Automorphism Groups of Cayley Graphs

Ted Dobson

Department of Mathematics & Statistics Mississippi State University [email protected] http://www2.msstate.edu/∼dobson/

Symmetries of Graphs and Networks

November 23–28, 2008

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs A circulant graph of order n is simply a Cayley graph of Zn.

Basic Deﬁnitions

Definition Let G be a group and S ⊂ G. We define the Cayley digraph of G with connection set S to be the digraph Γ = Γ(G, S) defined by V (Γ) = G and E(Γ) = {(g, gs): s ∈ S, g ∈ G}. If S = S−1, then Γ is a Cayley graph of G with connection set S. Note that GL = {x → gx : g ∈ G} ≤ Aut(Γ).

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Basic Deﬁnitions

A circulant graph of order n is simply a Cayley graph of Zn.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Observe that every vertex-transitive graph Γ of prime order is isomorphic to a circulant graph of prime order as Aut(Γ) contains a subgroup of order p, which, after a relabeling, we may assume is h(0, 1,..., p − 1)i = hx → x + 1i = (Zp)L. So circulant! Theorem (Burnside, 1901)

Let G ≤ Sp, p a prime, contain (Zp)L. Then ∗ G ≤ AGL(1, p) = {x → ax + b : a ∈ Zp, b ∈ Zp} or G is doubly-transitive. Note that if G ≤ Aut(Γ) is doubly-transitive, then Aut(Γ) = SX and Γ is complete or has no edges.

Automorphisms of prime order circulants

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Theorem (Burnside, 1901)

Automorphisms of prime order circulants

Observe that every vertex-transitive graph Γ of prime order is isomorphic to a circulant graph of prime order as Aut(Γ) contains a subgroup of order p, which, after a relabeling, we may assume is h(0, 1,..., p − 1)i = hx → x + 1i = (Zp)L. So circulant!

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Note that if G ≤ Aut(Γ) is doubly-transitive, then Aut(Γ) = SX and Γ is complete or has no edges.

Automorphisms of prime order circulants

Let G ≤ Sp, p a prime, contain (Zp)L. Then ∗ G ≤ AGL(1, p) = {x → ax + b : a ∈ Zp, b ∈ Zp} or G is doubly-transitive.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Automorphisms of prime order circulants

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs B. Alspach (1973) ﬁrst observed this, and went on to explicitly enumerate the vertex-transitive graphs of prime order. Burnside’s Theorem is equivalent to Theorem Let G ≤ Sp, p a prime, be transitive. Then either G contains a normal Sylow p-subgroup or G is doubly-transitive.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Burnside’s Theorem is equivalent to Theorem Let G ≤ Sp, p a prime, be transitive. Then either G contains a normal Sylow p-subgroup or G is doubly-transitive.

This gives an algorithm for determining the full automorphism group of a circulant graph Γ = Γ(Zp, S). Note that x → x + b is always contained in ∗ Aut(Γ), so we need only check which a ∈ Zp satisfy a · S = {as : s ∈ S} = S (we observe that AGL(1, p) is itself doubly-transitive, so if all such x → ax are in Aut(Γ), then Aut(Γ) = Sp). B. Alspach (1973) ﬁrst observed this, and went on to explicitly enumerate the vertex-transitive graphs of prime order.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Theorem Let G ≤ Sp, p a prime, be transitive. Then either G contains a normal Sylow p-subgroup or G is doubly-transitive.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs This gives an algorithm for determining the full automorphism group of a circulant graph Γ = Γ(Zp, S). Note that x → x + b is always contained in ∗ Aut(Γ), so we need only check which a ∈ Zp satisfy a · S = {as : s ∈ S} = S (we observe that AGL(1, p) is itself doubly-transitive, so if all such x → ax are in Aut(Γ), then Aut(Γ) = Sp). B. Alspach (1973) ﬁrst observed this, and went on to explicitly enumerate the vertex-transitive graphs of prime order. Burnside’s Theorem is equivalent to Theorem Let G ≤ Sp, p a prime, be transitive. Then either G contains a normal Sylow p-subgroup or G is doubly-transitive.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Theorem (D., D. Witte, 2002)

There are exactly 2p − 1 transitive p-subgroups P of Sp2 up to conjugation, and all but three have the property that if G ≤ Sp2 with Sylow p-subgroup P, then either P/G or G is doubly-transitive. Theorem (D., 2005)

Let P be a transitive p-subgroup of Spk , p an odd prime, k ≥ 1, such that every minimal transitive subgroup of P is cyclic. If G ≤ Spk with Sylow p-subgroup P, then either P/G or G is doubly-transitive. It is probable that many more generalizations are true, but it also seems likely that “most classes” of automorphism groups of graphs cannot be obtained in this way.

Can Burnside’s Theorem be generalized?

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Theorem (D., 2005)

Can Burnside’s Theorem be generalized?

