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Extending the well-studied concept of graphical regular representations to bipartite graphs, a Haar graphical representation (HGR) of a group $G$ is a bipartite graph whose automorphism group is isomorphic to $G$ and acts semiregularly with the orbits giving the bipartition. The question of which groups admit an HGR was inspired by a closely related question of Est\'elyi and Pisanski in 2016, as well as Babai's work in 1980 on poset representations, and has been recently solved by Morris and Spiga. In this paper, we introduce the $m$-Haar graphical representation ($m$-HGR) as a natural generalization of HGR to $m$-partite graphs for $m\geq2$, and explore the existence of $m$-HGRs for any fixed group. This inquiry represents a more robust version of the existence problem of G$m$SRs as addressed by Du, Feng and Spiga in 2020. Our main result is a complete classification of finite groups $G$ without $m$-HGRs.

By developing a tangible way to decompose unipotent radicals into irreducible submodules of Singer cycles, we achieve a classification of solvable factors of finite classical groups of Lie type. This completes previous work on factorizations of classical groups with a solvable factor. In particular, it resolves the final uncertain case in the long-standing problem of determining exact factorizations of almost simple groups. As a byproduct of the classification, we also obtain a new characterization of one-dimensional transitive groups, offering further insights into their group structure.

This paper represents a significant leap forward in the problem of enumerating vertex-transitive graphs. Recent breakthroughs on symmetry of Cayley (di)graphs show that almost all finite Cayley (di)graphs have the smallest possible automorphism group. Extending the scope of these results, we enumerate (di)graphs admitting a fixed semiregular group of automorphisms with m orbits. Moreover, we consider the more intricate inquiry of prohibiting arcs within each orbit, where the special case m = 2 is known as the problem of finding Haar graphical representations (HGRs). We significantly advance the understanding of HGRs by proving that the proportion of HGRs among Haar graphs of a finite nonabelian group approaches 1 as the group order grows. As a corollary, we obtain an improved bound on the proportion of DRRs among Cayley digraphs in the solution of Morris and the second author to the Babai-Godsil conjecture.

We classify the factorizations of finite classical groups with nonsolvable factors, completing the classification of factorizations of finite almost simple groups.

A Cayley graph on the symmetric group $S_n$ is said to have the Aldous property if its strictly second largest eigenvalue (that is, the largest eigenvalue strictly smaller than the degree) is attained by the standard representation of $S_n$. For $1\leq r < k < n$, let $C(n,k;r)$ be the set of $k$-cycles of $S_n$ moving every point in $\{1, \ldots, r\}$. Recently, Siemons and Zalesski [J. Algebraic Combin. 55 (2022) 989--1005] posed a conjecture which is equivalent to saying that for any $n \ge 5$ and $1\leq r

A Cayley digraph Cay(G,S) of a finite group $G$ with respect to a subset $S$ of $G$ is said to be a CI-digraph if for every Cayley digraph Cay(G,T) isomorphic to Cay(G,S), there exists an automorphism $\sigma$ of $G$ such that $S^\sigma=T$. A finite group $G$ is said to have the $m$-DCI property for some positive integer $m$ if all $m$-valent Cayley digraphs of $G$ are CI-digraphs, and is said to be a DCI-group if $G$ has the $m$-DCI property for all $1\leq m\leq |G|$. Let $\mathrm{Q}_{4n}$ be a generalized quaternion group of order $4n$ with an integer $n\geq 3$, and let $\mathrm{Q}_{4n}$ have the $m$-DCI property for some $1 \leq m\leq 2n-1$. It is shown in this paper that $n$ is odd, and $n$ is not divisible by $p^2$ for any prime $p\leq m-1$. Furthermore, if $n\geq 3$ is a power of a prime $p$, then $\mathrm{Q}_{4n}$ has the $m$-DCI property if and only if $p$ is odd, and either $n=p$ or $1\leq m\leq p$., Comment: 16

For positive integers $k$ and $n$, the shuffle group $G_{k,kn}$ is generated by the $k!$ permutations of a deck of $kn$ cards performed by cutting the deck into $k$ piles with $n$ cards in each pile, and then perfectly interleaving these cards following a certain permutation of the $k$ piles. For $k=2$, the shuffle group $G_{2,2n}$ was determined by Diaconis, Graham and Kantor in 1983. The Shuffle Group Conjecture states that, for general $k$, the shuffle group $G_{k,kn}$ contains $\mathrm{A}_{kn}$ whenever $k\notin\{2,4\}$ and $n$ is not a power of $k$. In particular, the conjecture in the case $k=3$ was posed by Medvedoff and Morrison in 1987. The only values of $k$ for which the Shuffle Group Conjecture has been confirmed so far are powers of $2$, due to recent work of Amarra, Morgan and Praeger based on Classification of Finite Simple Groups. In this paper, we confirm the Shuffle Group Conjecture for all cases using results on $2$-transitive groups and elements of large fixed point ratio in primitive groups.

