Cubic Cayley graphs with small diameter

In this paper we apply Polya's Theorem to the problem of enumerating Cayley graphs on permutation groups up to isomorphisms induced by conjugacy in the symmetric group. We report the results of a search of all three-regular Cayley graphs on permutation groups of degree at most nine for small diameter graphs. We explore several methods of constructing covering graphs of these Cayley graphs. Examples of large graphs with small diameter are obtained.


Introduction
The (∆, D) problem asks for the largest value n such that a graph on n vertices exists with diameter D and maximum vertex degree ∆.The Moore bound for the diameter D of a graph with n vertices and maximum vertex degree ∆ ≥ 3 gives n ≤ 1 + ∆ + (∆ − 1)∆ + (∆ − 1) 2 ∆ + • • • + (∆ − 1) D−1 ∆.Very few graphs satisfy equality in the Moore bound.The evaluation of n(∆, D), the largest integer such that a graph on n(∆, D) vertices with maximum vertex degree ∆ and diameter D exists, appears to be an intractable problem in general.Even the more modest goal of proving the existence of a family of graphs satisfying D ≤ log ∆−1 (n)+O(1) seems difficult in the case ∆ = 3.For random regular graphs D = log ∆−1 (n log n)+O(1), and similar results have been obtained for constructions with random components.For details see [BC66] and [BV82].Jerrum and Skyum [JS84] have obtained the best non-random constructive results known for an infinite family of cubic graphs.Their constructions yield graphs which satisfy approximately D = 1.47 log 2 n.Kantor [Kan92] has shown the existence of cubic Cayley graphs with diameter O(log n), however the constant c such that D ≤ c log 2 n implicit in his estimate is very large.In addition to the random graph results and the constructions of infinite families, there is much interest in the construction of specific large graphs with small diameters.For a table of the largest known graphs for ∆ ≤ 10 and D ≤ 10, see [Exc00].These graphs establish lower bounds for n(∆, D).
We examine all three-regular Cayley graphs on permutation groups of degree at most nine.While the graphs obtained with D ≤ 10 are not as large as those in the table, these Cayley graphs form the building block for constructing larger graphs which can easily be analyzed.By forming covering graphs of the Cayley graphs in various ways we obtain graphs of D ≤ 21 which give good lower bounds for n(3, D) for 11 ≤ D ≤ 21.Moreover several examples of bipartite graphs with low degree and diameter are obtained.
We consider only undirected graphs, so the generator set A of our Cayley graphs will have three elements, and A = A −1 where A −1 = {α −1 | α ∈ A}.We refer to such sets as Cayley sets.For three-regular 1365-8050 c 2001 Maison de l'Informatique et des Mathématiques Discrètes (MIMD), Paris, France graphs the Cayley sets will have the form A = {σ, σ −1 , τ} for some σ and τ where τ has order two and σ has order at least three, or A = {τ 1 , τ 2 , τ 3 } where τ i has order two for each i.If α ∈ S n let α β = β −1 αβ be the conjugate of α by β, and if A ⊂ S n let A β = {α β | α ∈ A}.If A and B are Cayley sets in S n and A β = B for some β ∈ S n , then the Cayley graphs generated by A and B are isomorphic.Isomorphism of Cayley graphs from non-conjugate Cayley sets is difficult to determine from the Cayley sets alone.Thus in searching three-regular Cayley graphs we first address the problem of enumerating the possible Cayley sets up to conjugation.Although we are primarily interested in the sets generating cubic graphs, we tackle this problem in the more general setting.

Enumeration of Cayley Sets up to Conjugacy
Any Cayley graph of vertex degree m has a generator set of the form . ., τ l } where each σ i has order at least three, each τ j is an involution, and 2k + l = m.We will call such a set a Cayley set of type (k, l).Let X n be the set of equivalence classes in S n under the equivalence relation x ∼ = y if and only if x = y or x = y −1 .We will denote these equivalence classes by any appropriate representative.There is a one-to-one correspondence between Cayley sets of type (k, l) and the subsets of X n the form {σ 1 , . . ., σ k , τ 1 , . . ., τ l }.We will call these subsets of X n Cayley sets of type (k, l) also.
The action of S n on S n by conjugation induces an action of S n on X n .Enumerating the Cayley graphs of vertex degree m on S n up to isomorphisms induced by conjugation in S n is equivalent to enumerating the sets of type (k, l) in X n with 2k + l = m up to the action of S n by conjugation.To accomplish this we compute the cycle index of the action of S n by conjugation on X n .We first recall some notation and basic facts.We refer to [PR73] for background on cycle indices and Pólya's Theorem.
