Vertex-primitive digraphs with large fixity

The relative fixity of a digraph $\Gamma$ is defined as the ratio between the largest number of vertices fixed by a nontrivial automorphism of $\Gamma$ and the number of vertices of $\Gamma$. We characterize the vertex-primitive digraphs whose relative fixity is at least $1/3$, and we show that there are only finitely many vertex-primitive digraphs of bounded out-valency and relative fixity exceeding a positive constant.


Introduction
Throughout this paper, we use the word digraph to denote a combinatorial structure Γ determined by a finite nonempty set of vertices V Γ and a set of arcs AΓ Ď V Γ ˆV Γ, sometimes also viewed as a binary relation on V Γ.If the set AΓ is symmetric (when viewed as a binary relation on V Γ), then the digraph Γ is called a graph and unordered pairs tu, vu such that pu, vq and pv, uq are arcs are called edges of Γ.
The fixity of a finite digraph Γ, denoted by FixpΓq, is defined as the largest number of vertices that are left fixed by a nontrivial automorphism of Γ, while the relative fixity of Γ is defined as the ratio RelFixpΓq " FixpΓq |V Γ| .
The notion of fixity of (di)graphs was introduced in a 2014 paper of L. Babai [2] (see also [4]), where several deep results regarding the fixity of strongly regular graphs were proved (these results were later used in his work on the graph isomorphism problem [3]).To convey the flavour of his work, let us mention [4,Theorem 1.6], which states that the relative fixity of a strongly regular graph (other then a complete bipartite graph or the line graph of a complete graph) is at most 7  8 .The study of the fixity of graphs continued in a series of papers [5,19,25] by P. Spiga and coauthors (including the authors of the present paper), where the problem was studied in the context of vertex-transitive graphs of fixed valency.
Let us mention that fixity is a well studied parameter in the slightly more general context of permutation groups, where, instead of fixity, it is more common to consider the dual notion of minimal degree of a permutation group G, defined by µpGq " min gPGzt1Gu |supppgq| , where supppgq denotes the set of all non-fixed points of g P G.Note that the fixity of a digraph Γ and the minimal degree of its automorphism group AutpΓq are related via the equality FixpΓq " |V pΓq| ´µpAutpΓqq .
A vast majority of papers on the topic of minimal degree of permutation groups (including the original work of Jordan on primitive permutation groups of minimal degree c for a fixed constant c) concentrates on primitive permutation groups (see, for example, [1,8,13,20,23,24]).It is thus natural to ask the following question: Question 1.What can be said about a digraph with large relative fixity whose automorphism group acts primitively on the vertex-set?
In this paper, we answer this question in the setting where the relative fixity is more than 1  3 .In our analysis, we rely heavily on the recent classification of primitive permutation groups of minimal degree at most 2  3 of the degree of the permutation group from [8].The essence of our work thus consists of determining the digraphs upon which the permutation groups from this classification act upon.
Before stating our main result, let us first introduce a few graph theoretical concepts and constructions.First, recall that the direct product of the family of digraphs Γ 1 , . . ., Γ r (sometimes also called the tensor product or the categorical product) is the digraph Γ 1 ˆ. . .ˆΓr whose vertex-set is the cartesian product V Γ 1 ˆ. . .ˆV Γ r and whose arc-set is ApΓ 1 ˆ. . .ˆΓr q " `pu 1 , . . ., u r q, pv 1 , . . ., v r q ˘ˇp u i , v i q P AΓ i for all i P t1, . . ., ru ( .
Recall also that a union of digraphs Γ 1 and Γ 2 is the digraph whose vertex-set and arc-set are the sets V Γ 1 Y V Γ 2 and AΓ 1 Y AΓ 2 , respectively.Note that when Γ 1 and Γ 2 share the same vertex-set, their union is then obtained simply by taking the union of their arc-sets.Further, for a positive integer m, let L m and K m denote the loop graph and the complete graph on a vertex-set V of cardinality m and with arc-sets tpv, vq : v P V u and tpu, vq : u, v P V, u " vu, respectively.We now have all the ingredients needed to present a construction yielding the digraph appearing in our main result.Construction 2. Let G " tΓ 0 , Γ 1 , . . ., Γ k u be a list of k `1 pairwise distinct digraphs sharing the same vertex-set ∆.Without loss of generality, we shall always assume that Γ 0 " L m with m " |∆|.Further, let r be a positive integer, and let J be a subset of the r-fold cartesian power X r , where X " t0, 1, . . ., ku.Given this input, construct the digraph Ppr, G, J q " ď pj1,j2,...,jrqPJ Γ j1 ˆΓj2 ˆ. . .ˆΓjr and call it the merged product action digraph.
Remark 3. We give some example to give a flavour of what can be obtained using Construction 2.
If r " 1, then Pp1, G, J q is simply the union of some digraphs from the set G.
If r " 2 and J " tp1, 0q, p0, 1qu, then Pp1, G, J q " L m ˆΓ1 Y Γ 1 ˆLm , which is, in fact, the Cartesian product Γ Γ. (This product is sometimes called the box product, and we refer to [14] for the definition of the Cartesian product.)More generally, if J " te i | i P t1, . . ., ruu, where e i " p0, . . ., 0, 1, 0, . . ., 0q is the r-tuple with 1 in the i-th component and zeroes elsewhere, then Ppr, G, J q " pΓ 1 q r , the r-th Cartesian power of the graph Γ 1 P G.More specifically, if Γ 1 " K m and J is as above, then Ppr, G, J q is the Hamming graph Hpr, mq " K r m .While J can be an arbitrary set of r-tuples in X r , we will be mostly interested in the case where J Ď X r is invariant under the induced action of some permutation group H ď Symprq on the set X r given by the rule pj 1 , j 2 , . . ., j r q h " pj 1h ´1 , j 2h ´1 , . . ., j rh ´1 q .