Theorem (D., D. Witte, 2002)

There are exactly 2p − 1 transitive p-subgroups P of Sp2 up to conjugation, and all but three have the property that if G ≤ Sp2 with Sylow p-subgroup P, then either P/G or G is doubly-transitive.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs It is probable that many more generalizations are true, but it also seems likely that “most classes” of automorphism groups of graphs cannot be obtained in this way.

Can Burnside’s Theorem be generalized?

Theorem (D., D. Witte, 2002)

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Can Burnside’s Theorem be generalized?

Theorem (D., D. Witte, 2002)

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs The following result of Gareth Jones can also be viewed as an extension of Burnside’s Theorem. Theorem (Jones, 1979)

Let G ≤ Sp2 be transitive with Sylow p-subgroup isomorphic to Zp × Zp. Then G contains a normal subgroup H such that H = H1 × H2 where Hi is a nonabelian simple group or Hi = Zp.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Theorem (Jones, 1979)

Let G ≤ Sp2 be transitive with Sylow p-subgroup isomorphic to Zp × Zp. Then G contains a normal subgroup H such that H = H1 × H2 where Hi is a nonabelian simple group or Hi = Zp.

Burnside’s Theorem can be restated as follows: Theorem Let G be a transitive group of prime degree. Then G contains a transitive normal subgroup which is either abelian or a nonabelian simple group. The following result of Gareth Jones can also be viewed as an extension of Burnside’s Theorem.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Burnside’s Theorem can be restated as follows: Theorem Let G be a transitive group of prime degree. Then G contains a transitive normal subgroup which is either abelian or a nonabelian simple group. The following result of Gareth Jones can also be viewed as an extension of Burnside’s Theorem. Theorem (Jones, 1979)

Let G ≤ Sp2 be transitive with Sylow p-subgroup isomorphic to Zp × Zp. Then G contains a normal subgroup H such that H = H1 × H2 where Hi is a nonabelian simple group or Hi = Zp.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Problem Let G ≤ Spk have an abelian Sylow p-subgroup P. Assume G admits a nontrivial complete block system B such that StabG (B)|B is primitive and not permutation isomorphic to a subgroup of AGL(r, p) for any r ≥ 1, but ﬁxG (B)|B is imprimitive, B ∈ B. Then G contains a normal subgroup that admits a nontrivial complete block system C such that C ≺ B.

Theorem (D., 2008)

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Theorem (D., 2008)

Let G ≤ Spk be transitive with an abelian Sylow p-subgroup P. Let t be the number of elementary divisors of P. If p > 2t−1, then G contains a normal subgroup permutation isomorphic to a direct product of cyclic groups and doubly-transitive nonabelian simple groups with the canonical action, with the number of factors in the direct product equal to the number of elementary divisors of P. Problem Let G ≤ Spk have an abelian Sylow p-subgroup P. Assume G admits a nontrivial complete block system B such that StabG (B)|B is primitive and not permutation isomorphic to a subgroup of AGL(r, p) for any r ≥ 1, but ﬁxG (B)|B is imprimitive, B ∈ B. Then G contains a normal subgroup that admits a nontrivial complete block system C such that C ≺ B.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs I Zpk , k ≥ 1. First proof using method of Schur by Klin and P¨oschel, in the early ’80’s. Another proof using result above. None of the proofs published...

I Zp × Zp, Dobson and D. Witte, 2002. I Zp × Zp2 , Dobson, manuscript 3 I Zp, Dobson and I. Kov´acs,manuscript

Using these results and other techniques, the full automorphism group of a Cayley (di)graph of the following groups are known:

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs First proof using method of Schur by Klin and P¨oschel, in the early ’80’s. Another proof using result above. None of the proofs published...

I Zp × Zp, Dobson and D. Witte, 2002. I Zp × Zp2 , Dobson, manuscript 3 I Zp, Dobson and I. Kov´acs,manuscript

Using these results and other techniques, the full automorphism group of a Cayley (di)graph of the following groups are known:

I Zpk , k ≥ 1.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Another proof using result above. None of the proofs published...

I Zp × Zp, Dobson and D. Witte, 2002. I Zp × Zp2 , Dobson, manuscript 3 I Zp, Dobson and I. Kov´acs,manuscript

Using these results and other techniques, the full automorphism group of a Cayley (di)graph of the following groups are known:

I Zpk , k ≥ 1. First proof using method of Schur by Klin and P¨oschel, in the early ’80’s.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs I Zp × Zp, Dobson and D. Witte, 2002. I Zp × Zp2 , Dobson, manuscript 3 I Zp, Dobson and I. Kov´acs,manuscript

Using these results and other techniques, the full automorphism group of a Cayley (di)graph of the following groups are known:

I Zpk , k ≥ 1. First proof using method of Schur by Klin and P¨oschel, in the early ’80’s. Another proof using result above. None of the proofs published...