If G is a permutation group acting on a set X with n elements and g ∈ G then the we denote the cycle type of g by the monomial m(g) = ∏ n 1 x c i i , where g has c i cycles of length i in its disjoint cycle decomposition.The cycle index of G is defined as Z(G) = 1 |G| ∑ g∈G m(g) and the cycle sum by ZS(G) = ∑ g∈G m(g).We use Z(G; w 1 , w 2 . . ., w n ) to denote Z(G) with w i substituted for x i for each i.We denote Z(G; x r , x 2r . . ., x nr ) by Z r (G).
Let P n be the set of partitions of n.We also denote the elements of P n by monomials ∏ n 1 x c i i where ∑ n 1 ic i = n.Let h(a) be the number of permutations on S n with cycle type a.
Let C n and D n be the cyclic and dihedral groups of degree n.Then regarding C n and D n as permutation groups on {1, 2, . . ., n} in the usual manner we have and where φ denotes the Euler Phi function.
For each permutation α ∈ S n we will need the cycle index of the centralizer C(α) = {β ∈ S n |α β = α} and the cycle index of the pseudo-centralizer Since these cycle indices will depend only on the cycle type m(α) we will sometimes use the notations the wreath product of S k with C j .Pólya [Pól37] showed that cycle index of the wreath product The centralizer of a permutation α of cycle type ∏ n 1 x c i i is isomorphic to the product of the centralizers of permutations α i of type x c i i in S ic i , hence To obtain the cycle index of the pseudo-centralizer PC(α), we need to consider separately the set F(α).
We identify F n with the set of permutations in S n mapping an n-cycle to its inverse under conjugation.Although F n is not a group for n ≥ 3 we may consider its cycle index, and By equations 2 and 3 we obtain explicitly that Now consider F(α) where α ∈ S k j has cycle type x k j .Denote the cycle-index of F(α) by Z(F(x k j )) as it depends only on the cycle type.Note that if g ∈ F(α) then g determines a permutation γ(g) ∈ S k of the k j-cycles w 1 , w 2 , . . ., w k of α.If (w t 1 , w t 2 , . . ., w t r ) is in an r-cycle of γ(g) then g r determines a permutation of the elements of each w t s .Thus g r determines an element of F j (r odd) or C j (r even) when we consider its action on the elements of w t 1 .Each l cycle of F j or C j determined by g r determines an rl cycle of g.Moreover there are j r−1 possible permutations of the r j elements of the r j-cycles w t 1 , w t 2 , . . ., w t r determining the same r-cycle (w t 1 , w t 2 , . . ., w t r ) and β in F j or C j .Therefore the contribution of each x r in the cycle-sum ZS(S k ) to the cycle sum ZS(F(x j k )) is given by j r−1 ZS r (F j ) = j r Z r (F j ) (r odd) or j r−1 ZS r (C j ) = j r Z r (C j ) (r even).Thus ZS(F(x j k )) = ZS(S k ; jZ 1 (F j ), j 2 Z 2 (C j ), j 3 Z 2 (F j ), j 4 Z 4 (C j ), . ..), and dividing both sides by j k k! yields Now if α has cycle type ∏ n 1 x c i i , there is an isomorphism between ∏ n 1 PC(α i ) and a subgroup of S n mapping the product ), so by 5 and 7 we obtain the cycle index of the pseudo-centralizer: If G is any subgroup of S n and b is a cycle type, we use G.b to denote G regarded as a permutation group acting on the elements of X n of cycle type b by conjugation.We will identify a cycle type b with the set of permutations in S n having that cycle type.Recall from equation 1 that h(b) is the number of permutations in S n with cycle type b.We use o(β) for the order of a permutation β ∈ S n , and as this depends only on cycle type we use o(b) for the order of any element of cycle type b.We use [t]Q to denote the coefficient of a monomial t in a polynomial Q.Also if α has cycle type a = ∏ n 1 x c i i then the cycle type of α d is given by , where (i, d) is the greatest common divisor of i and d.We may denote this cycle type as a d , as it depends only on a and d.We denote the Möbius function by µ.With these preliminaries, we are now in a position to calculate the cycle index of the action of S n by conjugation on X n .

