Báo cáo toán học: "Integral Cayley graphs defined by greatest common divisors" pot

Integral Cayley graphs defined by greatest common divisors Walter Klotz Institut f¨ur Mathematik Tech nische Universit¨at Clausthal, Germany klotz@math.tu-clausthal.de Torsten Sander Fakult¨at f¨ur Informatik Ostfalia Hochschule f¨ur angewandte Wissenschaften, Germany t.sander@ostfalia.de Submitted: Dec 6, 2010; Accepted: Apr 12, 2011; Published: Apr 21, 2011 Mathematics Subject Classification: 05C25, 05C50 Abstract An undirected graph is called integral, if all of its eigenvalues are integers. Let Γ = Z m 1 ⊗ · · · ⊗ Z m r be an abelian group represented as the direct product of cyclic groups Z m i of order m i such that all greatest common divisors gcd(m i , m j ) ≤ 2 for i = j. We prove that a Cayley graph Cay(Γ, S) over Γ is integral, if and only if S ⊆ Γ belongs to the the Boolean algebra B(Γ) generated by the subgroups of Γ. It is also shown th at every S ∈ B(Γ) can be characterized by greatest common divisors. 1 Introduction The greatest common divisor of nonnegative integers a and b is denoted by gcd(a, b). Let us agree upon gcd(0, b) = b. If x = (x 1 , . . . , x r ) and m = (m 1 , , m r ) are tuples of nonnegative integers, then we set gcd(x, m) = (d 1 , . . . , d r ) = d, d i = gcd(x i , m i ) for i = 1 , . . . , r. For an integer n ≥ 1 we denote by Z n the additive group, respectively the ring of integers modulo n, Z n = {0, 1, . . . , n−1} as a set. Let Γ be an (additive) abelian group represented as a direct product of cyclic groups. Γ = Z m 1 ⊗ · · · ⊗ Z m r , m i ≥ 1 for i = 1, . . . , r the electronic journal of combinatorics 18 (2011), #P94 1 Suppose that d i is a divisor of m i , 1 ≤ d i ≤ m i , for i = 1, . . . , r. For the divisor tuple d = (d 1 , . . . , d r ) of m = (m 1 , . . . , m r ) we define the gcd-set of Γ with respect to d, S Γ (d) = {x = (x 1 , . . . , x r ) ∈ Γ : gcd(x, m) = d}. If D = {d (1) , . . . , d (k) } is a set of divisor tuples of m, then the gcd-set of Γ with respect to D is S Γ (D) = k  j=1 S Γ (d (j) ). In Section 2 we realize tha t the g cd-sets of Γ constitute a Boolean subalgebra B gcd (Γ) of the Boolean algebra B(Γ) generated by the subgroups of Γ. The finite abelian group Γ is called a gcd-group, if B gcd (Γ) = B(Γ). We show that Γ is a gcd-group, if and only if it is cyclic or isomorphic to a group of the for m Z 2 ⊗ · · · ⊗ Z 2 ⊗ Z n , n ≥ 2. Eigenvalues of an undirected graph G are the eigenvalues of an arbitrary adja cency matrix of G. Harary and Schwenk [8] defined G to be integral, if all of its eigenvalues are integers. For a survey of integral graphs see [3]. In [2] the numb er of integral graphs on n vertices is estimated. Known characterizations of integral graphs are restricted to certain graph classes, see e.g. [1]. Here we concentrate on integral Cayley graphs over gcd-groups. Let Γ be a finite, additive group, S ⊆ Γ, 0 ∈ S, − S = {−s : s ∈ S} = S. The undirected Cayley graph over Γ with shift set S, Cay(Γ, S), has vertex set Γ. Vertices a, b ∈ Γ are adja cent, if and only if a − b ∈ S. For general properties of Cayley graphs we refer to Godsil and Royle [7] or Biggs [5]. We define a gcd-graph to b e a Cayley graph Cay(Γ, S) over an abelian group Γ = Z m 1 ⊗· · ·⊗Z m r with a gcd-set S of Γ. All gcd-graphs are shown to be integral. They can be seen