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs I Zp × Zp2 , Dobson, manuscript 3 I Zp, Dobson and I. Kov´acs,manuscript

Using these results and other techniques, the full automorphism group of a Cayley (di)graph of the following groups are known:

I Zpk , k ≥ 1. First proof using method of Schur by Klin and P¨oschel, in the early ’80’s. Another proof using result above. None of the proofs published...

I Zp × Zp, Dobson and D. Witte, 2002.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs 3 I Zp, Dobson and I. Kov´acs,manuscript

Using these results and other techniques, the full automorphism group of a Cayley (di)graph of the following groups are known:

I Zpk , k ≥ 1. First proof using method of Schur by Klin and P¨oschel, in the early ’80’s. Another proof using result above. None of the proofs published...

I Zp × Zp, Dobson and D. Witte, 2002. I Zp × Zp2 , Dobson, manuscript

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Using these results and other techniques, the full automorphism group of a Cayley (di)graph of the following groups are known:

I Zpk , k ≥ 1. First proof using method of Schur by Klin and P¨oschel, in the early ’80’s. Another proof using result above. None of the proofs published...

I Zp × Zp, Dobson and D. Witte, 2002. I Zp × Zp2 , Dobson, manuscript 3 I Zp, Dobson and I. Kov´acs,manuscript

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs A similar definition gives a graphical regular representation (GRR) of G In 1978, C. Godsil finished off the problem of determining which groups have DRR’s or GRR’s. Theorem All finite groups admit DRR’s with five exceptions of order at most 16. Theorem All finite groups admit GRR’s except abelian groups of exponent at least 3, generalized dicyclic groups, and 13 other groups of order at most 32.

GRR’s and DRR’s

Deﬁnition A Cayley digraph Γ of G is a digraphical regular representation (DRR) of G if Aut(Γ) = GL.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs In 1978, C. Godsil finished off the problem of determining which groups have DRR’s or GRR’s. Theorem All finite groups admit DRR’s with five exceptions of order at most 16. Theorem All finite groups admit GRR’s except abelian groups of exponent at least 3, generalized dicyclic groups, and 13 other groups of order at most 32.

GRR’s and DRR’s

Deﬁnition A Cayley digraph Γ of G is a digraphical regular representation (DRR) of G if Aut(Γ) = GL. A similar deﬁnition gives a graphical regular representation (GRR) of G

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Theorem All finite groups admit DRR’s with five exceptions of order at most 16. Theorem All finite groups admit GRR’s except abelian groups of exponent at least 3, generalized dicyclic groups, and 13 other groups of order at most 32.

GRR’s and DRR’s

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Theorem All ﬁnite groups admit GRR’s except abelian groups of exponent at least 3, generalized dicyclic groups, and 13 other groups of order at most 32.

GRR’s and DRR’s

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs GRR’s and DRR’s

Definition A Cayley digraph Γ of G is a digraphical regular representation (DRR) of G if Aut(Γ) = GL. A similar definition gives a graphical regular representation (GRR) of G In 1978, C. Godsil finished off the problem of determining which groups have DRR’s or GRR’s. Theorem All finite groups admit DRR’s with five exceptions of order at most 16. Theorem All finite groups admit GRR’s except abelian groups of exponent at least 3, generalized dicyclic groups, and 13 other groups of order at most 32.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Motivated by work on determining which groups are DRR’s and GRR’s, several people became convinced that the following conjecture is true: Conjecture Let G be a ﬁnite group of order g which is neither abelian of exponent at least 3 nor generalized dicyclic. Then

# of GRR0s of G lim = 1. g→∞ # of Cayley graphs of G

The conjecture is due to Imrich, Lov´asz,Babai, and Godsil in around 1982.

There is also the similar conjecture that “almost all” Cayley digraphs of G are DRR’s of G.

The Conjecture

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Conjecture Let G be a ﬁnite group of order g which is neither abelian of exponent at least 3 nor generalized dicyclic. Then

# of GRR0s of G lim = 1. g→∞ # of Cayley graphs of G

The conjecture is due to Imrich, Lov´asz,Babai, and Godsil in around 1982.

There is also the similar conjecture that “almost all” Cayley digraphs of G are DRR’s of G.

The Conjecture

Motivated by work on determining which groups are DRR’s and GRR’s, several people became convinced that the following conjecture is true:

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs The conjecture is due to Imrich, Lov´asz,Babai, and Godsil in around 1982.

There is also the similar conjecture that “almost all” Cayley digraphs of G are DRR’s of G.

The Conjecture

Motivated by work on determining which groups are DRR’s and GRR’s, several people became convinced that the following conjecture is true: Conjecture Let G be a ﬁnite group of order g which is neither abelian of exponent at least 3 nor generalized dicyclic. Then

# of GRR0s of G lim = 1. g→∞ # of Cayley graphs of G

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs There is also the similar conjecture that “almost all” Cayley digraphs of G are DRR’s of G.

