Circulant digraphs integral over number fields

Discrete Mathematics 313 (2013) 821–823

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Circulant digraphs integral over number fields Fei Li Institute of Applied Mathematics, School of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu, Anhui, 233030, PR China

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Article history: Received 30 July 2012 Received in revised form 25 December 2012 Accepted 26 December 2012 Available online 17 January 2013 Keywords: Circulant graph Integral graph Algebraic integer Number field

abstract A number field K is a finite extension of rational number field Q. A circulant digraph integral over K means that all its eigenvalues are algebraic integers of K . In this paper we give the necessary and sufficient condition for circulant digraphs which are integral over a number field K . We also solve Conjecture 3.3 in Xu and Meng (2011) [11] and find that it is affirmative. © 2013 Elsevier B.V. All rights reserved.

1. Introduction and main results Let G be a finite group and S be a subset of G. The Cayley digraph D = D(G, S ) of G with respect to S is a directed graph with vertex set G. For g1 , g2 ∈ G, there is an arc from g1 to g2 if and only if g2 g1−1 ∈ S. If G is the cyclic group Z/nZ with order n, the Cayley digraph D = D(Z/nZ, S ) = D(n, S ) is called a circulant digraph. Its spectrum consists of the eigenvalues of the adjacency matrix of D, namely, the roots of the characteristic polynomial of D, that is spec (D(n, S )) = {λ0 , λ1 , . . . , λn−1 }, 2π i

rs n , the primitive n-th root of unity (see [3,5]). It is obvious that where λr = s∈S ζn (0 6 r 6 n − 1) with ζn = e λr (0 6 r 6 n − 1) are algebraic integers in the cyclotomic field Q(ζn ). A circulant digraph integral over number field K means that all its eigenvalues are algebraic integers of K . The reader is referred to [8,10] for terms left undefined and for the general theory of number fields. Our main purpose in this paper is to characterize the circulant digraphs which are integral over a fixed number field K . We also prove Conjecture 3.3 in [11]. There is a lot of literature studying integral graphs (see, e.g., [1,2,4,6,9]). The main results in the present paper are stated as follows.



A. (See Theorem 1.) The circulant directed graph D(n, S ) is integral over K if and only if S is a union of some Mi ’s. B. (See Corollary 1.) There are at most 2r (n,K ) basic circulant digraphs integral over K on n vertices. Using the same notations as [11], we have the following corollary. C. (See Corollary 2.) (Conjecture 3.3 in [11]) Let D(Zn , S ) be a circulant digraph on n = 2t k(t > 2) vertices, then D(Zn , S ) is Gauss integral if and only if S is a union of the Mi ’s. 2. Constructing a partition of {1, 2, . . . , n − 1} Let L/M be a number field extension. We denote Gal(L/M ) the Galois group of L over M and [L : M ] the dimension of L as vector space over M. Let K be a number field, i.e., a finite extension of rational number  field Q, and d(n) be the set of all the positive proper divisors of n, and for p ∈ d(n), we write n = pg. Denote F = K Q(ζn ) and for p ∈ d(n) define

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Gn (p) = {g |1 6 g < n, gcd(g , n) = p}. It is clear that Gn (p) = pGg (1). Notice that F is an abelian field and D(n, S ) is integral over K if and only if D(n, S ) is integral over F .  Let H ′ = Gal(Q(ζn )/F ), H = Gal(F · Q(ζg )/F ), G′ = Gal(Q(ζg )/F Q(ζg )), and G = Gal(Q(ζg )/Q). By Galois theory (see [7, P. 266]), H ∼ = G′ ⊆ G. Denote π the translation operation of G′ on G defined by π (σ ′ , σ ) = σ ′ σ with σ ′ ∈ G′ , σ ∈ G and f : G −→ (Z/g Z)∗ the map defined by f : σ → a with σ (ζg ) = ζga . The map f is an isomorphism of groups, by which π also can be considered as an operation on (Z/g Z)∗ . For Gn (p) = p(Z/g Z)∗ , we can define an operation π ′ of G′ on Gn (p) by πg′ (x) = pπg (x′ ) with x = px′ ∈ Gn (p), g ∈ G′ and x′ ∈ (Z/g Z)∗ . It is easy to see that the number rp of all orbits under   π ′ is equal to [F Q(ζg ) : Q] and each orbit Mi (1 6 i 6 rp ) has [Q(ζg ) : F Q(ζg )] elements. By Galois theory, H ′ , when restricted to F · Q(ζg ), is equal to H and H ∼ = G′ . Hence π ′ can be regarded as an operation of H ′ on Gn (p), under which

rp

there are the same rp orbits. So Gn (p) is the disjoint union of the distinct  Mi ’s and we can write Gn (p) = i=1 Mi . It is clear that {1, 2, . . . , n − 1} has a partition written as {1, 2, . . . , n − 1} = p∈d(n) Gn (p). So {1, 2, . . . , n − 1} has a new partition

r (n,K ) i=1

Mi , where Mi is an orbit for some p ∈ d(n) and r (n, K ) =



p∈d(n) rp .