(Throughout this paper, in the indices, we choose to write ih ´1 instead of i h ´1 for improved legibility.)We shall say that J is an H-invariant subset of X r in this case.A subset J Ď X r which is H-invariant for some transitive subgroup of Symprq will be called homogeneous.
The last example of Remark 3 justifies the introduction of the following new family of graphs.
Definition 4. Let r, m be two positive integers, and let J Ď t0, 1u r be a homogeneous set.The graph P pr, tL m , K m u, J q is called generalised Hamming graph and is denoted by Hpr, m, J q.
Remark 5.The generalised Hamming graphs Hpr, m, J q, where J is H-invariant, are precisely the unions of orbital graphs for the group Sympmq wr H endowed with the product action (see Lemma 18 for further details).
Furthermore, a homogeneous set J is said to be Hamming if, for some nonnegative integers a, b such that a `b ď r and a transitive group H ď Symprq.It is said to be non-Hamming otherwise.
Remark 6.Let Ppr, G, J q be a merged product action digraph, where the digraphs in G have m vertices, and where J is a Hamming set.Build J 1 Ď t0, 1u r from J by substituting any nonzero entry of a sequence in J with 1. Then P pr, G, J q " P `r, tL m , K m u, J 1 ˘.
In particular, a generalised Hamming graph arises from Construction 2 if and only if J is a Hamming set.
Remark 7. The ordering of the Cartesian components in the definition of a Hamming set does not matter: indeed, a permutation of the components corresponds to a conjugation of the group H in Symprq, thus defining isomorphic digraphs in Construction 2.
We are ready to state our main result.
Theorem 8. Let Γ be a finite vertex-primitive digraph with at least one arc.Then RelFixpΓq ą 1 3 if and only if one of the following occurs: piq Γ is a generalised Hamming graph Hpr, m, J q, with m ě 4, and piiq Γ is a merged product action graph Ppr, G, J q, where r ě 1, where J is a non-Hamming subset of X r with X " t0, , where Γ 1 is a strongly regular graph listed in Section 4.4, Γ 2 is its complement, and RelFixpΓq " RelFixpΓ 1 q (the relative fixities are collected in Table 1).
Remark 9.Although we do not assume that a vertex-primitive digraph Γ in Theorem 8 is a graph, the assumption of large relative fixity forces it to be such.In other words, every vertex-primtive digraph of relative fixity larger than 1  3 is a graph.Remark 10.The relative fixity can be arbitrarily close to 1. Indeed, this can be achieved by choosing a generalised Hamming graph Hpr, m, J q with m arbitrarily large.
By analysing the vertex-primitive graphs of relative fixity more than 1  3 , one can notice that the out-valency of these graphs must grow as the number of vertices grows.More explicitly, a careful inspection of the families in Theorem 8 leads to the following result, the proof of which we leave out.
Remark 11.There exists a constant C such that every finite connected vertex-primitive digraph Γ with RelFixpΓq ą 1 3 satisfies valpΓq ě C log p|V Γ|q .
In particular, as both expressions are linear in r, a logarithmic bound in Remark 11 is the best that can be achieved.One of the consequences of Remark 11 is that for every positive integer d there exist only finitely many connected vertex-primitive digraphs of out-valency at most d and relative fixity exceeding 1  3 .As Theorem 12 and Corollary 13 show, this remains to be true if 1  3 is substituted by an arbitrary positive constant.We thank P. Spiga for providing us with the main ideas used in the proof.
Theorem 12. Let α and β be two positive constants, and let F be a family of quasiprimitive permutation groups G on Ω satisfying: paq µpGq ď p1 ´αq|Ω|; and pbq |G ω | ď β for every ω P Ω.Then F is a finite family.
Corollary 13.Let α be a positive constant, and let d be a positive integer.There are only finitely many vertex-primitive digraphs of out-valency at most d and relative fixity exceeding α.
The proof of Theorem 8 can be found in Section 5, while Theorem 12 and Corollary 13 are proved in Section 6.

Basic concepts and notations
2.1.Product action.We start by recalling the definition of a wreath product and its product action.By doing so, we also settle the notation for the rest of the paper.We refer to [12, Section 2.6 and 2.7] for further details.
Let H be a permutation group on a finite set Ω. Suppose that r " |Ω|, and, without loss of generality, identify Ω with the set t1, 2, . . ., ru.For an arbitrary set X, we may define a permutation action of H of rank r over X as the the action of H on the set X r given by the rule px 1 , x 2 , . . ., x r q h " px 1h ´1 , x 2h ´1 , . . ., x rh ´1 q .
Let K be a permutation group on a set ∆.We can consider the permutation action of H of rank r over K by letting pk 1 , k 2 , . . ., k r q h " pk 1h ´1 , k 2h ´1 , . . ., k rh ´1 q for all pk 1 , k 2 , . . ., k r q P K r , h P H .
If we denote by ϑ the homomorphism H Ñ AutpK r q corresponding to this action, then the wreath product of K by H, in symbols K wr H, is the semidirect product K r ⋊ ϑ H.We call K r the base group, and H the top group of this wreath product.