as a generalization o f unitary Cayley graphs and of circulant graphs, which have some remarkable properties and applications (see [4], [9], [11], [15]). In our paper [10] we proved for an abelian group Γ and S ∈ B(Γ), 0 ∈ S, that the Cayley graph Cay(Γ, S) is integral. We conjecture the conver se to be true fo r finite abelian groups in general. This can be confirmed for cyclic groups by a theorem of So [16]. In Section 3 we extend the result of So to gcd-groups. A Cayley graph Cay(Γ, S) over a gcd-g r oup Γ is integral, if and only if S ∈ B(Γ). 2 gcd-Groups Throughout this section Γ denotes a finite abelian group given as a direct product o f cyclic groups, Γ = Z m 1 ⊗ · · · ⊗ Z m r , m i ≥ 1 for i = 1, . . . , r. Theorem 1. The family B gcd (Γ) of gcd-sets of Γ constitutes a Boolean subalgebra of the Boolean algebra B(Γ) generated by the subgroups of Γ. the electronic journal of combinatorics 18 (2011), #P94 2 Proof. First we confirm t hat B gcd (Γ) is a Boolean algebra with respect to the usual set operations. From S Γ (∅) = ∅ we know ∅ ∈ B gcd (Γ). If D 0 denotes the set of all (positive) divisor tuples of m = (m 1 , . . . , m r ) then we have S Γ (D 0 ) = Γ, which implies Γ ∈ B gcd (Γ). As B gcd (Γ) is obviously clo sed under the set operations union, intersection and forming the complement, it is a Boolean algebra. In order to show B gcd (Γ) ⊆ B( Γ ), it is sufficient to prove for an arbitrary diviso r tuple d = (d 1 , . . . , d r ) of m = (m 1 , . . . , m r ) t hat S Γ (d) = {x = (x 1 , . . . , x r ) ∈ Γ : gcd(x, m) = d} ∈ B(Γ). Observe that d j = m j forces x j = 0 for x = (x i ) ∈ S Γ (d). If d i = m i for every i = 1, . . . , r then S Γ (d) = {(0, 0, . . . , 0)} ∈ B(Γ). So we may assume 1 ≤ d i < m i for at least one i ∈ {1, . . . , r}. For i = 1, . . . , r we define δ i = d i , if d i < m i , and δ i = 0, if d i = m i , δ = (δ 1 , . . . , δ r ). For a i ∈ Z m i we denote by [a i ] the cyclic group generated by a i in Z m i . One can easily verify the following representation of S Γ (d): S Γ (d) = [δ 1 ] ⊗ · · · ⊗ [δ r ] \  λ 1 , ,λ r ([λ 1 δ 1 ] ⊗ · · · ⊗ [λ r δ r ]). (1) In (1) we set λ i = 0 , if δ i = 0. For i ∈ {1, . . . , r} and δ i > 0 the r ange of λ i is 1 ≤ λ i < m i δ i such that gcd(λ i , m i δ i ) > 1 for at least one i ∈ {1, . . . , r}. As [δ 1 ]⊗· · ·⊗[δ r ] and [λ 1 δ 1 ]⊗· · ·⊗[λ r δ r ] are subgroups of Γ, (1) implies S Γ (d) ∈ B(Γ). A g cd-g r aph is a Cayley graph Cay(Γ, S Γ (D)) over an abelian group Γ = Z m 1 ⊗ · · · ⊗ Z m r with a gcd-set S Γ (D) as its shift set. In [10] we proved that for a finite abelian group Γ and S ∈ B(Γ), 0 ∈ S, the Cayley graph Cay(Γ, S) is integral. Therefore, Theorem 1 implies the following corollary. Corollary 1. Every gcd-graph Cay(Γ, S Γ (D)) is integral. We remind that we call Γ a gcd-group, if B gcd (Γ) = B(Γ). For a = (a i ) ∈ Γ we denote by [a] the cyclic subgroup of Γ generated by a. Lemma 1. Let Γ be the abelian group Z m 1 ⊗ · · · ⊗ Z m r , m = (m 1 , . . . , m r ). Then Γ is a gcd-group, if and only if for every a ∈ Γ, gcd(a, m) = d implies S Γ (d) ⊆ [a]. Proof. Let Γ b e a gcd-g r oup, B gcd (Γ) = B(Γ). Then every subgroup of Γ, especially every cyclic subgroup [a] is a gcd-set of Γ. This means [a] = S Γ (D) for a set D of divisor tuples of m. Now gcd(a, m) = d implies d ∈ D a nd therefore S Γ (d) ⊆ S Γ (D) = [a]. To