The algebraic degree of spectra of circulant graphs

Journal of Number Theory 208 (2020) 295–304

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Journal of Number Theory www.elsevier.com/locate/jnt

General Section

The algebraic degree of spectra of circulant graphs Katja Mönius Institute of Mathematics, Würzburg University, Emil-Fischer-Str. 40, 97074 Würzburg, Germany

a r t i c l e

i n f o

Article history: Received 27 May 2019 Received in revised form 29 August 2019 Accepted 29 August 2019 Available online 17 September 2019 Communicated by S.J. Miller Keywords: Circulant graphs Graph spectrum Graph eigenvalues Algebraic degree

a b s t r a c t We investigate the algebraic degree of circulant graphs, i.e. the dimension of the splitting field of the characteristic polynomial of the associated adjacency matrix over the rationals. Studying the algebraic degree of graphs seems more natural than characterizing graphs with integral spectra only. We prove that the algebraic degree of circulant graphs on n vertices is bounded above by ϕ(n)/2, where ϕ denotes Euler’s totient function, and that the family of cycle graphs provides a family of maximum algebraic degree within the family of all circulant graphs. Moreover, we precisely determine the algebraic degree of circulant graphs on a prime number of vertices. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Which graphs have integral spectra? This question was raised by Harary & Schwenk in their article from 1973/74 [2]. In the end of their article, they already remarked that this problem appears intractable. We consider the spectrum of a graph as the multiset of eigenvalues of its associated adjacency matrix. The quest of characterizing adjacency matrices for which all eigenvalues are integers seems to be a challenging project, there is

E-mail address: [email protected]. https://doi.org/10.1016/j.jnt.2019.08.002 0022-314X/© 2019 Elsevier Inc. All rights reserved.

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no satisfying answer so far. Thus, it is common to restrict to special families of graphs and to determine all graphs with integral spectra within this family. Since eigenvalues of graphs are algebraic integers, from a number-theoretical point of view, it seems more natural to ask the more general question: Which graphs have the same algebraic degree? This question was raised by Steuding, Stumpf and the author [6]. Given a graph G, the algebraic degree deg(G) is defined as the dimension of the splitting field of the characteristic polynomial of the associated adjacency matrix over the rationals. By definition, this splitting field is the smallest field that contains all eigenvalues of the spectrum of the graph. Within the family of circulant graphs, So [8] completely characterized those with integral spectra. A graph is said to be circulant if it has a circulant adjacency matrix. Since each row is a cyclic shift of the first row, such a matrix is completely determined by specifying its first row. Therefore, with every circulant graph, we can associate a set S ⊆ Zn (where Zn denotes the ring of integers modulo n) of the positions of non-zero entries of the first row of the adjacency matrix of the graph. Respectively, we denote by Sn the corresponding graph and call S the connection set of Sn . These notations are adopted from Mans et al. [4,5]. Two vertices x, y ∈ Zn are adjacent in Sn if and only if x − y ∈ S. All graphs in this paper are assumed to be simple, i.e. undirected and without loops or multi-edges. Note that a circulant graph Sn is undirected if and only if S is symmetric, i.e. S ≡ −S mod n, and Sn has no loops if and only if 0 ∈ / S. In view of this, we call a set S ⊆ Zn a connection set if S ≡ −S mod n and 0 ∈ / S. The complete graph Kn ∼ = {1, . . . , n − 1}n and the cycle graph Cn ∼ = {1, n − 1}n are well-known examples of circulant graphs. The spectrum of a circulant graph Sn is given by  spec(Sn ) =

   s k e |0≤k ≤n−1 , n

(1)

s∈S

where here and in the following e (x) denotes exp(2πix). A proof for this can be found in the book by Zhang [9]. In particular, circulant graphs are Cayley graphs on cyclic groups, i.e. Sn ∼ = Cay(Zn , S). In this paper, we precisely determine the algebraic degree of Sp , where S ⊆ Zp is a connection set and p is a prime number. We show that the algebraic degree of Sp is bounded from below by a function depending on the number of elements in S only. We also give bounds for the algebraic degree of circulant graphs on n vertices where n is any natural number. Furthermore, we show that the family of cycle graphs provides a family of maximum algebraic degree within the family of all circulant graphs, and give a lower bound for the number of non-isomorphic circulant graphs on a prime number of vertices and of maximum algebraic degree. Graphs of maximum algebraic degree can be considered as a counterpart of graphs with integral spectra. Throughout this paper, by ϕ we denote Euler’s totient function and Z∗n denotes the group of units in Zn .