Note that the base and the top group are both embedded into K wr H via the monomorphisms In this way, we may view the base and the top group as subgroups of the wreath product and identify an element ppk 1 , k 2 , . . ., k r q, hq P K wr H with the product pk 1 , k 2 , . . ., k r qh of pk 1 , k 2 , . . ., k r q P K r and h P H (both viewed as elements of the group K wr H).
We recall the condition for a wreath product endowed with product action to be primitive.
Lemma 14 ([12, Lemma 2.7A]).Let K be a permutation group on ∆ and let H be a permutation group on Ω.The wreath product K wr H endowed with the product action on ∆ r is primitive if and only if H is transitive and K is primitive but not regular.
We now introduce some notation to deal with any subgroup G of Symp∆q wr SympΩq endowed with product action on ∆ r .
By abuse of notation, we identify the set ∆ with Moreover, recalling that every element of G can be written uniquely as gh, for some g P Symp∆q r and some h P SympΩq, we can define the group homomorphism This map defines a new permutational representation of G acting on Ω.We denote by G Ω the permutation group corresponding to the faithful action that G defines on Ω, that is, Recall that a primitive group G, according to the O'Nan-Scott classification (see, for instance, [22,IIIpbqpiq]), is said to be of product action type if there exists a transitive group H ď SympΩq and a primitive almost simple group K ď Symp∆q with socle T such that, for some integer r ě 2, where T r is the socle of G, thus contained in the base group K r .A detailed description of primitive groups of product action type was given by L. G. Kovács in [18].
Remark 15.By [26, Theorem 1.1 pbq], a group G of product action type is permutationally isomorphic to a subgroup of G ∆ ∆ wr G Ω .Therefore, up to a conjugation in Symp∆ r q, the group K can always be chosen as G ∆ ∆ , and H as G Ω .2.2.Groups acting on digraphs.We give a short summary of standard notations for digraphs and graphs.
If a subgroup G ď AutpΓq is primitive on V Γ, we say that Γ is G-vertex-primitive.In a similar way, if G is transitive on AΓ, we say that Γ is G-arc-transitive.The analogue notions can be defined for graphs, and when G " AutpΓq we drop the prefix G.
For any vertex v P VΓ, we denote by Γpvq its out-neighbourhood, that is, the set of vertices u P Γ such that pv, uq P AΓ.The size of the out-neighbourhood of a vertex v, |Γpvq|, is called out-valency of v.If Γ is G-vertex-primitive, for some group G, then the out-valency in independent of the choice of the vertex v, thus we will refer to it as the out-valency of Γ, in symbols valpΓq.Whenever Γ is a graph, neighbourhood and valency can be defined in the same way.
An orbital for G is an orbit of G in its induced action on Ω ˆΩ.An orbital digraphs for G is a digraph whose vertex-set is Ω, and whose arc-set is an orbital for G.An example of orbital for G is the diagonal orbital pω, ωq G , whose corresponding disconnected orbital graph is called diagonal orbital graph.We refer to [12, Section 3.2] for further details.
Note that an orbital graph for G is always G-arc-transitive, and, conversely, every G-arc-transitive digraph is an orbital graph for G. Furthermore, if G ď AutpΓq is a group of automorphism for a given digraph Γ, then Γ is a union of orbitals for G acting on VΓ.
The number of distinct orbital digraphs for G is called the permutational rank of G.In particular, 2-transitive permutation groups are precisely those of permutational rank 2.
If A Ď Ω ˆΩ is an orbital for G, then so is the set A ˚" tpβ, αq | pα, βq P Au.If A " A ˚, then the orbital A is called self-paired.Similarly, an orbital digraph is self-paired if its arc-set is a self-paired orbital.Note that any G-arc-transitive graph is obtained from a self-paired orbital digraph for G.

Orbital digraphs for wreath products in product action
We are interested in reconstructing the orbital digraphs of a wreath product K wr H endowed with product action once the orbital digraphs of K are known.Lemma 16.Let K wr H be a wreath product endowed with the product action on ∆ r , and let G " tΓ 0 , Γ 1 , . . ., Γ k u be the complete list of the orbital digraphs for K. Then any orbital digraph is a merged product action digraph of the form P `r, G, pj 1 , j 2 , . . ., j r q H ˘, for a sequence of indices pj 1 , j 2 , . . ., j r q P X r , where X " t0, 1, . . ., ku.
Proof.Let Γ be an orbital digraph for K wr H. Suppose that pu, vq P AΓ, where u " pu 1 , u 2 , . . ., u r q and v " pv 1 , v 2 , . . ., v r q.We aim to compute the K wr H-orbit of pu, vq, and, in doing so, proving that there is a sequence of indices pj 1 , j 2 , . . ., j r q P X r such that AΓ " AP `r, G, pj 1 , j 2 , . . ., j r q H ˘.
We start by computing the K r -orbit of pu, vq (where by K r we refer to the base group of K wr H).Since this action is componentwise, we obtain that pu, vq K r " A pΓ j1 ˆΓj2 ˆ. . .ˆΓjr q , where pu i , v i q is an arc of Γ ji for all i " 1, 2, . . ., r.
The top group H acts by permuting the components, so that pu, vq K wr H " ď the arc-sets of Γ and P `r, G, pj 1 , j 2 , . . ., j r q H ˘coincide.As their vertex-sets are both ∆ r , the proof is complete.Now that we know how to build the orbital digraphs for a permutation group in product action, we ask ourselves what can we say about the orbital digraphs of its subgroups.