The Conjecture

# of GRR0s of G lim = 1. g→∞ # of Cayley graphs of G

The conjecture is due to Imrich, Lov´asz,Babai, and Godsil in around 1982.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs The Conjecture

# of GRR0s of G lim = 1. g→∞ # of Cayley graphs of G

The conjecture is due to Imrich, Lov´asz,Babai, and Godsil in around 1982.

There is also the similar conjecture that “almost all” Cayley digraphs of G are DRR’s of G.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Theorem Let G be a group of prime-power order with no homomorphism onto Zp o Zp (a full Sylow p-subgroup of Sp2 ). Then almost all Cayley digraphs of G are DRR’s of G. Theorem Let G be a non-abelian group of prime-power order with no homomorphism onto Zp o Zp (a full Sylow p-subgroup of Sp2 ). Then almost all Cayley graphs of G are GRR’s of G.

Known Results

The ﬁrst results were obtained by Godsil in 1981:

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Theorem Let G be a non-abelian group of prime-power order with no homomorphism onto Zp o Zp (a full Sylow p-subgroup of Sp2 ). Then almost all Cayley graphs of G are GRR’s of G.

Known Results

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Known Results

The ﬁrst results were obtained by Godsil in 1981: Theorem Let G be a group of prime-power order with no homomorphism onto Zp o Zp (a full Sylow p-subgroup of Sp2 ). Then almost all Cayley digraphs of G are DRR’s of G. Theorem Let G be a non-abelian group of prime-power order with no homomorphism onto Zp o Zp (a full Sylow p-subgroup of Sp2 ). Then almost all Cayley graphs of G are GRR’s of G.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Theorem Let G be a nilpotent group of odd order g. Let Γ be a random Cayley digraph or Cayley graph of G. In the undirected case, assume additionally that G is not abelian. Then the probability that Aut(Γ) 6= G is less than √ (0.91 + o(1)) g . Theorem Let G be an abelian group of order g ≡ −1 (mod 4). Then, for almost all Cayley graphs Γ of G, |Aut(Γ)| = 2|G|.

These results were improved by Babai and Godsil in 1982:

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Theorem Let G be an abelian group of order g ≡ −1 (mod 4). Then, for almost all Cayley graphs Γ of G, |Aut(Γ)| = 2|G|.

These results were improved by Babai and Godsil in 1982: Theorem Let G be a nilpotent group of odd order g. Let Γ be a random Cayley digraph or Cayley graph of G. In the undirected case, assume additionally that G is not abelian. Then the probability that Aut(Γ) 6= G is less than √ (0.91 + o(1)) g .

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs These results were improved by Babai and Godsil in 1982: Theorem Let G be a nilpotent group of odd order g. Let Γ be a random Cayley digraph or Cayley graph of G. In the undirected case, assume additionally that G is not abelian. Then the probability that Aut(Γ) 6= G is less than √ (0.91 + o(1)) g . Theorem Let G be an abelian group of order g ≡ −1 (mod 4). Then, for almost all Cayley graphs Γ of G, |Aut(Γ)| = 2|G|.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Corollary Let k be a positive integer, and p a prime such that p > 2k−1. Let G ≤ Spk be the automorphism group of a Cayley digraph of some abelian group P. Then one of the following is true: 1. G contains an automorphism α of P of order p, or 2. G has a normal Sylow p-subgroup which is isomorphic to P and abelian, or

3. G contains a transitive subgroup isomorphic to Sp × A, where A ≤ Spk−1 has an abelian Sylow p-subgroup.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Theorem Let G be an abelian group of odd prime-power order. Then almost every Cayley graph of G has automorphism group of order 2 · |G|.

Automorphism groups of graphs

The following result extends Babai and Godsil’s result for graphs, provided that the order of the group is an odd prime power.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Automorphism groups of graphs

The following result extends Babai and Godsil’s result for graphs, provided that the order of the group is an odd prime power. Theorem Let G be an abelian group of odd prime-power order. Then almost every Cayley graph of G has automorphism group of order 2 · |G|.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Deﬁnition A Cayley (di)graph Γ of G is a normal Cayley (di)graph of G if GL/Aut(Γ). Xu also proposed a weaker conjecture to the one of Imrich, Lov´asz,Babai, and Godsil: Conjecture Almost every Cayley (di)graph is a normal Cayley digraph.

Normal Cayley Graphs

Ming-Yao Xu introduced the notion of a “normal Cayley graph” in 1998:

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Xu also proposed a weaker conjecture to the one of Imrich, Lov´asz,Babai, and Godsil: Conjecture Almost every Cayley (di)graph is a normal Cayley digraph.

Normal Cayley Graphs

Ming-Yao Xu introduced the notion of a “normal Cayley graph” in 1998: Deﬁnition A Cayley (di)graph Γ of G is a normal Cayley (di)graph of G if GL/Aut(Γ).

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Conjecture Almost every Cayley (di)graph is a normal Cayley digraph.