prove the converse assume that the co ndition in Lemma 1 is satisfied. Let H be an arbitrary subgroup of Γ. We show H ∈ B gcd (Γ). Let a ∈ H, gcd(a, m) = d. Then our assumption implies a ∈ S Γ (d) ⊆ [a] ⊆ H, H =  d∈D S Γ (d) = S Γ (D) ∈ B gcd (Γ), where D = {gcd(a, m) : a ∈ H}. the electronic journal of combinatorics 18 (2011), #P94 3 For integers x, y, n we express by x ≡ y mod n that x is congruent to y modulo n. Lemma 2. Every cyclic group Γ = Z n , n ≥ 1, is a gcd-group. Proof. As the lemma is trivially true for n = 1, we assume n ≥ 2. Let a ∈ Γ, 0 ≤ a ≤ n−1, gcd(a, n) = d. According to Lemma 1 we have to show S Γ (d) ⊆ [a]. Again, to avoid the trivial case, assume a ≥ 1. From gcd(a, n) = d < n we deduce a = αd, 1 ≤ α < n d , gcd (α, n d ) = 1. As the order of a ∈ Γ is ord(a) = n/d, the cyclic group generated by a is [a] = {x ∈ Γ : x ≡ (λα)d mod n, 0 ≤ λ < n d }. Finally, we conclude [a] ⊇ {x ∈ Γ : x ≡ (λα)d mod n, 0 ≤ λ < n d , gcd(λ, n d ) = 1} = { x ∈ Γ : x ≡ µd mod n, 0 ≤ µ < n d , gcd(µ, n d ) = 1} = S Γ (d). Lemma 3. If Γ = Z m 1 ⊗ · · · ⊗ Z m r , r ≥ 2, is a gcd-group, then gcd(m i , m j ) ≤ 2 for every i = j, i, j = 1, . . . , r. Proof. Without loss of generality we concentrate on gcd(m 1 , m 2 ). We may assume m 1 > 2 and m 2 > 2. Consider a = (1, 1, 0, . . . , 0) ∈ Γ and b = (m 1 − 1, 1, 0, . . . , 0) ∈ Γ. For m = (m 1 , . . . , m r ) we have gcd(a, m) = (1, 1, m 3 , . . . , m r ) = gcd(b, m). By Lemma 1 the element b must belong to the cyclic group [a]. This requires the existence of an integer λ, b = λa in Γ, or equivalently λ ≡ −1 mod m 1 and λ ≡ 1 mod m 2 . Therefore, integers k 1 and k 2 exist satisfying λ = −1 + k 1 m 1 and λ = 1 + k 2 m 2 , which implies k 1 m 1 − k 2 m 2 = 2 and gcd(m 1 , m 2 ) divides 2. The next two lemmas will enable us to prove the converse of Lemma 3. Lemma 4. Let a 1 , . . . , a r , g 1 , . . . , g r be integers, r ≥ 2, g i ≥ 2 for i = 1, . . . , r. Moreover, assume gcd(g i , g j ) = 2 for every i = j, i, j = 1, . . . , r. The system of congruences x ≡ a 1 mod g 1 , . . . , x ≡ a r mod g r (2) is solvable, if and only if a i ≡ a j mod 2 for every i, j = 1, . . . , r. (3) If the system is solvable, then the so l ution consists of a unique residue class modulo (g 1 g 2 · · · g r )/2 r−1 . the electronic journal of combinatorics 18 (2011), #P94 4 Proof. Suppo se that x is a solution of (2). As every g i is even, the necessity of condition (3) follows by a i ≡ x mod 2 for i = 1, . . . , r. Assume now that condition (3) is satisfied. We set κ = 0, if every a i is even, and κ = 1, if every a i is odd. By x ≡ a i mod 2 we have x = 2y + κ for an integer y. The congruences ( 2) can be equivalently transformed to y ≡ a 1 − κ 2 mod g 1 2 , . . . , y ≡ a r − κ 2 mod g r 2 . (4) As gcd((g i /2), (g j /2)) = 1 for i = j, i, j = 1, . . . , r, we know by the Chinese remainder theorem [14] that the system (4) has a unique solution y ≡ h mod (g 1 · · · g r )/2 r . This implies for the solution x of (2): x = 2y + κ ≡ 2h + κ mod g 1 · · · g r 2 r−1 . Lemma 5. Let a 1 , . . . , a r , m 1 , . . . , m r be integers, r ≥ 2, m i ≥ 2 f or i = 1 , . . . , r. More- over, assume gcd(m i , m j ) ≤ 2 for ev ery i = j, i, j = 1, . . . , r. The system of congruences x ≡ a 1 mod m 1 , . . . , x ≡ a r mod m r (5) is solvable, if and only if a i ≡ a j mod 2 for every i = j, m i ≡ m j ≡ 