Theorem 17.Let G ď Symp∆q wr SympΩq be a primitive group of product action type, and let T be the socle of G ∆ ∆ .Suppose that T and G ∆ ∆ share the same orbital digraphs.Then the orbital digraphs for G coincide with the orbital digraphs for G ∆ ∆ wr G Ω , or, equivalently, for T wr G Ω .Proof.Since G is a primitive group of product action type, we can suppose that G is a subgroup of G ∆ ∆ wr G Ω with socle T r , where r " |Ω|.Further, we set K " G ∆ ∆ , H " G Ω .As G ď K wr H, the partition of ∆ r ˆ∆r in arc-sets of orbital digraphs for K wr H is coarser than the one for G. Hence, our aim is to show that a generic orbital digraph for K wr H is also an orbital digraph for G.

Let
G " tΓ 0 , Γ 1 , . . ., Γ k u be the complete list of orbital digraphs for T acting on ∆, and let X " t0, 1, . . ., ku.Observe that the set of orbital digraphs for T r can be identified with the Cartesian product of r copies of G: indeed, we can bijectively map the generic orbital digraph T r , say Γ j1 ˆΓj2 ˆ. . .ˆΓjr , for some pj 1 , j 2 , . . ., j r q P X r , to the generic r-tuple of the Cartesian product G r of the form pΓ j1 , Γ j2 , . . ., Γ jr q.This point of view explains why H can act on the set of orbital digraphs for T r with its action of rank r.
Observe that the set of orbital digraphs for T r is equal to the set of orbital digraphs for K r .Moreover, T r is a subgroup of G, and K r is a subgroup of K wr H. Thus the orbital digraphs for G and for K wr H are obtained as a suitable unions of the elements of G r .
By Lemma 16, the orbital digraphs for K wr H are of the form for some pj 1 , j 2 , . . ., j r q P X r .Aiming for a contradiction, suppose that Γ j1 ˆΓj2 ˆ. . .ˆΓjr and Γ i1 ˆΓi2 ˆ. . .ˆΓir are two distinct orbital digraphs for T r that are merged under the action of top group H, but they are not under the action of G.The first portion of the assumption yields that there is an element h P H such that pΓ j1 ˆΓj2 ˆ. . .ˆΓjr q h " Γ i1 ˆΓi2 ˆ. . .ˆΓir .
By definition of H " G Ω , there is an element in G of the form pg 1 , g 2 , . . ., g r qh P G.
Therefore, the merging among these orbital graphs is also realised under the action of G, a contradiction.
By the initial remark, the proof is complete.

Daily specials
The aim of this section is to give a descriptions of the digraphs appearing in Theorem 8.
4.1.Generalised Hamming graphs.In this section, we clarify Remark 5 and we compute the relative fixity of the generalised Hamming graphs.
Lemma 18.Let H ď Symprq be a transitive permutation group, let G " Altp∆q wr H endowed with the product action on ∆ r , and let Γ be a digraph with vertex-set V Γ " ∆ r .Then G ď AutpΓq if and only if Γ is a generalised Hamming graph Hpr, m, J q, where |∆| " m and J Ď t0, 1u r is H-invariant.
Proof.By applying Lemma 16 and taking the union of the resulting orbital digraphs, we obtain the left-to-right direction of the equivalence.Let us now deal with the converse implication.Let Γ " Hpr, m, J q, where |∆| " m and J Ď t0, 1u r is H-invariant.By Construction 2 and Definition 4, for some non negative integers a, b such that a `b ď r.As each component of the graphs in parenthesis is either Moreover, as J is H-invariant, the action of rank r that H induces on ∆ r preserves the arc-set of Hpr, m, J q.As G is generated by Altpmq r and this H in their actions on ∆ r , this implies that G ď AutpΓq, as claimed.
Instead of directly computing the relative fixity of Hpr, m, J q, we prove the following stronger result.
Lemma 19.Let K wr H be a wreath product endowed with the product action on ∆ r , and let Γ be a digraph with vertex set ∆ r .Suppose that K wr H ď AutpΓq.Then RelFixpΓq " 1 ´µ pAutpΓq X Symp∆q r q |V Γ| .
In particular, the relative fixity of a generalised Hamming graph is RelFix pHpr, m, J qq " 1 ´2 m .
We claim that the automorphism that realizes the minimal degree must be contained in AutpΓq X Sympmq r (where Sympmq r is the base group of Sympmq wr Symprq).Indeed, upon choosing an element of minimal degree in Kˆtiduˆ. . .tidu and a transposition from the top group in Sympmq wr Symprq, we obtain the inequalities µ pAutpΓq X Sympmq r q ď µpKqm r´1 ď pm ´1qm r´1 ď min t|supppgq| | g P AutpΓqzSympmq r u This is enough to prove the first portion of the statement.
In particular, to compute the relative fixity of Hpr, m, J q, it is enough to look at the action of Sympmq on a single component.Thus, upon choosing a transposition in Sympmq ˆtidu ˆ. . .tidu, we obtain RelFix pHpr, m, J qq " 1 ´2m r´1 m r " 1 ´2 m .

Distance-i Johnson graphs.