Normal Cayley Graphs

Ming-Yao Xu introduced the notion of a “normal Cayley graph” in 1998: Deﬁnition A Cayley (di)graph Γ of G is a normal Cayley (di)graph of G if GL/Aut(Γ). Xu also proposed a weaker conjecture to the one of Imrich, Lov´asz,Babai, and Godsil:

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Normal Cayley Graphs

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Theorem Let G be an abelian group of prime-power order pk . Then almost every Cayley digraph of G that is not a DRR is a normal Cayley digraph of G. In particular,

|NorCayDi(G) − DRR(G)| lim = 1. p→∞ |CayDi(G) − DRR(G)|

Theorem Let G be an abelian group of prime-power order pk . Then almost every Cayley graph of G that does not have automorphism group of order 2 · |G| is a normal Cayley graph of G.

We prove something more interesting, but not more surprising.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Theorem Let G be an abelian group of prime-power order pk . Then almost every Cayley graph of G that does not have automorphism group of order 2 · |G| is a normal Cayley graph of G.

We prove something more interesting, but not more surprising. Theorem Let G be an abelian group of prime-power order pk . Then almost every Cayley digraph of G that is not a DRR is a normal Cayley digraph of G. In particular,

|NorCayDi(G) − DRR(G)| lim = 1. p→∞ |CayDi(G) − DRR(G)|

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs We prove something more interesting, but not more surprising. Theorem Let G be an abelian group of prime-power order pk . Then almost every Cayley digraph of G that is not a DRR is a normal Cayley digraph of G. In particular,

|NorCayDi(G) − DRR(G)| lim = 1. p→∞ |CayDi(G) − DRR(G)|

Theorem Let G be an abelian group of prime-power order pk . Then almost every Cayley graph of G that does not have automorphism group of order 2 · |G| is a normal Cayley graph of G.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Conjecture Almost every Cayley (di)graph whose automorphism group is not as small as possible is a normal Cayley (di)graph. It is diﬃcult to determine the automorphism group of a (di)graph, so the main way to obtain examples of vertex-transitive graphs is to construct them. An obvious construction is that of a Cayley (di)graph, and the conjecture of Imrich, Lov´asz,Babai, and Godsil says that when performing this construction, additional automorphism are almost never obtained. The obvious way of constructing a Cayley (di)graph of G that does not have automorphism group as small as possible is to choose an automorphism α of G and make the connection set a union of orbits of α. The above conjecture in some sense says that this construction almost never yields additional automorphisms other than the ones given by the construction.

Some Problems and Conjectures We would like to make the following (not terribly surprising) conjecture:

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs It is diﬃcult to determine the automorphism group of a (di)graph, so the main way to obtain examples of vertex-transitive graphs is to construct them. An obvious construction is that of a Cayley (di)graph, and the conjecture of Imrich, Lov´asz,Babai, and Godsil says that when performing this construction, additional automorphism are almost never obtained. The obvious way of constructing a Cayley (di)graph of G that does not have automorphism group as small as possible is to choose an automorphism α of G and make the connection set a union of orbits of α. The above conjecture in some sense says that this construction almost never yields additional automorphisms other than the ones given by the construction.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs An obvious construction is that of a Cayley (di)graph, and the conjecture of Imrich, Lov´asz,Babai, and Godsil says that when performing this construction, additional automorphism are almost never obtained. The obvious way of constructing a Cayley (di)graph of G that does not have automorphism group as small as possible is to choose an automorphism α of G and make the connection set a union of orbits of α. The above conjecture in some sense says that this construction almost never yields additional automorphisms other than the ones given by the construction.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs The obvious way of constructing a Cayley (di)graph of G that does not have automorphism group as small as possible is to choose an automorphism α of G and make the connection set a union of orbits of α. The above conjecture in some sense says that this construction almost never yields additional automorphisms other than the ones given by the construction.

Some Problems and Conjectures We would like to make the following (not terribly surprising) conjecture: Conjecture Almost every Cayley (di)graph whose automorphism group is not as small as possible is a normal Cayley (di)graph. It is diﬃcult to determine the automorphism group of a (di)graph, so the main way to obtain examples of vertex-transitive graphs is to construct them. An obvious construction is that of a Cayley (di)graph, and the conjecture of Imrich, Lov´asz,Babai, and Godsil says that when performing this construction, additional automorphism are almost never obtained.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs The above conjecture in some sense says that this construction almost never yields additional automorphisms other than the ones given by the construction.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Some Problems and Conjectures We would like to make the following (not terribly surprising) conjecture: Conjecture Almost every Cayley (di)graph whose automorphism group is not as small as possible is a normal Cayley (di)graph. It is diﬃcult to determine the automorphism group of a (di)graph, so the main way to obtain examples of vertex-transitive graphs is to construct them. An obvious construction is that of a Cayley (di)graph, and the conjecture of Imrich, Lov´asz,Babai, and Godsil says that when performing this construction, additional automorphism are almost never obtained. The obvious way of constructing a Cayley (di)graph of G that does not have automorphism group as small as possible is to choose an automorphism α of G and make the connection set a union of orbits of α. The above conjecture in some sense says that this construction almost never yields additional automorphisms other than the ones given by the construction.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs - such graphs are not normal Cayley graphs provided p 6= 2. Deﬁnition A Cayley graph Γ of an abelian group G is a semiwreath product if there exist subgroups H ≤ K < G such that S − K is a union of cosets of H. Note that if H = K, a semiwreath product is in fact a wreath product. Deﬁnition A Cayley graph Γ of an abelian group G is a deleted wreath product if Γ = (Γ1 o K¯m) − mΓ1, where Γ1 is a Cayley graph of an abelian group of order |G|/m, and mΓ1 is m vertex-disjoint copies of Γ1.