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2. Results Let Sn be a circulant graph on n ∈ N vertices with n > 2 and let ω denote a primitive n-th root of unity. From (1) we observe that every eigenvalue λ of Sn is contained in the field Q(ω) since λ is a linear combination of powers of ω. Thus, the algebraic degree of Sn is smaller or equal to ϕ(n) = [Q(ω) : Q]. In fact, since Sn is undirected, i.e. the adjacency matrix of Sn is symmetric, all eigenvalues of Sn are real and, therefore, λ is contained in the maximal real subfield F of Q(ω). It is well-known that F is given by Q(ω + ω −1 ) and that [Q(ω) : Q(ω + ω −1 )] = 2. Hence, it follows that deg(Sn ) ≤ [Q(ω + ω −1 ) : Q] = ϕ(n)/2. Furthermore, the following proposition gives a more precise upper bound for the algebraic degree of arbitrary circulant graphs: Proposition 2.1. Let n ∈ N, n > 2 and let S ⊆ Zn be a connection set with d := gcd(S, n). Then, deg(Sn ) ≤

ϕ(n/d) . 2

Proof. Since gcd(S, n) = d, every eigenvalue of Sn can be rewritten as   s k   s/d k e = e . n n/d s∈S

s∈S

Thus, if ω denotes a primitive (n/d)-th root of unity, every eigenvalue of Sn appears as a linear combination of powers of ω, and, since spec(Sn ) ⊆ R, it follows that spec(Sn ) ⊆ Q(ω + ω −1 ). Thus, deg(Sn ) ≤ [Q(ω + ω −1 ) : Q] = ϕ(n/d)/2. 2 Note that, as Boesch and Tindell [1] have proven, Sn is connected if and only if d = 1. The theorem below shows that, indeed, the upper bound in Proposition 2.1 is sharp. Theorem 2.2. The family of cycle graphs {Cn | 2 < n ∈ N} provides a family of circulant graphs of maximum algebraic degree within the family of all circulant graphs. Proof. We show that if {±s} is a connection set and d := gcd(s, n), we have that deg({±s}n ) = ϕ(n/d)/2. From Proposition 2.1 we deduce that deg({±s}n ) ≤ ϕ(n/d)/2. The eigenvalues of {±s}n are given by

λk := e

 s k n

 +e

−s n

k

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298

for k = 0, . . . , n − 1. We may write λ1 = ω + ω −1 for ω := e

  s

s

s/d . Since e = e n n n/d ,

we observe that ω is a primitive (n/d)-th root of unity, and Q(ω + ω −1 ) = Q(λ1 ). Hence, it follows that deg({±s}n ) ≥ [Q(ω + ω −1 ) : Q] = ϕ(n/d)/2. Since, for example, Cn ∼ = {1, n − 1}n for all n > 2, the statement follows. 2 Having the question of Harary and Schwenk [2] in mind, graphs of maximum algebraic degree seem to be interesting for future studies, as they somehow can be considered as a counterpart of integral graphs. For the sake of determining the algebraic degree of circulant graphs other than cycle graphs or unions thereof, the next lemma provides some information about the minimal polynomial of any eigenvalue of an arbitrary circulant graph. Lemma 2.3. Let n ∈ N, S ⊆ Zn and k ∈ Zn . Then, all elements in the multiset Cn (k) =

   s k e | k  ∈ Zn , gcd(k  , n) = gcd(k, n) n s∈S

are conjugates of

s∈S

e

s k n

and all conjugates of

Proof. Let Ψk denote the minimal polynomial of

s∈S

e

n

are contained in Cn (k).