The nomenclature dealing with possible generalizations of the Johnson graph is as lush as confusing.In this paper, we are adopting the one from [16].Let m, k, i be integers such that m ě 1, 1 ď k ď m and 0 ď i ď k.A distance-i Johnson graph, denoted by Jpm, k, iq is a graph whose vertex-set is the family of k-subsets of t1, 2, . . ., mu, and such that two k-subsets, say X and Y , are adjacent whenever |X X Y | " k ´i.The usual Johnson graph is then Jpm, k, 1q, and two subsets X and Y are adjacent in Jpm, k, iq if and only if they are at distance-i in Jpm, k, 1q.
Lemma 20.Let m, k be two positive integers such that m ě 2k `2.The orbital digraphs of Altpmq and of Sympmq in their action on k-subsets are the distance-i Johnson graphs Jpm, k, iq, one for each choice of i P t0, 1, . . ., ku.
Proof.Suppose that two k-subsets X and Y are such that pX, Y q is an arc of the considered orbital digraph and |X X Y | " k ´i, for a nonnegative integer i ď k.Since Altpmq is pm ´2q-transitive and 2k ď m ´2, the Altpmq-orbit of the arc pX, Y q contains all the pairs pZ, W q, where Z and W are k-subsets with |Z X W | " k ´i.Therefore, the statement is true for the alternating group.The same proof can be repeated verbatim for Sympmq.
Lemma 21.Let m, k, i be three positive integers such that m ě 2k `2 and i ‰ k.Then the relative fixity of the distance-i Johnson graphs Jpm, k, iq is RelFixpJpm, k, iqq " 1 ´2kpm ´kq mpm ´1q .
Proof.Under our assumption, by [

Squashed distance-i Johnson graphs.
A usual construction in the realm of distance transitive graphs consist in obtaining smaller example starting from a distance transitive graph and identifying vertices at maximal distance.We need to apply this idea to a family of generalised Johnson graphs.Consider the distance-i Johnson graph Jp2m, m, iq, for some integers m and i, with m positive and 0 ď i ď m.We say that two vertices of Jp2m, m, iq are disjoint if they have empty intersection as m-subset.Observe that being disjoint is an equivalence relation, and our definition coincides with the usual notion of antipodal for Jp2m, m, 1q seen as a metric space.We can build a new graph QJp2m, m, iq whose vertex-set is the set of equivalence classes of the disjoint relation, and such that, if rXs and rY s are two generic vertices, then prXs, rY sq is an arc in QJp2m, m, iq whenever there is a pair of representatives, say X 1 P rXs and Y 1 P rY s, such that pX 1 , Y 1 q is an arc in Jp2m, m, iq.We call QJp2m, m, iq an squashed distance-i Johnson graph.
Observe that Jp2m, m, iq is a regular double cover of QJp2m, m, iq.Furthermore, QJp2m, m, iq and QJp2m, m, m ´iq are isomorphic graphs, thus we can restrict the range of i to t0, 1, . . ., tm{2uu.
Lemma 22.Let m ě 3 be an integer.The orbital digraphs of Altp2mq and of Symp2mq in their primitive actions with stabilizer pSympmq wr C 2 q X Altp2mq and Sympmq wr C 2 respectively are the squashed distance-i Johnson graphs Jpm, k, iq, one for each choice of i P t0, 1, . . ., tm{2uu.
Proof.To start, we note that the set Ω on which the groups are acting can be identified with the set of partitions of the set t1, 2, . . ., 2mu with two parts of equal size m.Suppose that tX 1 , X 2 u and tY 1 , Y 2 u are two such partitions and that ptX 1 , X 2 u, tY 1 , Y 2 uq is an arc of the orbital digraph we are building, with mint|X 1 X Y 1 |, |X 1 X Y 2 |u " m ´i , for a nonnegative integer i ď tm{2u.To determine the image of ptX 1 , X 2 u, tY 1 , Y 2 uq under the group action, it is enough to know the images of X 1 and Y 2 , that is, of at most 2m ´rm{2s ď 2m ´2 distinct points.By the p2m ´2q-transitivity of Altp2mq, the Altp2mq-orbit of ptX 1 , X 2 u, tY 1 , Y 2 uq contains all the arc of the form ptZ 1 , Z 2 u, tW 1 , W 2 uq, where tZ 1 , Z 2 u, tW 1 , W 2 u P Ω and To conclude, observe that Ω is the set of m-subsets of t1, 2, . . ., 2mu in which two elements are identified if they are disjoint, and that is the adjacency condition in an squashed distance-i Johnson graph.As in Lemma 20, the same reasoning can be exteneded to Symp2mq.Therefore, the orbital digraphs of Altp2mq and of Symp2mq in these primitive actions are the squashed distance-i Johnson graphs QJp2m, m, iq, for some i P t0, 1, . . ., tm{2uu.
Lemma 23.Let m, i be two positive integers such that m ě 3 and i ‰ tm{2u.Then the relative fixity of the distance-i Johnson graphs QJp2m, m, iq is RelFixpQJp2m, m, iqq " 1 ´2kpm ´kq mpm ´1q .