0 mod 2, i, j = 1, . . . , r. (6) Proof. If at most o ne of the integers m i , i = 1, . . . r, is even then gcd(m i , m j ) = 1 for every i = j, i, j = 1, . . . , r, and system (5) is solvable. Therefore, we may assume that m 1 , . . . , m k are even, 2 ≤ k ≤ r, and m k+1 , . . . , m r are o dd, if k < r. Now we split system (5) into two systems. x ≡ a 1 mod m 1 , . . . , x ≡ a k mod m k (7) x ≡ a k+1 mod m k+1 , . . . , x ≡ a r mod m r (8) By Lemma 4 the solvability of (7) requires (6). If this condition is satisfied, then (7) has a unique solution x ≡ b mod (m 1 · · · m k )/2 k−1 by Lemma 4. System (8) has a unique solution x ≡ c mod (m k+1 · · · m r ) by the Chinese remainder theorem, because gcd(m i , m j ) = 1 for i = j, i, j = k + 1, . . . , r. So the orig inal system (5) is equivalent to x ≡ b mod m 1 · · · m k 2 k−1 and x ≡ c mod (m k+1 · · · m r ). (9) As gcd((m 1 · · · m k ), (m k+1 · · · m r )) = 1, the Chinese remainder theorem can be applied once more to arrive at a unique solution x ≡ h mod (m 1 · · · m r )/2 k−1 of (9) and (5). the electronic journal of combinatorics 18 (2011), #P94 5 Theorem 2. The abelian group Γ = Z m 1 ⊗ · · · ⊗ Z m r is a gcd-group, if and only if gcd(m i , m j ) ≤ 2 for ev ery i = j, i, j = 1, . . . , r. (10) Proof. As every cyclic g r oup is a gcd-group by Lemma 2, we may assume r ≥ 2. Then (10) necessarily holds for every gcd-group Γ by L emma 3. Suppose now that Γ sa t isfies (10). Let a = (a 1 , . . . , a r ) and b = (b 1 , . . . , b r ) be elements of Γ, m = (m 1 , . . . , m r ), and gcd(a, m) = d = (d 1 , . . . , d r ) = gcd(b, m). (11) According to Lemma 1 we have to show that b belongs to the cyclic group [a] generated by a. Now b ∈ [a] is equivalent to the existence of an integer λ which solves the f ollowing system of congruences: b 1 ≡ λa 1 mod m 1 , . . . , b r ≡ λa r mod m r . (12) If d i = m i then a i = b i = 0 and the congruence b i ≡ λa i mod m i becomes trivial. Therefore, we assume 1 ≤ d i < m i for every i = 1, . . . , r. By (11) we have gcd(a i , m i ) = gcd(b i , m i ) = d i , which implies the existence of integers µ i , ν i satisfying a i = µ i d i , 1 ≤ µ i < m i d i , gcd (µ i , m i d i ) = 1; b i = ν i d i , 1 ≤ ν i < m i d i , gcd(ν i , m i d i ) = 1. (13) Inserting a i and b i for i = 1, . . . , r from (13) in (12) yields ν 1 d 1 ≡ λµ 1 d 1 mod m 1 , . . . , ν r d r ≡ λµ r d r mod m r . We divide the i-th congruence by d i and multiply with κ i , the multiplicative inverse of µ i modulo m i /d i . Thus each congruence is solved for λ and we arrive a t the following system equivalent to (12). λ ≡ κ 1 ν 1 mod m 1 d 1 , . . . , λ ≡ κ r ν r mod m r d r (14) To prove the solvability of (14) by Lemma 5 we first notice that gcd (m i , m j ) ≤ 2 for i = j implies gcd((m i /d i ), (m j /d j )) ≤ 2 for i, j = 1, . . . , r. Suppose now that m i /d i is even. As gcd(µ i , (m i /d i )) = 1, see (13), µ i must be odd. Also κ i is odd because of gcd(κ i , (m i /d i )) = 1. If for i = j both m i /d i and m j /d j are even, then both κ i ν i and κ j ν j are o dd, because all involved integers κ i , ν i , κ j , ν j are o dd. We conclude now by Lemma 5 that (14) is solvable, which finally confirms b ∈ [a]. Lemma 6. Let Γ = Z m 1 ⊗ · · · ⊗ Z m r be isomorphic to Γ ′ = Z n 1 ⊗ · · · ⊗ Z n s , Γ ≃ Γ ′ . Then Γ is a gcd-group, if an d only if Γ ′ is a gcd-group. Proof. We may assume m i ≥ 2 fo r i = 1, . . . , r and n j ≥ 2 for j = 1, . . . , s. For the following isomorphy and more basic facts about abelian groups we refer to Cohn [6]. Z pq ≃ Z p ⊗ Z q , if gcd(p, q) = 1 (15) the electronic journal of combinatorics 18 (2011), #P94 6 If the positive integer m is written as a product of pairwise coprime prime powers, m = u 1 · · · u h , then Z m ≃ Z u 1 ⊗ · · · ⊗ Z u h . (16) We apply the decomposition (16) to every factor Z m i , i = 1, . . . , r, of Γ a nd to every factor Z n j , j = 1, . . . , s, of Γ ′ . So we obtain the “prime power representation” Γ ∗ , which is the same for Γ and fo r Γ ′ , if the factors are e. g. arranged in ascending order. Γ ≃ Γ ∗ = Z q 1 ⊗ · · · ⊗ Z q t ≃ Γ ′ , q j a prime power for j = 1, . . . , t The following equivalences are easily checked. gcd(m i , m j ) ≤ 2 for every i = j, i, j = 1, . . . , r ⇔ gcd(q k , q l ) ≤ 2 for every k = l, k, l = 1, . . . , t ⇔ gcd(n i , n j ) ≤ 2 for every i = j, i, j = 1, . . . , s (17) Theorem 2 and (17) imply that Γ is a gcd-group, if and only if Γ ∗ , respectively Γ ′ , is a gcd-group. Every finite abelian group ˜ Γ can be represented as the direct product of cyclic groups. ˜ Γ ≃ Z m 1 ⊗ · · · ⊗ Z m r = Γ (18) We define ˜ Γ to be a gcd-group, if Γ is a gcd-group. Although the representation (18) may not be unique, this definition is correct by Lemma 6. Theorem 3. The finite abelia n group Γ is a g cd-group, if and only i f Γ is cyclic or Γ is isomorphic to a group Γ ′ of the form Γ ′ = Z 2 ⊗ · · · ⊗ Z 2 ⊗ Z n , n ≥ 2. Proof. If Γ is isomorphic to a group Γ ′ as stated in the theorem, then Γ is a gcd-group by Theorem 2. To prove the converse, let Γ be a gcd-group. We may assume that Γ is not cyclic. The prime power representation Γ ∗ of Γ is established as described in the proof of Lemma 6. We start this representation with those orders which are a power of 2, followed possibly by odd orders. Γ ≃ Γ ∗ = Z 2 ⊗ · · · ⊗ Z 2 ⊗ Z 2 α ⊗ Z u 1 ⊗ · · · ⊗ Z u s , α ≥ 1 , u i odd for i = 1, . . . , s (19) Theorem 2 implies that there is at most one order 2 α with α ≥ 2. Moreover, all odd orders u 1 , . . . , u s must be pairwise coprime. As 2 α , u 1 , . . . , u s are pairwise coprime integers, we deduce from (15) that Z 2 α ⊗ Z u 1 ⊗ · · · ⊗ Z u s ≃ Z n for n = 2 α u 1 · · · u s . Now (19) implies Γ ≃ Γ ′ = Z 2 ⊗ · · · ⊗ Z 2 ⊗ Z n . the electronic journal of combinatorics 18 (2011), #P94 7 3 Integral Cayley graphs over gcd-groups The following method to determine the eigenvectors and eigenvalues of Cayley graphs over abelian groups is due to Lov´asz [13], see also our description in [10]. We outline the main features of this method, which will be applied in this section. The finite, additive, abelian group Γ, |Γ| = n ≥ 2, is represented as the direct product of cyclic groups, Γ = Z m 1 ⊗ · · · ⊗ Z m r , m i ≥ 2 for 1 ≤ i ≤ r. (20) We consider the elements x ∈ Γ as elements of the cartesian product Z m 1 × · · · × Z m r , x = (x i ), x i ∈ Z m i = { 0 , 1, . . . , m i − 1}, 1 ≤ i ≤ r. Addition is coordinatewise modulo m i . A character ψ of Γ is a homomorphism from Γ into the multiplicative group of complex n-th roots of unity. Denote by e i the unit vector with entry 1 in position i and entry 0 in every position j = i. A character ψ of Γ is uniquely determined by its values ψ(e i ), 1 ≤ i ≤ r. x = (x i ) = r  i=1 x i e i , ψ(x) = r  i=1 (ψ(e i )) x i (21) The value of ψ(e i ) must be an m i -th root of unity. There are m i possible choices for this value. Let ζ i be a fixed primitive m i -th root of unity fo r every i, 1 ≤ i ≤ r. For every α = (α i ) ∈ Γ a character ψ α can be uniquely defined by ψ α (e i ) = ζ α i i , 1 ≤ i ≤ r. (22) Combining (21) and (22) yields ψ α (x) = r  i=1 ζ α i x i i for α = (α i ) ∈ Γ and x = (x i ) ∈ Γ. (23) Thus all |Γ| = m 1 · · · m r = n characters of the abelian group Γ can be obtained. Lemma 7. Let ψ 0 , . . . , ψ n−1 be the distinct characters of the additive abelian g roup Γ = {w 0 , . . . , w n−1 }, S ⊆ Γ, 0 ∈ S, −S = S. Assume that A(G) = A = (a i,j ) is the adjacency matrix of G = Cay(Γ, S) with respect to the given ordering of the vertex se t V (G) = Γ. a i,j =  1, if w i is a djacent to w j 0, if w i and w j are not adjacent , 0 ≤ i ≤ n − 1, 0 ≤ j ≤ n − 1 Then the vectors (ψ i (w j )) j=0, ,n−1 , 0 ≤ i ≤ n − 1, represent an orthogonal basis of C n consisting of eigenvectors of A. To the eigenvector (ψ i (w j )) j=0, ,n−1 belongs the eig envalue ψ i (S) =  s∈S ψ i (s). the electronic journal of combinatorics 18 (2011), #P94 8 There is a unique character ψ w i associated with every w i ∈ Γ according to (23). So we may assume in Lemma 7 that ψ i = ψ w i for i = 0, . . . , n − 1. Let us call the n × n-matrix H(Γ) = (ψ w i (w j )), 0 ≤ i ≤ n − 1, 0 ≤ j ≤ n − 1, the character matrix of Γ with resp ect to the given ordering of the elements of Γ. Here we always assume t hat Γ is represented by (20) as a direct product of cyclic groups and that the elements of Γ are ordered lexicographically increasing. Then w 0 is the zero element of Γ. Moreover, by (23) the character matrix H(Γ) becomes the Kronecker product of the character matrices of the cyclic factors of Γ, Γ = Z m 1 ⊗ · · · ⊗ Z m r implies H(Γ ) = H(Z m 1 ) ⊗ · · · ⊗ H(Z m r ). (24) We remind that the K ronecker product A ⊗B of matrices A and B is defined by replacing the entry a i,j of A by a i,j B for all i, j. For every Cayley graph G = Cay(Γ, S) the rows of H(Γ) represent an orthogonal basis o f C n consisting of eigenvectors of G, respectively A(G). The corresponding eigenvalues are obtained by H(Γ)c S,Γ , the product of H(Γ) and the characteristic (column) vector c S,Γ of S in Γ, c S,Γ (i) =  1, if w i ∈ S 0, if w i ∈ S , 0 ≤ i ≤ n − 1. Consider the situation, when Γ is a cyclic group, Γ = Z n , n ≥ 2. Let ω n be a primitive n-th root of unity. Setting r = 1 a nd ζ 1 = ω n in (23) we establish the character matrix H(Z n ) = F n according to the natural ordering of the elements 0, 1, . . . , n − 1. F n = ((ω n ) ij ), 0 ≤ i ≤ n − 1, 0 ≤ j ≤ n − 1 Observe that all entries in the first r ow and in the first column of F n are equal to 1. For a divisor δ of n, 1 ≤ δ ≤ n, we simplify the notatio n of t he characteristic vector of the gcd-set S Z n (δ) in Z n to c δ,n , c δ,n (i) =  1, if gcd(i, n) = δ 0, otherwise , 0 ≤ i ≤ n − 1. For δ < n we have 0 ∈ S Z n (δ). So the Cayley graph Cay(Z n , S Z n (δ)) is well defined. It is integral by Corollary 1. The eigenvalues of this gr aph are the entries of F n c δ,n . Therefore, this vector is integral, which is a lso trivially true for δ = n, F n c δ,n ∈ Z n for every p ositive divisor δ of n. (25) The only quadratic primitive root is −1. This implies that H(Z 2 ) = F 2 is the elemen- tary Hadamard matrix (see [12]) F 2 =  1 1 1 −1  . the electronic journal of combinatorics 18 (2011), #P94 9 By (24) the character matrix of the r-fold direct product Z 2 ⊗ · · · ⊗ Z 2 = Z r 2 is H(Z r 2 ) = F 2 ⊗ · · · ⊗ F 2 = F (r) 2 , the r-fold Kronecker product of F 2 with itself, which is also a Hadamard matrix consisting of orthogonal rows with entries ±1. From now on let Γ be a gcd-group. By Theorem 3 we may assume Γ = Z r 2 ⊗ Z n , r ≥ 0, n ≥ 2 . (26) If we set p = n − 1 and q = 2 r − 1 , then we have |Γ| − 1 = 2 r n − 1 = qn + p. We order the elements of Z r 2 , a nd Γ lexicographically increasing. Z r 2 = {a 0 , a 1 , . . . , a q }, a 0 = (0 , . . . , 0, 0), a 1 = (0, . . . , 0, 1), . . . , a q = (1, . . . , 1, 1); Γ = {w 0 , w 1 , . . . , w qn+p }, w 0 = (a 0 , 0), w 1 = (a 0 , 1), . . . , w p = (a 0 , p), . . . . . . w qn = (a q , 0), w qn+1 = (a q , 1), . . . , w qn+p = (a q , p). (27) The character matrix H(Γ) with respect to the given ordering of elements becomes the Kronecker product of the character matrix F (r) 2 of Z r 2 and the character matrix F n of Z n , H(Γ) = F (r) 2 ⊗ F n . This means that H(Γ) consists of disjoint submatrices ±F n , because F (r) 2 has only entries ±1. The structure of H(Γ) is displayed in Figure 1. Rows and columns are labelled with the elements of Γ. Observe that a label α at a row stands for the unique character ψ α . The sign ǫ(j, l) ∈ {1, −1} of a submatrix F n is the entry of F (r) 2 in position (j, l), 0 ≤ j ≤ q, 0 ≤ l ≤ q. (a 0 , 0) · · · (a 0 , p) · · · (a l , 0) · · · (a l , p) · · · (a q , 0) · · · (a q , p) (a 0 , 0) · · · · · · · · · ǫ(0, 0)F n · · · ǫ(0, l)F n · · · ǫ(0, q)F n (a 0 , p) · · · · · · · · · · · · · · · · · · · · · · · · (a j , 0) · · · · · · · · · ǫ(j, 0)F n · · · ǫ(j, l)F n · · · ǫ(j, q)F n (a j , p) · · · · · · · · · · · · · · · · · · · · · · · · (a q , 0) · · · · · · · · · ǫ(q, 0 )F n · · · ǫ(q, l)F n · · · ǫ(q, q)F n (a q , p) · · · · · · Figure 1: The structure of H(Z r 2 ⊗ Z n ). the electronic journal of combinatorics 18 (2011), #P94 10 [...]... are exactly 6 nonisomorphic gcd -graphs over Γ, we conclude that there are at least (p + 2) − 6 = p − 4 nonisomorphic integral Cayley graphs over Γ, which are not gcd -graphs An interesting task would be to determine for every prime number p the number of all nonisomorphic integral Cayley graphs over Γ = Zp ⊗ Zp References [1] Abdollahi, A., and Vatandoost, E Which Cayley graphs are integral? Electronic... J Which graphs have integral spectra? Lecture Notes in Mathematics 406, Springer Verlag (1974), 45–50 ´ [9] Ilic, A The energy of unitary cayley graphs Linear Algebra and its Applications 431 (2009), 1881–1889 [10] Klotz, W., and Sander, T Integral Cayley graphs over abelian groups Electronic J Combinatorics 17 (2010), R81, 1–13 [11] Klotz, W., and Sander, T Some properties of unitary Cayley graphs. .. gcd-set of Γ′ by Lemma 10, S ′ ∈ Bgcd (Γ′ ) = B(Γ′ ) The group isomorphism ϕ provides a bijection between the sets in B(Γ′ ) and in B(Γ) So we conclude S ∈ B(Γ) the electronic journal of combinatorics 18 (2011), #P94 13 Example We have shown that for a gcd-group Γ the integral Cayley graphs over Γ are exactly the gcd -graphs over Γ For an arbitrary group Γ the number of integral Cayley graphs over Γ... Zp has order p except