s k

and let d := gcd(k, n).  k/d 1 Furthermore, for the primitive (n/d)-th root of unity ω := e n = e n/d , let σy be y the Q-automorphism of Q(ω) given by σy (ω) = ω for some y ∈ Zn with gcd(y, n/d) = 1. Then, σy

s∈S

e

s k

n

1 k

   s k     s k    s ky e = e . = σy e n n n s∈S

s∈S

s∈S

Thus,    s ky     s k 

   s k 

0 = σy (0) = σy Ψk e e e = Ψk σy = Ψk n n n s∈S

s∈S

s∈S

ky

and, therefore, s∈S e ns is also a root of Ψk , where gcd(k, n) = gcd(k , n) for k := ky. Since all Q-automorphisms on Q(ω) are of this form, Ψk does not have any other roots. 2 In order to determine the algebraic degree of Sn , we first have to identify the degrees of the irreducible factors of the characteristic polynomial of the adjacency matrix of Sn . These factors correspond to the minimal polynomials of the eigenvalues of Sn . From Lemma 2.3 we deduce that the number of conjugates, i.e. the degree of the minimal

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k

polynomial, of s∈S e ns equals the number of distinct elements in Cn (k). In order to get this number, we have to determine all solutions to the equation   s k1   s k2 e = e n n s∈S

(2)

s∈S

for k1 , k2 ∈ Zn with gcd(k1 , n) = gcd(k2 , n) = gcd(k, n). Since, in general, this problem is hard, in the following, we restrict us to the case where n = p is a prime number. In this case, Equation (2) holds if and only if k1 S ≡ k2 S mod p resp. S ≡ k2 k1−1 S mod p (i.e. only in the trivial case that the summands on the left-hand side of Equation (2) are equal to the summands on the right-hand side). The following lemma provides a characterization of such equivalences: Lemma 2.4. Let p > 2 be a prime, S ⊆ Z∗p with M := #S and k ∈ Zp . Then, S ≡ kS mod p if and only if there exists a common divisor m of M and p − 1 such that km ≡ 1 mod p and S may be written as 

M/m

S=

Si

i=1 m with #Si = m and sm i,1 ≡ · · · ≡ si,m mod p for si,j ∈ Si and i = 1, . . . , M/m.

Proof. Let S ≡ kS mod p. Then, 



s ≡ kM

s∈S

s

mod p

s∈S

and, therefore, kM ≡ 1 mod p. Thus, we get that ordp (k)|M and km ≡ 1 mod p for m := ordp (k). By Fermat’s little theorem, we also get that m|p − 1. Now, let s1 ∈ S. Since S ≡ kS mod p, we observe that kj s1 mod p ∈ S for j = 0, . . . , m − 1 and that these elements are pairwise distinct, since S is a set, s1 = 0, gcd(k, p) = 1 and kj ≡ 1 mod p for j = 1, . . . , m − 1. Thus, by defining S1 := {s1 , ks1 , k2 s1 , . . . , km−1 s1

mod p}

and Si := {si , ksi , k2 si , . . . , km−1 si

mod p} with

si ∈ S\

i−1  j=1

for i = 2, . . . , M/m, it follows that 

M/m

S=

i=1

Si

Sj

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with #Si = m and m 2 m m−1 sm si )m i ≡ (ksi ) ≡ (k si ) ≡ . . . ≡ (k

mod p

resp. m m m 2 m m m−1 m sm si i ≡ k si ≡ (k ) si ≡ . . . ≡ (k )

mod p

for i = 1, . . . , M/m, since km ≡ 1 mod p. For the other direction, since S is a set, we observe that for i = 1, . . . , M/m the elements of Si are the m distinct solutions to the equation xm ≡ ci mod p for a constant ci ∈ Zp . On the other hand, the elements ksi,1 , ksi,2 , . . . , ksi,m are all incongruent modulo p as well and are also solutions to xm ≡ ci mod p, since km ≡ 1 mod p. Thus, Si ≡ kSi mod p for i = 1, . . . , M/m and, therefore, S ≡ kS mod p. 2 The lemma shows that Equation (2) holds only if the connection set S bears some special structure. In the following, for a common divisor m of M and p − 1, we say that a set S ⊆ Z∗p with M := #S is m-decomposable if S can be written as 

