, i ∈ I. So V ⊂ W. Let ω ∈ W andω = (ω1, ω2, . . . , ωn−1)T , µ = Γω = (µ1, µ2, . . . , µn−1)T . First, we show thatµu = µv if hu, hv are in the same orbit Hgi. Because hu, hv ∈ Hgi, thereexists an element σ ∈ H such that σ(hu) = hv, i.e., σ(hu1, hu2, . . . , hum) =(hv1, hv2, . . . , hvm). In fact,µu = σ(µu) = σ(n−1∑k=1ωkχu(hk)) =n−1∑k=1ωkσ(χu(hk))=n−1∑k=1ωkm∏l=1σ(ζhulhklnl ) =n−1∑k=1ωkm∏l=1ζσ(hul)hklnl=n−1∑k=1ωkm∏l=1ζhvlhklnl =n−1∑k=1ωkχv(hk) = µv,which implies that Γ(W ) ⊂ V. Notice that the matrix Γ is nonsingular byLemma 2.4 and V ⊂W. Hence dimW = dimV, and W = V. Now it comes to prove Proposition 2.3.Proof of Proposition 2.3. SinceD(G,S) is algebraically integral overK,Γτ ∈Kn−1. We have τ ∈ W. By Lemma 2.5, τ ∈ V and τ = ∑i∈I ciτi for somecoefficients ci ∈ K. By the construction of τ and the τi’s, we conclude thatS is the union of the Hgi’s with ci = 1. The proof is completed. Merging Proposition 2.2 and Proposition 2.3 together, we obtain the resultof Theorem 2.1.3. Calculating the number of algebraically integral CayleydigraphsLet Gni(1 ≤ i ≤ m) be the set of all the orbits of Z/niZ under H.Denote by PG the collection of Cartesian product {P = p1 × p2 × · · · × pm |P 6= 0, pi ∈ Gni}. For a Cartesian product P ∈ PG, choose one elementρ ∈ P, ρ = (a1, a2, . . . , am). Denote by Q(P ) the cyclotomic fieldQ(ζa1n1 , ζa2n2 , · · · , ζamnm) = Q(ζa1n1) ·Q(ζa2n2) · · ·Q(ζamnm),and by [Q(P ) : Q(P ) ∩ K] the dimension of Q(P ) as a vector space overQ(P ) ∩ K. Here |P | represents the cardinal number of P. It is easy to seethat Q(P ) is well-defined. By Galois theory (see [10]) in number fields, wehave |P | = ∏mi=1[Q(ζaini) : Q(ζaini) ∩K].Let f be the map Gal(Q(P )/Q(P )∩K)→∏mi=1 Gal(Q(ζaini)/Q(ζaini)∩K)by restriction, namely, σ 7→ (σ |Q(ζa1n1 ), σ |Q(ζa2n2 ), . . . , σ |Q(ζamnm )). It is easy tosee that f is injective. So every orbit contained in the Cartesian product Phas [Q(P ) : Q(P )∩K] elements, which implies that P can be divided equallyinto |P |/[Q(P ) : Q(P ) ∩K] orbits under H. Totally, we get the followingLemma.152 FEI LILemma 3.1. Under the operation of H, the Cartesian product P describedas above is divided into ∏mi=1[Q(ζaini) : Q(ζaini) ∩K][Q(P ) : Q(P ) ∩K]orbits.Letr(G,K) = ΣP∈PG∏mi=1[Q(ζaini) : Q(ζaini) ∩K][Q(P ) : Q(P ) ∩K] .By Lemma 3.1, r(G,K) is the orbit number of group operation of H onG \ {0}. So we obtain the following proposition by Theorem 2.1.Proposition 3.2. For G = (Z/n1Z)⊕(Z/n2Z)⊕ · · ·⊕(Z/nmZ), thereare at most 2r(G,K) Cayley digraphs algebraically integral over K.References1. R. C. Alperin and B. L. Peterson, Integral sets and Cayley graphs of finite groups,Electron. J. Combin. 19 (2012), #P44.2. K. Babai, Spectra of Cayley graphs, J. Comb. Theory Ser. B 27 (1979), 180–189.3. K. Balinska, D. Cvetkovic, Z. Radosavljevic, S. Simic, and D. Stevanovic, A survey onintegral graphs, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 13 (2002), 42–65.4. N. Biggs, Algebraic graph theory, North–Holland, Amsterdam, 1985.5. W. G. Bridges and R. A. Mena, Rational G-matrices with rational eigenvalues, J.Comb. Theory Ser. A 32 (1982), no. 2, 264–280.6. F. Esser and F. Harary, Digraphs with real and Gaussian spectra, Discrete Appl. Math.2 (1980), 113–124.7. C. Godsil and G. Royle, Algebraic graph theory, Springer–Verlag, New York, 2001.8. F. Harary and A. Schwenk, Which graphs have integral spectra, Graphs Combin.(R. Bari and F. Harary, eds.), Springer–Verlag, Berlin, 1974, pp. 45–51.9. X. D. Hou, On the G-matrices with entries and eigenvalues in Q(i), Graphs Combin.8 (1992), 53–64.10. S. Lang, Algebra, 3rd ed., Springer–Verlag, New York, 2002.11. F. Li, Circulant digraphs integral over number fields, Discrete Math. 313 (2013), 821–823.12. J. Sander and T. Sander, The exact maxamal energy of integral circulant graphs withprime power order, Contrib. Discrete Math. 8 (2013), no. 2, 19–40.13. J. P. Serre, Linear representations of finite groups, Springer–Verlag, New York, 1977.14. W. So, Integral circulant graphs, Discrete Math. 306 (2005), 153–158.15. Y. Xu and J. X. Meng, Gaussian integral circulant digraphs, Discrete Math. 311(2011), 45–50.School of Statistics and Applied Mathematics, Anhui University of Financeand Economics, Bengbu, 233030, Anhui, P. R. ChinaE-mail address: cczxlf@163.com

A method to determine algebraically integral Cayley digraphs on finite abelian group

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A method to determine algebraically integral Cayley digraphs on finite Abelian group

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