for the zero element (0, 0) Denote by [a] the cyclic subgroup generated by a There are nonzero elements a1 , , ap+1 in Γ such that Γ = U1 ∪ · · · ∪ Up+1 , Ui = [ai ], Ui ∩ Uj = {(0, 0)} for i = j The sets S0 = ∅, Si = (U1 ∪ · · · ∪ Ui )\{(0, 0)}, 1 ≤ i ≤ p + 1, belong to the Boolean algebra B(Γ) Therefore, the Cayley graphs Gi = Cay(Γ, Si), 0 ≤ i ≤ p + 1, are integral They... isomorphic gcd -graphs over Γ, so do D5 and D6 Therefore, we cancel D4 and D6 The cardinalities |SΓ (Di )| for i ∈ {1, 2, 3, 5, 7, 8} = M are in ascending order: 0, p − 1, 2(p − 1), (p − 1)2 , p(p − 1), p2 − 1 These are the degrees of regularity of the corresponding gcd -graphs Cay(Γ, SΓ(Di )), i ∈ M As the above degree sequence is strictly increasing for p ≥ 5, there are exactly 6 nonisomorphic gcd -graphs. .. dim(A) = |D| r Lemma 10 Let Γ = Z2 ⊗Zn , S ⊆ Γ, 0 ∈ S, −S = S The Cayley graph G = Cay(Γ, S) is integral, if and only S = ∅ or if there are positive divisor tuples d(1) , , d(k) of m = (2, , 2, n) such that S = SΓ (D) for D = {d(1) , , d(k)} Proof For S = SΓ (D) the Cayley graph G = Cay(Γ, S) is a gcd-graph, which is integral by Corollary 1 To prove the converse, we skip the trivial case of... (d(k) ) = SΓ (D) Theorem 4 Let Γ be a gcd-group, S ⊆ Γ, 0 ∈ S, − S = S The Cayley graph G = Cay(Γ, S) is integral, if and only if S belongs to the Boolean algebra B(Γ) generated by the subgroups of Γ Proof In [10] we showed that S ∈ B(Γ) implies that G is integral To prove the converse, we assume S = ∅ and G = Cay(Γ, S) integral By Theorem 3 r we know that there is a group Γ′ = Z2 ⊗Zn and a group isomorphism... Graphs with integral spectrum Linear Alg Appl 430 (2009), 547–552 ´ ´ ´ [3] Balinska, K., Cvetkovic, D., Rodosavljevic, Z., Simic, S., and Ste´, D A survey on integral graphs Univ Beograd, Publ Elektrotehn Fak vanovic Ser Mat 13 (2003), 42–65 the electronic journal of combinatorics 18 (2011), #P94 14 ´ ´ ´ [4] Basic, M., Petkovic, M., and Stevanovic, D Perfect state transfer in integral circulant graphs. .. of combinatorics 18 (2011), #P94 12 This implies λd cd,Γ ∈ D u = ˜ d∈D Lemma 9 With the notations introduced for Lemma 8 and its corollary we have D = A Proof By (29) D is a subspace of the linear space A ⊆ Q|Γ| Consider the mapping ∆ defined by ∆(v) = Hv for v ∈ A Corollary 2 shows that ∆ maps A in D As the rows of H are pairwise orthogonal and nonzero, this matrix is regular Therefore, ∆ is bijective,... Spectra of graphs with transitive groups Priodica Mathematica Hungarica 6 (1975), 191–195 [14] Rose H E A course in number theory Oxford Science Publications Oxford University Press, 1994 [15] Saxena, N., Severini, S., and Shparlinski, I Parameters of integral circulant graphs and periodic quantum dynamics Intern Journ of Quantum Information 5 (2007), 417–430 [16] So, W Integral circulant graphs Discrete . Integral Cayley graphs defined by greatest common divisors Walter Klotz Institut f¨ur Mathematik Tech nische Universit¨at. at every S ∈ B(Γ) can be characterized by greatest common divisors. 1 Introduction The greatest common divisor of nonnegative integers a and b is denoted by gcd(a, b). Let us agree upon gcd(0,. integral Cayley graphs over Γ are exactly the gcd -graphs over Γ . For an arbitrary group Γ the number of integral Cayley graphs over Γ may be considerably larger than the number of gcd -graphs over