M/m

S=

Si

i=1 m with #Si = m such that sm i,1 ≡ · · · ≡ si,m mod p for si,j ∈ Si and i = 1, . . . , M/m. In order to find all solutions k ∈ Zp to the equation S ≡ kS mod p, it suffices to consider the maximum number m ∈ Z such that S is m-decomposable: Assume that S is m1 -decomposable and m2 -decomposable for m1 = m2 . Then, by Lemma 2.4, we get that S ≡ kl S mod p for all kl satisfying klml ≡ 1 mod p for l = 1, 2. In particular, this holds true for elements kl of order ml . Thus, we have that S ≡ k1i k2j S mod p for all i, j ∈ Zp , and there exists an element k1i k2j of order lcm(m1 , m2 ) in Zp , since k1 , k2 generate a cyclic group of order lcm(m1 , m2 ). Now, again by Lemma 2.4, it follows that S is m-decomposable for m := lcm(m1 , m2 ) ≥ m1 , m2 . Taking this as a basis, we are now able to determine the algebraic degree of circulant graphs whose number of vertices is a prime number:

Theorem 2.5. Let p > 2 be a prime number and S ⊆ Zp be a connection set with M := #S. Furthermore, let m be the maximum common divisor of M and p − 1 such that S is m-decomposable. Then, deg(Sp ) =

p−1 . m

k

Proof. The p eigenvalues of Sp are given by λk = s∈S e ns for k = 0, . . . , p − 1. We have that λ0 = M ∈ Z. Furthermore, by Lemma 2.3, we observe that every two

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eigenvalues of the remaining eigenvalues are conjugated resp. equal. Let ω be a primitive p-th root of unity and k ∈ {1, . . . , p − 1}, then p − 1 = [Q(ω) : Q(λk )][Q(λk ) : Q]. The Q-automorphisms on Q(ω) are induced by σy : ω → ω y for y = 1, . . . , p − 1. By the Fundamental Theorem of Galois Theory, the number of elements in the subgroup

{σ ∈ Aut Q(ω)|Q | σ(λk ) = λk } equals [Q(ω) : Q(λk )]. We observe that σy (λk ) = λk holds only if S ≡ yS mod p (this follows, for example, from Redei [7, Satz 13]). Since S is a connection set and S is m-decomposable, from Lemma 2.4 we deduce that S ≡ yS mod p holds for all y ∈ Zp satisfying y m ≡ 1 mod p. On the other hand, these are all y ∈ Zp satisfying S ≡ yS mod p since m was chosen maximally. As m divides p − 1,

there exist exactly m solutions to y m ≡ 1 mod p, i.e. #Aut Q(ω)|Q(λk ) = [Q(ω) : Q(λk )] = m. The Galois group of Q(ω)|Q is given by the group of units in Zp . Given that Z∗p is cyclic, for every divisor d of #Z∗p , there exists exactly one subgroup of Z∗p of order d. Hence, by the Fundamental Theorem of Galois Theory, there also exists only one intermediate field of Q(ω)|Q of degree d over Q. Since [Q(ω) : Q(λk )] = m holds for every k ∈ {1, . . . , p − 1}, we therefore observe that Q(λ1 ) = . . . = Q(λp−1 ), i.e. λ1 , . . . , λp−1 ∈ Q(λk ). Thus, deg(Sp ) = [Q(λk ) : Q] = p−1 m . 2 Corollary 2.6. Let p > 2 be a prime number and S ⊆ Zp be a connection set with M := #S. Then, p−1 p−1 ≤ deg(Sp ) ≤ . M 2 In particular, Sp is integral if and only if M = p − 1, i.e. Sp ∼ = Kp . Proof. The second statement follows directly from Theorem 2.5. Since S is symmetric, we may write S⊆





