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On groups all of whose undirected Cayley graphs of bounded valency are integral Istvn Kovcs University of Primorska, Slovenia istvan.kovacs@upr.si Joint work with Istvn Estlyi Modern Trends in Algebraic Graph Theory Villanova


  1. On groups all of whose undirected Cayley graphs of bounded valency are integral István Kovács University of Primorska, Slovenia istvan.kovacs@upr.si Joint work with István Estélyi Modern Trends in Algebraic Graph Theory Villanova University, June 2-5, 2014 István Kovács Integral Cayley graphs of bounded valency June 4, 2014 1 / 16

  2. Setting For a group G and subset S ⊆ G , 1 / ∈ S , the Cayley digraph Cay ( G , S ) is the digraph whose vertex set is G and ( x , y ) is an arc if and only if yx − 1 ∈ S . We regard Cay ( G , S ) as an undirected graph when S = S − 1 , and use the term Cayley graph. The spectrum of a matrix is the set of its eigenvalues. The spectrum of a graph is the spectrum of its adjacency matrix. Definition A group G is called Cayley integral if every undirected Cayley graph Cay ( G , S ) of G has integral spectrum. István Kovács Integral Cayley graphs of bounded valency June 4, 2014 2 / 16

  3. Motivation Finite abelian Cayley integral groups have been determined: Theorem (Klotz, Sander 2010) If G is an abelian Cayley integral group, then G is isomorphic to one of the following: E 2 n , E 3 n , Z n 4 , E 2 m × E 3 n , E 2 m × Z n 4 , ( m ≥ 1 , n ≥ 1 ) . W HAT ARE THE FINITE NON - ABELIAN CAYLEY INTEGRAL GROUPS ? Theorem (Abdollahi and Jazaeri 2014; Ahmadi et al. 2014+) The only finite non-abelian Cayley integral groups are D 6 , Dic 12 and Q 8 × E 2 n , where n ≥ 0 . István Kovács Integral Cayley graphs of bounded valency June 4, 2014 3 / 16

  4. The main result H OW TO GENERALIZE C AYLEY INTEGRAL GROUPS FURTHER ? Let us study groups G for which we require Cay ( G , S ) to be integral only when | S | is bounded by a constant. Formally, for k ∈ N , we set Definition � � G k = G : Cay ( G , S ) is integral whenever | S | ≤ k . Theorem (Estélyi, K., 2014+) Every class G k consists of the Cayley integral groups if k ≥ 6 . Furthermore, G 4 and G 5 are equal, and consist of the following groups: (1) the Cayley integral groups, (2) the generalized dicyclic groups Dic ( E 3 n × Z 6 ) , where n ≥ 1 . István Kovács Integral Cayley graphs of bounded valency June 4, 2014 4 / 16

  5. Generalized dicyclic groups Let A be an abelian group with a unique involution x ∈ A . Definition The generalized dicyclic group over A is Dic ( A ) = � A , y � , where y 2 = x and a y = a − 1 for all a ∈ A . Some important special cases: A = Z n gives rise to the dicyclic group Dic 2 n . A = Z 2 n gives rise to the generalized quaternion group Q 2 n + 1 . In particular if A = Z 4 = � i � , then we get Q 8 = � i , j � , the quaternion group. István Kovács Integral Cayley graphs of bounded valency June 4, 2014 5 / 16

  6. Basic properties of G k Lemma The following hold for every G ∈ G k if k ≥ 2 . (i) For every x ∈ G, the order of x is in { 1 , 2 , 3 , 4 , 6 } . (ii) For every subgroup H ≤ G , H ∈ G k . (iii) For every N � G such that | N | | k , G / N ∈ G l , where l = k / | N | . István Kovács Integral Cayley graphs of bounded valency June 4, 2014 6 / 16

  7. One further property of G k Unlike in the case of Cayley integral groups, the class G k is not closed under taking homomorphic images: For example, G = Z 4 ⋊ Z 4 = � a � ⋊ � b � , where a b = a − 1 , is in G 2 , while G / � b 2 � ∼ = D 8 is not. Lemma Let G ∈ G k , and N � G , N is abelian and | N | is odd. Then G / N ∈ G k . István Kovács Integral Cayley graphs of bounded valency June 4, 2014 7 / 16

  8. Spectrum of graphs with semiregular groups Let Γ be a graph, and let H ≤ Aut Γ an abelian semiregular group of automorphisms with m orbits on the vertex set. Fix m verices v 1 , . . . , v m , a complete set of representatives of H -orbits. Definition The symbol of Γ relative to H and the m -tuple ( v 1 , . . . , v m ) is the m × m array S = ( S ij ) i , j ∈{ 1 ,..., m } , where S ij = { x ∈ H : v i ∼ v x j in Γ } . Definition For an irreducible character χ of H let χ ( S ) be the m × m complex matrix defined by �� s ∈ S ij χ ( s ) if S ij � = ∅ ( χ ( S )) ij = i , j ∈ { 1 , . . . , m } . 0 otherwise, István Kovács Integral Cayley graphs of bounded valency June 4, 2014 8 / 16