Proof.Consider Jp2m, m, iq, the regular double covering of QJp2m, m, iq.In view of [ (see [8,Theorem 4]).Thus, we find that RelFixpQJp2m, m, iqq " 1 2 4.4.Strongly regular graphs.We list all the strongly regular graphs appearing as Γ 1 in Theorem 8 pcq.We divide them according to the socle L of the almost simple group that acts on them.Further, the present enumeration corresponds to the one of the groups that act on these graphs as listed in (the soon to be enunciated) Theorem 24 peq.piq L " U 4 pqq, q P t2, 3u, acting on totally singular 2-dimensional subspaces of the natural module, two vertices of Γ are adjacent if there is a third 2-dimensional subspace that intersect both vertices in a 1-dimensional subspace (see [7, Section 2.2.12]); piiq L " Ω 2m`1 p3q, m ě 2, acting on the singular points of the natural module, two vertices of Γ are adjacent if they are orthogonal (see [7, Theorem 2.2.12]); piiiq L " Ω 2m`1 p3q, m ě 2, acting on the nonsingular points of the natural module, two vertices of Γ are adjacent if the line that connects them is tangent to the quadric where the quadratic form vanishes (see [7,Section 3.1.4]);pivq L " PΩ ε 2m p2q, ε P t`, ´u, m ě 3, acting on the singular points of the natural module, two vertices of Γ are adjacent if they are orthogonal (see [7, 1 collects the usual parameters of a strongly regular graph, pv, d, λ, µq, and their relative fixity.Recall that v is the number of vertices, d is the valency of the graph, λ is the number of common neighbours between two adjacent vertices, and µ is the number of common neighbours between two nonadjacent vertices.As µpGq can be found in [8,Theorem 4] Table 1.Parameters of strongly regular graphs with large fixity.

Proof of Theorem 8
The primitive permutation groups we are concerned with were classified by T. Burness and R. Guralnick in [8].We report their result here.For the sake of our proof, we explicitly write the permutational rank of the almost simple groups of Lie type.This information can be easily obtained combining the complete list of 2-transitive finite permutation groups, first described by P. J. Cameron in [9, Section 5], and the complete list of classical finite permutation groups of permutational rank 3, compiled by W. M. Kantor  Then one of the following holds: paq Altpmq ď G ď Sympmq, for some m ě 3, in its action on k-subsets, for some k ă m{2; pbq G " Symp2mq, for some m ě 2, in its primitive action with stabilizer G α " Sympmq wr C 2 ; pcq G " M 22 : 2 in its primitive action of degree 22 with stabilizer G α " L 3 p4q.2 2 ; pdq G is an almost simple group of socle L and permutational rank 2, and one of the following occurs: piq L " L m p2q, m ě 3, in its natural action; piiq L " L m p3q, m ě 3, in its natural action, and G contains an element of the form p´I n´1 , I 1 q; piiiq L " Sp 2m p2q, m ě 3, in its action on the singular points of the natural module; pivq L " Sp 2m p2q, m ě 3, in its action on the right cosets of SO 2m p2q; pvq L " Sp 2m p2q, m ě 3, in its action on the right cosets of SO 2m p2q; peq G is an almost simple group of socle L and permutational rank 3, and one of the following occurs: piq L " U 4 pqq, q P t2, 3u, in its primitive action on totally singular 2-dimensional subspaces, and G contains the graph automorphism τ ; piiq L " Ω 2m`1 p3q in its action on the singular points of the natural module, and G contains an element of the form p´I 2m , I 1 q with a `-type p´1q-eigenspace; piiiq L " Ω 2m`1 p3q in its action on the nonsingular points of the natural module whose orthogonal complement is an orthogonal space of ´-type, and G contains an element of the form p´I 2m , I 1 q with a ´-type p´1q-eigenspace; pivq L " PΩ ε 2m p2q, ε P t`, ´u, in its action on the singular points on the natural module, and G " SO ε 2m p2q; pvq L " PΩ ε 2m p2q, ε P t`, ´u, in its action on the nonsingular points on the natural module, and G " SO ε 2m p2q; pviq L " PΩ 2m p3q in its action on the nonsingular points on the natural module, and G contains an element of the form p´I 2m´1 , I 1 q such that the discriminant of the 1dimensional 1-eigenspace is a nonsquare; pviiq L " PΩ 2m p3q in its action on the singular points on the natural module, and G contains an element of the form p´I 2m´1 , I 1 q; pviiiq L " PΩ 2m p3q in its action on the nonsingular points on the natural module, and G contains an element of the form p´I 2m´1 , I 1 q such that the discriminant of the 1dimensional 1-eigenspace is a square; pf q G ď K wr Symprq is a primitive group of product action type, where K is a permutation group appearing in points paq ´peq, the wreath product is endowed with the product action, and r ě 2; pgq G is an affine group with a regular normal socle N , which is an elementary abelian 2subgroup.
Proof of Theorem 8.The proof is split in two independent chunks.First, we prove that every vertexprimitive digraph of relative fixity exceeding 1 3 belongs to one of the families appearing in Theorem 8.Then, we tackle the problem of computing the relative fixities of the graphs appearing in Theorem 8, thus showing that they indeed all have relative fixity larger than 1  3 .Assume that Γ is a digraph on n vertices with at least one arc and with RelFixpΓq ą 1  3 such that G " AutpΓq is primitive.If Γ is disconnected, then the primitivity of G implies that Γ -L n .Hence we may assume that Γ is connected.Moreover, RelFixpΓq ą 1  3 implies that µpGq ă 2n 3 .Hence G is one of the groups determined in [8] and described in Theorem 24.Suppose that G is an almost simple group.Then G is one of the groups appearing in parts paq ´peq of Theorem 24.Since any G-vertex-primitive group is a union of orbital digraphs for G, the digraphs arising from these cases will be merged product action digraphs Pp1, G, J q (see Remark 3).Hence, our goal is to consider these almost simple groups in turn and compile their list of orbitals digraphs G.