 ±s

s≤(p−1)/2

with s2 ≡ (−s)2 mod p for all s ∈ S. Thus, by Theorem 2.5, deg(Sp ) ≤ p−1 2 (this also p−1 follows directly from Proposition 2.1). Moreover, it is clear that M ≤ deg(Sp ) since M is the maximum divisor of M . 2 Note that (p − 1)/M is not necessarily an integer. More precisely, we have that (p − 1)/M  ≤ deg(Sp ), where x denotes the least integer greater or equal to x. In some special cases, we can already deduce that Sp is of maximum algebraic degree from considering the numbers M and p − 1 only:

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K. Mönius / Journal of Number Theory 208 (2020) 295–304

Corollary 2.7. Let p > 2 be a prime number and S ⊆ Zp be a connection set with M := #S. If gcd(M, p − 1) = 2, then deg(Sp ) = p−1 2 . Proof. Since S ≡ −S mod p and p is a prime number not equal to 2, it is clear that gcd(M, p − 1) ≥ 2. Thus, if gcd(M, p − 1) = 2, in view of Theorem 2.5 and Corollary 2.6, we have that m = 2, i.e. deg(Sp ) = p−1 2 . 2 This corollary also yields a lower bound for the number of non-isomorphic circulant graphs on a prime number of vertices and of maximum algebraic degree:

Corollary 2.8. Let p > 2 be a prime number. Then there exist at least ϕ p−1 non2 isomorphic circulant graphs on p vertices and of maximum algebraic degree within the family of all circulant graphs. Proof. By Corollary 2.7, a circulant graph Sp is of maximum algebraic degree whenever

gcd(M, p − 1) = 2 for M = #S. Since #{M | M ∈ Zp , gcd(M, p − 1) = 2} = ϕ p−1 2 and circulant graphs can only be isomorphic if their connection sets contain the same number of elements, the statement follows. 2 Example 2.9. Let p = 31 and S = {±1, ±2, ±3, ±4, ±5, ±6, ±7}. Since gcd(M, p − 1) = gcd(14, 30) = 2, we deduce from Corollary 2.7 that deg(S31 ) = 30/2 = 15, i.e. S31 is of maximum algebraic degree within the family of all circulant graphs (and is not isomorphic to a cycle graph, in particular). Example 2.10. Let p = 31 and S = {±2, ±3, ±10, ±12, ±13, ±15}. Since gcd(M, p − 1) = gcd(12, 30) = 6, we first check whether S is 6-decomposable. We observe that (±2)6 ≡ (±10)6 ≡ (±12)6 ≡ 2 mod 31 and (±3)6 ≡ (±13)6 ≡ (±15)6 ≡ 16 mod 31. Thus, by Theorem 2.5, the algebraic degree of S31 equals 30/6 = 5. Indeed, the factorized characteristic polynomial of the adjacency matrix of Sn is given by χ = −(−12 + x)(67 + 16x − 44x2 − 17x3 + 2x4 + x5 )6 with Galois group Gal(χ|Q) ∼ = Z5 . Example 2.11. Corollary 2.6 shows that the number of edges in a circulant graph on a prime number of vertices gives a lower bound for its algebraic degree. However, there exist also graphs with same number of vertices and edges but different algebraic degree: Let p = 13 and S1 = {±1, ±5}. Since gcd(M, p − 1) = gcd(4, 12) = 4, we first check whether S1 is 4-decomposable. Indeed, we observe that (±1)4 ≡ (±5)4 ≡ 1 mod 13. Thus, by Theorem 2.5, the algebraic degree of S1 13 equals 12/4 = 3. Now, let S2 = {±1, ±6}. Again, we first try whether S2 is 4-decomposable, but 1 ≡ 4 1 ≡ 64 ≡ 9 mod 13. Thus, S2 is only 2-decomposable and, therefore, by Theorem 2.5, the algebraic degree of S2 13 equals 12/2 = 6.