There are two additional families of (di)graphs that can be considered, namely “semiwreath products” and “deleted wreath products”

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Deﬁnition A Cayley graph Γ of an abelian group G is a semiwreath product if there exist subgroups H ≤ K < G such that S − K is a union of cosets of H. Note that if H = K, a semiwreath product is in fact a wreath product. Deﬁnition A Cayley graph Γ of an abelian group G is a deleted wreath product if Γ = (Γ1 o K¯m) − mΓ1, where Γ1 is a Cayley graph of an abelian group of order |G|/m, and mΓ1 is m vertex-disjoint copies of Γ1.

There are two additional families of (di)graphs that can be considered, namely “semiwreath products” and “deleted wreath products” - such graphs are not normal Cayley graphs provided p 6= 2.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Note that if H = K, a semiwreath product is in fact a wreath product. Deﬁnition A Cayley graph Γ of an abelian group G is a deleted wreath product if Γ = (Γ1 o K¯m) − mΓ1, where Γ1 is a Cayley graph of an abelian group of order |G|/m, and mΓ1 is m vertex-disjoint copies of Γ1.

There are two additional families of (di)graphs that can be considered, namely “semiwreath products” and “deleted wreath products” - such graphs are not normal Cayley graphs provided p 6= 2. Deﬁnition A Cayley graph Γ of an abelian group G is a semiwreath product if there exist subgroups H ≤ K < G such that S − K is a union of cosets of H.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Deﬁnition A Cayley graph Γ of an abelian group G is a deleted wreath product if Γ = (Γ1 o K¯m) − mΓ1, where Γ1 is a Cayley graph of an abelian group of order |G|/m, and mΓ1 is m vertex-disjoint copies of Γ1.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs There are two additional families of (di)graphs that can be considered, namely “semiwreath products” and “deleted wreath products” - such graphs are not normal Cayley graphs provided p 6= 2. Deﬁnition A Cayley graph Γ of an abelian group G is a semiwreath product if there exist subgroups H ≤ K < G such that S − K is a union of cosets of H. Note that if H = K, a semiwreath product is in fact a wreath product. Deﬁnition A Cayley graph Γ of an abelian group G is a deleted wreath product if Γ = (Γ1 o K¯m) − mΓ1, where Γ1 is a Cayley graph of an abelian group of order |G|/m, and mΓ1 is m vertex-disjoint copies of Γ1.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs The preceding conjecture is known to be true if G is cyclic. The preceding conjecture is known to be false for at least some nonabelian groups G.

Conjecture Let Γ be a Cayley (di)graph of an abelian group G. Then one of the following is true:

I Γ is a normal Cayley graph of G,

I Γ is a semiwreath product, or

I the automorphism group of Γ is same as the automorphism group of a deleted wreath product.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs The preceding conjecture is known to be false for at least some nonabelian groups G.

Conjecture Let Γ be a Cayley (di)graph of an abelian group G. Then one of the following is true:

I Γ is a normal Cayley graph of G,

I Γ is a semiwreath product, or

I the automorphism group of Γ is same as the automorphism group of a deleted wreath product. The preceding conjecture is known to be true if G is cyclic.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Conjecture Let Γ be a Cayley (di)graph of an abelian group G. Then one of the following is true:

I Γ is a normal Cayley graph of G,

I Γ is a semiwreath product, or

I the automorphism group of Γ is same as the automorphism group of a deleted wreath product. The preceding conjecture is known to be true if G is cyclic. The preceding conjecture is known to be false for at least some nonabelian groups G.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Problem For an abelian group G, does there exist a natural collection F of families of Cayley (di)graphs of G and a partial order on F such that every Cayley (di)graph of G is contained in some element of F and if F1 F2 and there is no F3 such that F1 F3 F2, then almost every Cayley (di)graph of G that is not in F1 is in F2?