Let G be a group as described in Theorem 24 paq.Lemma 20 states the orbital digraphs for G are the distance-i Johnson graph Jpm, k, iq.
Assume that k " 1, that is, consider the natural action of either Altpmq or Sympmq of degree m.Since this action is 2-transitive, their set of orbital digraphs is G " tL m , K m u.In particular, Pp1, G, J q " Hp1, m, J q.This case exhausts the generalized Hamming graphs with r " 1, which appear in Theorem 8 piq.Therefore, in view of Remark 6, for as long as we suppose r " 1, we can also assume that J is a non-Hamming homogeneous set.Observe m ě 4, otherwise, we go against our assumption on the relative fixity.
Going back to distance-i Johnson graphs, to guarantee that J is non-Hamming, we have to take k ě 2. Thus, G " tJpm, k, iq | i P t0, 1, . . ., kuu , which corresponds to Theorem 8 piiqpaq.
Let G " Symp2mq be a permutation group from Theorem 24 pbq.If m " 2, the degree of G is 3, and the relative fixity of any action of degree 3 can either be 0 or 1  3 .Hence, we must suppose that m ě 3: by Lemma 22, the orbital digraphs for G are the squashed distance-i Johnson graph QJp2m, m, iq.We obtain that G " tQJp2m, m, iq | i P t0, 1, . . ., tm{2uuu , as described in Theorem 8 piiqpbq.
Let G " M 22 : 2 in the action described in Theorem 24 pcq.Consulting the list of all the primitive groups of degree 22 in Magma [6] (which is based on the list compiled in [11]), we realize that they are all 2-transitive.Hence, the set of orbital digraphs is G " tK 22 , L 22 u.In particular, all the graphs are generalised Hamming graphs.
Let G be an almost simple of Lie type appearing in Theorem 24 pdq.Since all these groups are 2-transitive with a 2-transitive socle L, their orbital digraphs are either K m or L m , where m ě 7 is the degree of G. Once again, we obtain only generalise Hamming graphs.
Let G be an almost simple of Lie type described in Theorem 24 peq.Any group of permutational rank 3 defines two nondiagonal orbital digraphs, and, as such digraphs are arc-transitive and one the complement of the other, they are strongly regular digraphs (see, for instance, [7, Section 1.1.5]).The set of orbital digraphs is of the form G " tL m , Γ 1 , Γ 2 u, where we listed the possible Γ 1 in Section 4.4, and where m " |V Γ 1 |.The graphs described in this paragraph appear in Theorem 8 piiqpcq.
We have exhausted the almost simple groups from Theorem 24.Hence, we pass to Theorem 24 pf q.Suppose that G ď K wr Symprq is a primitive group of product action type.We want to apply Theorem 17 to G. The only hypothesis we miss is that T and G ∆ ∆ share the same set of orbital digraphs.
We claim that T and K induces the same set of orbital digraphs.If K is either alternating or symmetric, the claim follows from Lemmas 20 and 22.If K is 2-transitive, then we can observe that its socle L is also 2-transitive: the socle of M 22 : 2 is T " M 22 in its natural 3-transitive action, while the socle T of the almost simple groups of Lie type of rank 2 is 2-transitive by [9, Section 5].In particular, K and T both have G " tL m , K m u as their set of orbital graphs.Finally, suppose that K is an almost simple group of permutational rank 3. We have that its socle T is also of permutational rank 3 by [17,Theorem 1.1].Observe that, since any orbital digraph for T is a subgraph of an orbital digraph for G, the fact that G and L have permutational rank 3 implies that they share the same set of orbital digraphs.Therefore, the claim is true.
By our claim together with the double inclusion we obtain that T, G ∆ ∆ and K all induce the same set of orbital digraphs.Therefore, we can apply Theorem 17 to G: we obtain that G shares its orbital graphs with T wr G Ω .
Therefore, all the G-vertex-primitive digraphs are union of orbital digraphs for T wr H, with T socle type of G and H transitive permutation group isomorphic to G Ω .In other words, we found all the graphs Ppr, G, J q with r ě 2 described in Theorem 8. (Recall that, by Definition 4, among the graphs Ppr, G, J q, we find all the generalised Hamming graphs.)Suppose that G is an affine group with a regular normal socle N , which is an elementary abelian 2-subgroup.We have that G can be written as the split extension N : H, where H is a group of matrices that acts irreducibly on N .It follows that G We have completed the first part of the proof, showing that the list of vertex-primitive digraphs appearing in Theorem 8 is exhaustive.As all the orbital digraphs in G are actually graphs, the same property is true for the graphs in the list, as we have underlined in Remark 9.
We can now pass to the second part of the proof, that is, we can now tackle the computation of relative fixities.We already took care of the generalised Hamming graphs in Lemma 18.Thus, we can suppose that Γ is a merged product action graph Ppr, G, J q appearing in Theorem 8 piiq.
Suppose that r " 1, that is, Γ is a union of graphs for some primitive almost simple group K. (We are tacitely assuming that K is maximal among the groups appearing in a given part of Theorem 24.)In view of [21,Theorem], we have that K is a maximal subgroup of either Altp|V Γ|q or Symp|V Γ|q.Therefore, there are just two options for AutpΓq: either it is isomorphic to K or it contains Altp|V Γ|q.In the latter scenario, as Altp|V Γ|q is 2-transitive on the vertices, we obtain that Γ P tL m , K m , L m Y K m u.All those graphs are generalised Hamming graphs, against our assumption on Γ.Therefore, we have K " AutpΓq.In particular, the relative fixity for Γ are computed in Lemma 21, Lemma 23 or Table 1 given that G is described in Theorem 8 piiqpaq, piiqpbq or piiqpcq respectively.