Enumeration Theorem:
[a]Z(PC(b)). Proof: Let F n (x, y) be the polynomial obtained from Z(S n .X n ) by substituting (1 + x i ), (1 + y i ) or 1 for each x (b,i) according to whenever o(b) ≥ 3, o(b) = 2 or o(b) = 1.By Pólya's Theorem the coefficient of x k y l in F n (x, y) will give the number of inequivalent (k, l) subsets of X n .Thus these polynomials are the generating functions for the number of (k, l) subsets of X n .We give the terms of F n (x, y) in Table 1 below up to O(y 7 ) for 3 ≤ n ≤ 10, considering x as O(y 2 ).We have investigated the three-regular Cayley graphs on subgroups of S 9 .We see from F 9 (x, y) that there are 2641 of these with generating set of type (1,1) and 12022 of type (0,3) for a total of 14,663 Cayley sets.When generating a complete list of 14,663 inequivalent Cayley sets it is desirable to know how many there are with each possible set of cycle-types for the generators.This information is contained in the polynomial Z(S n .X n ).For example the substitution of (1 ) and 1 for every other variable in Z(S 9 .X 9 ) enables us to read the number of inequivalent Cayley sets with three generators of type x 3 1 x 3 2 or x 1 x 4 2 in S 9 .Alternatively we could use the above techniques to compute the action of C(α) on the set of permutations of types x 3 1 x 3 2 or x 1 x 4 2 in X 9 by conjugation, where α has type x 3 1 x 3 2 .
One further approach to enumeration of the Cayley sets with a specified set of cycle types for the generators is worthy of note.Let us consider the case of counting the number of Cayley sets in S 9 with three permutations of type x 1 x 4 2 .The type x 1 x 4 2 may be identified with an unlabeled graph Γ on 9 points with 4 disjoint edges.Any permutation α of type x 1 x 4 2 may be identified with a labeling of the graph Γ with numbers {1, 2, . . ., 9}.Equivalence up to relabeling in the graph context corresponds to equivalence up to conjugation with permutations.The number of unlabeled superpositions of n differently colored copies of Γ gives the number of inequivalent ordered n-tuples of permutations of type x 1 x 4 2 .If we regard the colors as interchangeable then the number of unlabeled superpositions of n differently colored copies of Γ gives the number of inequivalent n-multisets of permutations of type x 1 x 4 2 .We wish to count the number of three-sets.The three-multisets which do not have three distinct elements are in one-to-one correspondence with ordered pairs.Palmer and Robertson [PR73] show how to count superpositions of colored graphs.Their techniques cover the cases of interchangeable and non-interchangeable colors.There are 548 superpositions of three differently colored copies of Γ where the colors are interchangeable, and 12 superpositions of two copies of Γ where the colors are not interchangeable.Therefore we obtain 548 − 12 = 536 inequivalent Cayley sets with three permutations of type x 1 x 4 2 in S 9 .The graph analogy needs some modification for permutations which are not involutions, however the same technique remains applicable.
We refer to a Cayley graph with generator set of type (0, 3) or type (1, 1) in X 9 as a G(3, 9) graph.We examined every possible generating set for G(3, 9) up to equivalence under conjugation, using the enumeration results as a check on the correctness of the lists obtained.We performed an exhaustive examination of these graphs and tabulated below examples of graphs of diameter D such that no larger G(3, 9) graph of diameter D exists.For most values of D the largest G(3, 9) graph of diameter D is not unique, and for each such D we have selected one example rather that listing them all.While the G(3, 9) graphs do not yield any new lower bounds for n(3, D) with D ≤ 10, it is natural to explore further with the examples which have relatively low diameter.Given a generator set {σ 1 , τ 1 } for a G(3, 9) we may consider generator sets {σ 1 σ 2 , τ 1 τ 2 } where σ 2 and τ 2 are permutations fixing {1, 2, . . ., 9}, and τ 2 has order at most 2. The resulting graph will be a covering graph of the original graph.Taking σ 2 to be a k-cycle and τ 2 to be the identity yields a d-fold cover of the initial graph for some d dividing k.
There are many examples where this leads to a larger graph with little or no corresponding increase in the diameter.Taking σ 2 = τ 2 = (n + 1 n + 2) yields a bipartite double cover of the initial graph, assuming it was not bipartite to begin with.The girth calculations were performed by comparing the sequence of sphere sizes about the identity in the initial graph and comparing it to the corresponding sequence for the bipartite double cover.We also pursued taking combinations {σ 1 σ 2 , τ 1 τ 2 } where both {σ 1 , τ 1 } and {σ 2 , τ 2 } generated low diameter graphs.Likewise we investigated covering graphs of G(3, 9) graphs with generating sets {τ 1 , τ 2 , τ 3 } where each τ i is an involution for 1 ≤ i ≤ 3 by graphs with generating sets {τ 1 τ 4 , τ 2 τ 5 , τ 3 τ 6 }, where each τ j is an involution or the identity for 4 ≤ j ≤ 6.The best examples obtained are in Table 3 below.We include only examples which improve over the degree nine results.
The calculations pertaining to this paper were performed using Mathematica [Wol99].Searches of the GAP [Gro99] small group and transitive permutation group libraries did not yield any larger cubic graphs of given diameter.However we emphasize that these searches were not exhaustive.