3. Proofs of the main results If S is a union of some Mi ’s, we have σ S = S for each σ ∈ H ′ by the construction of Mi ’s. So σ (λr ) = λr (0 6 r 6 n − 1), which shows that λr ∈ F ⊆ K and D(n, S ) is integral over K . We have the following theorem. Theorem 1. The circulant directed graph D(n, S ) is integral over K if and only if S is a union of some Mi ’s. From the discussion above, it suffices to prove the necessity. Before proving, we need the following lemma. We denote by vi the (n − 1)-dimensional vector corresponding to the orbit Mi with 1 at j-th entry for all j ∈ Mi and 0 otherwise and M the set {vi |1 6 i 6 r (n, K )}. Let Γ = (γst ) be an (n − 1)-order square matrix defined by γst = ζnst . Notice that Γ is invertible and Γ vi ∈ F n−1 (1 6 i 6 r (n, K )). Let V = {v ∈ Qn−1 |Γ v ∈ F n−1 } and W be the vector space over Q spanned by the vectors belonging to M. We have the following results. Lemma 1. We have V = W and M is an orthogonal basis of V . Proof. Firstly, we claim that if v ∈ V , then Γ v ∈ W

 Q

F . Let v = (w1 , w1 , . . . , wn−1 )T ∈ V and u = Γ v = (u1 , u1 , . . . ,

un−1 )T ∈ F n−1 . It is enough to show uk = ul if k, l in the same Mi . Recall that Mi is an orbit obtained from Gn (p) for some n−1 n−1 h k wh ζnkh . Because p ∈ d(n) and n = pg. Let f (x) = uk − h=1 wh x be a polynomial in F [x]. So f (ζn ) = 0 for uk = h=1  k, l ∈ Mi , ζnk and ζnl are two primitive g-th roots of unity. So ζnk and ζnl ∈ Q(ζg ), which shows that f (x) ∈ (F Q(ζg ))[x]. By  the construction of Mi , there exists an element σ ∈ Gal(Q(ζg )/F Q(ζg )) such that σ (ζnk ) = ζnl , which means that ζnk is a   conjugate element of ζnl over F Q(ζg ) in Q(ζg ). Hence ζnk and ζnl have the same minimal polynomial over F Q(ζg ), which

lh divides f (x). So f (ζnl ) = 0, that is ul = h=1 wh ζn = uk . Using the result above, it is easy to get Γ (V Q F) ⊆ W   which shows that V F = W F . Therefore V = W and M is an orthogonal basis of V . The proof is completed. Q Q

n−1



 Q

F,



Now we prove Theorem 1. Proof. Consider the vector v ∈ Qn−1 such that it is 1 at the j-th entry for all j ∈ S and 0 otherwise. Since D(n, S ) is a circulant r (n,K ) digraph integral over K , Γ v = (λ1 , λ2 , . . . , λn−1 )T ∈ F n−1 . Hence v ∈ V and by Lemma 1 v = ci vi for some rational i =1 coefficients ci . By the construction of M and v , S is a union of those Mi ’s with ci = 1. The proof is completed.  By Theorem 1, we obtain the following two corollaries. r (n,K )   Given a fixed pair (n, K ), we have the partition {1, 2, . . . , n − 1} = i=1 Mi , where r (n, K ) = Q(ζ n ) : Q]. p∈d(n) [K p r (n,K ) Therefore there are 2 distinct S for circulant digraph integral over K . Corollary 1. There are at most 2r (n,K ) basic circulant digraphs integral over K on n vertices. Replacing K by the specific Gauss field Q(i) and letting n = 2t k(t > 2), it is easy to check that the partition of {1, 2, . . . , r (n,K ) n − 1} described in Part 3 of paper [11] is the same as i=1 Mi as provided above. By Theorem 1, we infer that Conjecture 3.3 in [11] is affirmative. Using the same notations as [11], we have the following corollary. Corollary 2 (Conjecture 3.3 in [11]). Let D(Zn , S ) be a circulant digraph on n = 2t k(t > 2) vertices, then D(Zn , S ) is Gauss integral if and only if S is a union of the Mi ’s. References [1] Omran Ahmadi, Noga Alon, Ian F. Blake, Igor E. Shparlinski, Graphs with integral spectrum, Linear Algebra and its Applications 430 (2009) 547–552. [2] K. Balinska, D. Cvetkovic, Z. Radosavljevic, S. Simic, D. Stevanovic, A survey on integral graphs, Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika 13 (2002) 42–65.

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