Suppose now that r ě 2. The automorphism group of Γ either embeds into Sympmq wr Symprq, where m " |V Γ i | for any Γ i P G, or, by maximality of Sympmq wr Symprq, AutpΓq " Sympm r q.In the latter scenario, Γ P tL m , K m , L m Y K m u.All these graphs can be written as a merged product graph where r " 1 and J is a Hamming set.This goes against our assumption on Γ, thus we must suppose AutpΓq ‰ Sympm r q.
As a consequence, we obtain that, for some almost simple group K listed in Theorem 24 paq ´peq, and for some transitive group H ď Symprq, K wr H ď AutpΓq.Note that, as K ď AutpΓq ∆ ∆ , by [21,Theorem], AutpΓq ∆ ∆ is either K or it contains Altpmq.If the latter case occurs, then Altpmq r wr H ď AutpΓq.By Lemma 18, Γ is a generalised Hamming graph, which contradicts our choice of Γ.Therefore, AutpΓq ď K wr Symprq.
Observe that we can apply Lemma 19.We obtain that RelFixpΓq " 1 ´µpKqm r´1 m r " 1 ´µpKq m " RelFix `Pp1, G, J 1 q ˘, for some non-Hamming homogeneous set J 1 .In particular, the relative fixities for r ě 2 coincides with those we have already computed for r " 1.This complete the proof.

Proof of Theorem 12
Recall that a permutation group G on Ω is quasiprimitive if all its normal subgroups are transitive on Ω.Clearly, any primitive group is quasiprimitive.Moreover, recall that, by repeating the proof of Cauchy-Frobenius Lemma (see [12,  Proof of Theorem 12. (We would like to thank P. Spiga again for pointing out the key ingredients for this proof.)Let G be a quasiprimitive permutation group on a set Ω, and let x P Gzt1u be an element achieving |supppxq| ď p1 ´αq|Ω|.For any point ω P Ω, we obtain It follows that |x G | ď α ´1β.Now consider the normal subgroup of G defined by Recall that |G : C G pxq| " |x G |. Observe that G acts by conjugation on the set tC G px g q | g P Gu , it defines a single orbit of size |x G |, and N is the kernel of this action.Therefore that is, N is a bounded index subgroup of G. Since G is quasiprimitive, either N is trivial or N is transitive.Aiming for a contradiction, we suppose that N is transitive.Since rN, xs " 1, for any ω P Ω and for any n P N , ω nx " ω xn " ω n .
The transitivity of N implies that x " 1, against our choice of x.Therefore, N is trivial.It follows that Since there are finitely many abstract groups of bounded size, the proof is complete.
An equivalent formulation of Sims' Conjecture states that if G is a primitive permutation group and the minimal out-valency among its nondiagonal orbital digraphs is at most d, then the size of a point stabilizer is bounded from above by a function f pdq depending only on the positive integer d.An answer in the positive to this conjecture was given in [10].
Proof of Corollary 13.Let Γ be a vertex-primitive digraphs of out-valency at most d and relative fixity exceeding α, and let G " AutpΓq.The hypothesis on the out-valency implies that, for any v P V Γ, |G v | ď f pdq, where f pdq is the function that solves Sims' Conjecture.The result thus follows by choosing β " f pdq in Theorem 12.
We conclude the paper by observing that, as f pdq ě pd ´1q!, from Corollary 13 we cannot obtain a bound as sharp as that in Remark 11.
is 2-transitive on N , hence, as |N | ě 4, the graphs arising in this scenario are L |N | , K |N | and L |N | Y K |N | , which are generalised Hamming graphs.
Theorem 1.7A]) on the conjugacy class of a permutation x P G, we get fixpxq|x G | " |x G X G ω | where fixpxq " |Ω| ´|supppxq| is the number of fixed points of x.
15, Theorem 2 paq], the automorphism group of Jpm, k, iq is Sympmq in its action on k subsets.Its minimal degree is achieved by any transposition (see[13, 15, Theorem 2 peq], the automorphism group of Jp2m, m, iq is Symp2mq ˆSymp2q, where the central involution swaps pairs disjoint vertices.It follows that the automorphism group of QJp2m, m, iq is Symp2mq.Now, we can immediately verify that the stabilizer of the vertex tt1, 2, . . ., mu, tm 1, m `2, . . ., 2muu is Sympmq wr C 2 .The minimal degree of the primitive action of Symp2mq with stabilizer Sympmq wr C 2 is Theorem 2.2.12]); pvq L " PΩ ε 2m p2q, ε P t`, ´u, m ě 2, acting on the nonsingular points of the natural module, two vertices of Γ are adjacent if they are orthogonal (see [7, Section 3.1.2]);pviq L " PΩ 2m p3q, m ě 2 acting on the nonsingular points of the natural module, two vertices of Γ are adjacent if they are orthogonal (see [7, Section 3.1.3]);pviiq L " PΩ 2m p3q, m ě 3 acting on the singular points of the natural module, two vertices of Γ are adjacent if they are orthogonal (see [7, Theorem 2.2.12]); pviiiq L " PΩ 2m p3q, m ě 2 acting on the nonsingular points of the natural module, two vertices of Γ are adjacent if they are orthogonal (see [7, Section 3.1.3]).Table