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Fig. 1. Circulant graphs with same number of vertices and edges but different algebraic degree.

Hence, S1 13 and S2 13 have the same number of vertices and edges, but different algebraic degree. The two graphs are shown in Fig. 1. Note that this example is a minimal example with respect to the number p of vertices. Example 2.12 (Paley graphs). It is well-known that the spectrum of the Paley graph P (q) for some prime power q ≡ 1 mod 4 and vertex set V = Fq is given by 

q−1 , 2

√

 √ q − 1 [(q−1)/2] − q − 1 [(q−1)/2] , , 2 2

where the multiplicity of each eigenvalue is written within square brackets. Thus, the eigenvalues of P (q) are of degree at most 2. If q is a prime number, P (q) is a circulant graph, i.e. P (q) ∼ = Sq for some S ⊆ Zq , and P (q) is of algebraic degree 2. We now want to prove this again with Theorem 2.5. Since P (q) is (q − 1)/2-regular, #S = (q − 1)/2. By definition, there exists an edge between two vertices i, j in P (q) if and only if i − j is a quadratic residue modulo q. Since, in particular, P (q) is circulant, i − j ∈ S holds as well. Considering any vertex i and the vertex 0 yields that every element in S is a quadratic residue modulo q. Thus, for every s ∈ S there exists as ∈ {1, . . . , q − 1} such that s ≡ a2s mod q. Hence, for all s ∈ S it follows from Euler’s Theorem that s(q−1)/2 ≡ (a2s )(q−1)/2 = aq−1 ≡ 1 mod q. s Therefore, by Theorem 2.5, we get that the algebraic degree of P (q) equals

q−1 (q−1)/2

= 2.

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3. Outlook In future work we aim at generalizing our result to circulant graphs Sn for any n ∈ N. So far, this seems to be a much more difficult task, even if we restrict ourselves to the cases where Z∗n is cyclic. Besides that, we want to consider the more general Cayley graphs over arbitrary groups. Klotz and Sander [3] gave a complete characterization of integral Cayley graphs over abelian groups. They proved that a Cayley graph Cay(G, S) (for G being an abelian group) is integral if and only if the set S bears special algebraical structure. Thus, it seems an interesting question how deviation from structure is encoded in the algebraic degree of the eigenvalues (which measure the distance to rational integers). Acknowledgments The author is grateful to the anonymous referee for his or her kind remarks and would also like to thank Dominik Barth for many helpful discussions. References [1] A. Boesch, R. Tindell, Circulants and their connectivities, J. Graph Theory 8 (1984) 487–499. [2] F. Harary, A.J. Schwenk, Which Graphs Have Integral Spectra? Springer, Berlin Heidelberg, Berlin, Heidelberg, 1974, pp. 45–51. [3] W. Klotz, T. Sander, Integral Cayley graphs defined by greatest common divisors, Electron. J. Combin. 18 (1) (2011) #P94. [4] B. Litow, B. Mans, A note on the Ádám conjecture for double loops, Inform. Process. Lett. 66 (3) (1998) 149–153. [5] B. Mans, F. Pappalardi, I. Shparlinski, On the spectral Ádám property for circulant graphs, Discrete Math. 254 (2002) 309–329. [6] K. Mönius, J. Steuding, P. Stumpf, Which graphs have non-integral spectra?, Graphs Combin. 34 (6) (Nov 2018) 1507–1518. [7] L. Rédei, Natürliche Basen des Kreisteilungskörpers, Teil I, Abh. Math. Semin. Univ. Hambg. 23 (1959) 180–200. [8] W. So, Integral circulant graphs, Discrete Math. 306 (2005) 153–158. [9] F. Zhang, Matrix Theory, 2nd edition, Springer-Verlag, New York, 2011.