  9. Spectrum of graphs with a semiregular group Proposition (K., Marušiˇ c, Malniˇ c, Miklaviˇ c, 2014+) The spectrum of Γ is the union of eigenvalues of χ ( S ) , where χ runs over the set of all irreducible characters of H. Using this theorem we have proved: Lemma Let G ∈ G k , and N � G , N is abelian and | N | is odd. Then G / N ∈ G k . Lemma The group Dic ( E 3 n × Z 6 ) is in G 5 for every n ≥ 0 . István Kovács Integral Cayley graphs of bounded valency June 4, 2014 9 / 16

  10. Nilpotent groups in G k , k ≥ 4 Proposition Every p-group in G k is Cayley integral if k ≥ 4 . Namely, they are one of the following: E 3 m , E 2 n × Z m 4 , Q 8 × E 2 n , where m , n ≥ 0 . In order to prove this first we show that the minimal non-abelian subgroup of such a group can only be Q 8 .Then we use the following theorem: Theorem (Janko, 2007) If G is a 2-group whose minimal nonabelian subgroups are isomorphic to Q 8 , then G ∼ = Q 2 m × E 2 n , where m ≥ 3 , n ≥ 0 . Since every nilpotent group is the direct product of its Sylow subgroups, we have obtained the following corollary: Corollary Every nilpotent group in G k is Cayley integral if k ≥ 4 . István Kovács Integral Cayley graphs of bounded valency June 4, 2014 10 / 16

  11. Minimal non-abelian p -groups in G k , k ≥ 4 A finite group G is said to be minimal non-abelian if all proper subgroups of G are abelian. Theorem (Rédei, 1947) Let G be a minimal non-abelian p-group. Then G is one of the following: (i) Q 8 ; a , b | a p m = b p n = 1 , a b = a 1 + p m − 1 � � (ii) , where m ≥ 2 (metacyclic); a , b , c | a p m = b p n = c p = 1 , [ a , b ] = c , [ c , a ] = [ c , b ] = 1 � � (iii) , where m + n ≥ 3 if p = 2 (non-metacyclic). István Kovács Integral Cayley graphs of bounded valency June 4, 2014 11 / 16

  12. Minimal non-abelian p -groups in G k , k ≥ 4 Corollary The minimal non-abelian groups of exponent at most 4 are the following groups: (i) Q 8 ; a , b | a 4 = b 2 = 1 , a b = a − 1 � � (ii) D 8 = , a , b | a 4 = b 4 = 1 , a b = a − 1 � � H 2 = (metacyclic); a , b , c | a 4 = b 2 = c 2 = 1 , [ a , b ] = c , [ c , a ] = [ c , b ] = 1 � � (iii) H 16 = , a , b , c | a 4 = b 4 = c 2 = 1 , [ a , b ] = c , [ c , a ] = [ c , b ] = 1 � � H 32 = , a , b , c | a 3 = b 3 = c 3 = 1 , [ a , b ] = c , [ c , a ] = [ c , b ] = 1 � � H 27 = (non-metacyclic). István Kovács Integral Cayley graphs of bounded valency June 4, 2014 12 / 16

  13. Non-niloptent groups in G k , k ≥ 4 Proposition Suppose that G ∈ G k , k ≥ 4 , and G is not nilpotent. Then G ∼ = D 6 or Dic ( E 3 n × Z 6 ) for some n ≥ 0 . In order to prove this we used the following lemma: Lemma Suppose that G ∈ G k , k ≥ 4 , and 3 | | G | . Then G has a normal Sylow 3 -subgroup. István Kovács Integral Cayley graphs of bounded valency June 4, 2014 13 / 16

  14. Proof of the main theorem Let G ∈ G k , k ≥ 4. If G is nilpotent, then G is Cayley integral by Corollary Every nilpotent group in G k is Cayley integral if k ≥ 4 . If G is not nilpotent, then we apply an earlier Proposition Suppose that G ∈ G k , k ≥ 4 , and G is not nilpotent. Then G ∼ = D 6 or Dic ( E 3 n × Z 6 ) for some n ≥ 0 . As seen earlier, these groups are in G 5 . However, they are not in G k , k ≥ 6, except for D 6 and Dic ( Z 6 ) = Dic 12 . István Kovács Integral Cayley graphs of bounded valency June 4, 2014 14 / 16

  15. What about G 3 ? This classes of groups may even be too wide for a "nice" characterization, since For example, all 3-groups of exponent 3 are in G 3 . For 2-groups in G 3 we have proved the following proposition: Proposition Let G be a non-abelian 2 -group of exponent 4 . Then G ∈ G 3 if and only if every minimal non-abelian subgroup of G is isomorphic to Q 8 , H 2 or H 32 . István Kovács Integral Cayley graphs of bounded valency June 4, 2014 15 / 16

  16. Bibliography I W. Klotz, T. Sander, Integral Cayley graphs over abelian groups EJC (2012). A. Abdollahi, M. Jazaeri, Groups all of whose undirected Cayley graphs are integral Europ. J. Combin. 38 (2014), 102–109. A. Ahmady, J. P . Bell, B. Mohar, Integral Cayley graphs and groups, preprint arXiv:1209.5126v1 [math.CO] 2013. I. Estélyi, I. Kovács, On groups all of whose undirected Cayley graphs of bounded valency are integral, preprint arXiv:1403.7602 [math.GR] 2014. István Kovács Integral Cayley graphs of bounded valency June 4, 2014 16 / 16

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