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Theorem Let G ≤ Sn contain a regular cyclic subgroup hρi. Then one of the following statements holds: (2) ∼ 1. G = G1 × G2 × · · · × Gr , where r ≥ 1, each Gi = Sni , or Gi contains a normal regular cyclic group of order ni , such that gcd(ni , nj ) = 1 and n = n1n2 ··· nr , or 2. if G ≤ Aut(Γ) for a circulant digraph Γ, then Γ is a semiwreath product. The same result was obtained by Dobson and J. Morris for square-free integers n in 2005.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs The same result was obtained by Dobson and J. Morris for square-free integers n in 2005.

Automorphism Groups of Circulant Digraphs The following result is a group-theoretic translation of several papers on Schur rings by Leung and Man (1996 and 1998), and Evdomikov and Ponomarenko (2002), and appears in a paper by Li (2005). Theorem Let G ≤ Sn contain a regular cyclic subgroup hρi. Then one of the following statements holds: (2) ∼ 1. G = G1 × G2 × · · · × Gr , where r ≥ 1, each Gi = Sni , or Gi contains a normal regular cyclic group of order ni , such that gcd(ni , nj ) = 1 and n = n1n2 ··· nr , or 2. if G ≤ Aut(Γ) for a circulant digraph Γ, then Γ is a semiwreath product.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Automorphism Groups of Circulant Digraphs The following result is a group-theoretic translation of several papers on Schur rings by Leung and Man (1996 and 1998), and Evdomikov and Ponomarenko (2002), and appears in a paper by Li (2005). Theorem Let G ≤ Sn contain a regular cyclic subgroup hρi. Then one of the following statements holds: (2) ∼ 1. G = G1 × G2 × · · · × Gr , where r ≥ 1, each Gi = Sni , or Gi contains a normal regular cyclic group of order ni , such that gcd(ni , nj ) = 1 and n = n1n2 ··· nr , or 2. if G ≤ Aut(Γ) for a circulant digraph Γ, then Γ is a semiwreath product. The same result was obtained by Dobson and J. Morris for square-free integers n in 2005.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs while Dobson and Morris have found an algorithm to determine the full automorphism group of a circulant digraph of square-free order in time O(n3 log n).

This may seem to end the story, but it does not. There are many interesting classes of circulant digraphs for which more precise information can be obtained, as there are usually many different possible automorphism groups of semiwreath circulants. We give several examples, the first due to I. Kovácsand independently to C. H. Li (2005): Theorem Let Γ be a connected arc transitive circulant of order n which is not a complete graph. Then either (1) Γ is a normal circulant, or (2) there exists an arc transitive circulant Γ1 of order m such that n = mb with b, m > 1 and Γ = Γ1 o K¯b, or Γ1 o K¯b − bΓ1 with (b, m) = 1.

Ponomarenko also gave a polynomial time algorithm (nO(1)) to determine the full automorphism group of a circulant digraph in 2006,

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs There are many interesting classes of circulant digraphs for which more precise information can be obtained, as there are usually many different possible automorphism groups of semiwreath circulants. We give several examples, the first due to I. Kovácsand independently to C. H. Li (2005): Theorem Let Γ be a connected arc transitive circulant of order n which is not a complete graph. Then either (1) Γ is a normal circulant, or (2) there exists an arc transitive circulant Γ1 of order m such that n = mb with b, m > 1 and Γ = Γ1 o K¯b, or Γ1 o K¯b − bΓ1 with (b, m) = 1.

Ponomarenko also gave a polynomial time algorithm (nO(1)) to determine the full automorphism group of a circulant digraph in 2006, while Dobson and Morris have found an algorithm to determine the full automorphism group of a circulant digraph of square-free order in time O(n3 log n).

This may seem to end the story, but it does not.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs We give several examples, the ﬁrst due to I. Kov´acsand independently to C. H. Li (2005): Theorem Let Γ be a connected arc transitive circulant of order n which is not a complete graph. Then either (1) Γ is a normal circulant, or (2) there exists an arc transitive circulant Γ1 of order m such that n = mb with b, m > 1 and Γ = Γ1 o K¯b, or Γ1 o K¯b − bΓ1 with (b, m) = 1.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs the ﬁrst due to I. Kov´acsand independently to C. H. Li (2005): Theorem Let Γ be a connected arc transitive circulant of order n which is not a complete graph. Then either (1) Γ is a normal circulant, or (2) there exists an arc transitive circulant Γ1 of order m such that n = mb with b, m > 1 and Γ = Γ1 o K¯b, or Γ1 o K¯b − bΓ1 with (b, m) = 1.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Ponomarenko also gave a polynomial time algorithm (nO(1)) to determine the full automorphism group of a circulant digraph in 2006, while Dobson and Morris have found an algorithm to determine the full automorphism group of a circulant digraph of square-free order in time O(n3 log n).

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Theorem Let D be a circulant digraph of order n such that D is regular of degree at most 2p − 2, where p is the smallest prime divisor of n. Then one of the following is true:

1. D is a connected normal circulant digraph of Zn, 2. D has connection set (u + H) − {h}, where ∗ H = {0, n/p, 2n/p,..., (p − 1)n/p}, u ∈ Zn, and h ∈ u + H such 2 that h ≡ 0 (mod p), p does not divide n, and Aut(D) = Zn/p × Sp (note that it is possible in this case that n/p = 1),

3. D has no edges or every edge is a loop and Aut(D) = Sn, or 0 0 4. D is disconnected, D = K¯m o D , where D is a connected circulant digraph of order k, mk = n (and so D0 is one of the digraphs listed 0 above), and Aut(D) = Sm o Aut(D ).

The following results are in a manuscript by J.Ara´ujo,Dobson, J. Konieczny, and J. Morris.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs 2. D has connection set (u + H) − {h}, where ∗ H = {0, n/p, 2n/p,..., (p − 1)n/p}, u ∈ Zn, and h ∈ u + H such 2 that h ≡ 0 (mod p), p does not divide n, and Aut(D) = Zn/p × Sp (note that it is possible in this case that n/p = 1),

The following results are in a manuscript by J.Ara´ujo,Dobson, J. Konieczny, and J. Morris. Theorem Let D be a circulant digraph of order n such that D is regular of degree at most 2p − 2, where p is the smallest prime divisor of n. Then one of the following is true:

1. D is a connected normal circulant digraph of Zn,

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs 3. D has no edges or every edge is a loop and Aut(D) = Sn, or 0 0 4. D is disconnected, D = K¯m o D , where D is a connected circulant digraph of order k, mk = n (and so D0 is one of the digraphs listed 0 above), and Aut(D) = Sm o Aut(D ).

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs 0 0 4. D is disconnected, D = K¯m o D , where D is a connected circulant digraph of order k, mk = n (and so D0 is one of the digraphs listed 0 above), and Aut(D) = Sm o Aut(D ).

3. D has no edges or every edge is a loop and Aut(D) = Sn, or

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs The following results are in a manuscript by J.Ara´ujo,Dobson, J. Konieczny, and J. Morris. Theorem Let D be a circulant digraph of order n such that D is regular of degree at most 2p − 2, where p is the smallest prime divisor of n. Then one of the following is true:

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs 1.Γ is a normal digraph of Zn, 2.Γ=Γ 1 o K¯`, where `|n and Γ1 is a unit circulant digraph of order n/` that cannot be written as a nontrivial wreath product. Thus Aut(Γ) = Aut(Γ1) o S`. Furthermore, if p | ` is prime, then p | n/` as well,or

3.Γ=Γ 1 o K¯p − pΓ1, where p is prime, gcd(n/p, p) = 1, and Γ1 is a unit circulant digraph of order n/p that cannot be written as a nontrivial wreath product. Thus Γ is a deleted wreath product, and Aut(Γ) = Aut(Γ1) × Sp.

Theorem Let Γ be a unit circulant digraph of order n. Then one of the following is true:

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs 2.Γ=Γ 1 o K¯`, where `|n and Γ1 is a unit circulant digraph of order n/` that cannot be written as a nontrivial wreath product. Thus Aut(Γ) = Aut(Γ1) o S`. Furthermore, if p | ` is prime, then p | n/` as well,or

Theorem Let Γ be a unit circulant digraph of order n. Then one of the following is true:

1.Γ is a normal digraph of Zn,

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs or

Theorem Let Γ be a unit circulant digraph of order n. Then one of the following is true:

1.Γ is a normal digraph of Zn, 2.Γ=Γ 1 o K¯`, where `|n and Γ1 is a unit circulant digraph of order n/` that cannot be written as a nontrivial wreath product. Thus Aut(Γ) = Aut(Γ1) o S`. Furthermore, if p | ` is prime, then p | n/` as well,

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Theorem Let Γ be a unit circulant digraph of order n. Then one of the following is true:

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs It should be the case that for circulant graphs, the asymptotic behavior of non-normal circulant digraphs should be obtainable.

Theorem Almost every circulant graph has automorphism group as small as possible.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs Theorem Almost every circulant graph has automorphism group as small as possible.

It should be the case that for circulant graphs, the asymptotic behavior of non-normal circulant digraphs should be obtainable.

Ted Dobson Mississippi State University Automorphism Groups of Cayley Graphs 2 I all vertex-transitive graphs of order p (circulants by Klin and Pöschel,1978, Cayley graphs of Zp × Zp by D. and Witte, 2002) I all vertex-transitive graphs of order pq (D. 2005, many others involved) I Zp × Zp2 , p a prime (D. ) 3 I Zp (D. and I. Kovács) I circulants of odd prime-power order (Klin, Pöschel (Schur rings), with another later group theoretic proof by D.) I circulants of order a power of 2 (Klin, Najmark, and Pöschel(Schur rings)) I structure theorem for all circulants (Leung, Ma, Evdomikov and Ponomarenko 1996-2002 (Schur rings)) I polynomial time algorithm for determining full automorphism groups of all circulant digraphs (Ponomarenko, 2006)