For groups $G$ that can be generated by an involution and an element of odd prime order, this paper gives a sufficient condition for a certain Cayley graph of $G$ to be a graphical regular representation (GRR), that is, for the Cayley graph to have full automorphism group isomorphic to $G$. This condition enables one to show the existence of GRRs of prescribed valency for a large class of groups, and in this paper, $k$-valent GRRs of finite nonabelian simple groups with $k\geq5$ are considered.

We study the normal Cayley graphs $\mathrm{Cay}(S_n, C(n,I))$ on the symmetric group $S_n$, where $I\subseteq \{2,3,\ldots,n\}$ and $C(n,I)$ is the set of all cycles in $S_n$ with length in $I$. We prove that the strictly second largest eigenvalue of $\mathrm{Cay}(S_n,C(n,I))$ can only be achieved by at most four irreducible representations of $S_n$, and we determine further the multiplicity of this eigenvalue in several special cases. As a corollary, in the case when $I$ contains neither $n-1$ nor $n$ we know exactly when $\mathrm{Cay}(S_n, C(n,I))$ has the Aldous property, namely the strictly second largest eigenvalue is attained by the standard representation of $S_n$, and we obtain that $\mathrm{Cay}(S_n, C(n,I))$ does not have the Aldous property whenever $n \in I$. As another corollary of our main results, we prove a recent conjecture on the second largest eigenvalue of $\mathrm{Cay}(S_n, C(n,\{k\}))$ where $2 \le k \le n-2$., Comment: 27 pages

We estimate the number of graphical regular representations (GRRs) of a given group with large enough order. As a consequence, we show that almost all finite Cayley graphs have full automorphism groups 'as small as possible'. This confirms a conjecture of Babai-Godsil-Imrich-Lovasz on the proportion of GRRs, as well as a conjecture of Xu on the proportion of normal Cayley graphs, among Cayley graphs of a given finite group.

A pair of graphs $(\Gamma,\Sigma)$ is said to be stable if the full automorphism group of $\Gamma\times\Sigma$ is isomorphic to the product of the full automorphism groups of $\Gamma$ and $\Sigma$ and unstable otherwise, where $\Gamma\times\Sigma$ is the direct product of $\Gamma$ and $\Sigma$. In this paper, we reduce the study of the stability of any pair of regular graphs $(\Gamma,\Sigma)$ with coprime valencies and vertex-transitive $\Sigma$ to that of $(\Gamma,K_2)$. Since the latter is well studied in the literature, this enables us to determine the stability of any pair of regular graphs $(\Gamma,\Sigma)$ with coprime valencies in the case when $\Sigma$ is vertex-transitve and the stability of $(\Gamma,K_2)$ is known., Comment: 9 pages

This is the fifth one in a series of papers classifying the factorizations of almost simple groups with nonsolvable factors. In this paper we deal with orthogonal groups of plus type., Comment: arXiv admin note: text overlap with arXiv:2109.05459, arXiv:2107.09630

The fact that the adjacency matrix of every finite graph is diagonalizable plays a fundamental role in spectral graph theory. Since this fact does not hold in general for digraphs, it is natural to ask whether it holds for digraphs with certain level of symmetry. Interest on this question dates back to early 1980s, when P.~J.~Cameron asked for the existence of arc-transitive digraphs with non-diagonalizable adjacency matrix. This was answered in the affirmative by L.~Babai in 1985. Then Babai posed the open problem of constructing a 2-arc-transitive digraph and a vertex-primitive digraph whose adjacency matrices are not diagonalizable. In this paper, we solve Babai's problem by constructing an infinite family of $s$-arc-transitive digraphs for each integer $s\geq2$, and an infinite family of vertex-primitive digraphs, respectively, both of whose adjacency matrices are non-diagonalizable.

In 2017, Giudici, Li and the third author constructed the first known family of vertex-primitive $2$-arc-transitive digraphs of valency at least $2$. The smallest digraph in this family admits $\mathrm{PSL}_3(49)$ acting $2$-arc-transitively with vertex-stabilizer $\mathrm{A}_6$ and hence has $30758154560$ vertices. In this paper, we prove that this digraph is the vertex-primitive $2$-arc-transitive digraph of valency at least $2$ with fewest vertices.

Aldous' spectral gap conjecture states that the second largest eigenvalue of any connected Cayley graph on the symmetric group Sn with respect to a set of transpositions is achieved by the standard representation of Sn. This celebrated conjecture, which was proved in its general form in 2010, has inspired much interest in searching for other families of Cayley graphs on Sn with the property that the largest eigenvalue strictly smaller than the degree is attained by the standard representation of Sn. In this paper, we prove three results on normal Cayley graphs on Sn possessing this property for sufficiently large n, one of which can be viewed as a generalization of the "normal" case of Aldous' spectral gap conjecture., Comment: Final version published in European Journal of Combinatorics

A graphical regular representation (GRR) of a group $G$ is a Cayley graph of $G$ whose full automorphism group is equal to the right regular permutation representation of $G$. In this paper we study cubic GRRs of $\mathrm{PSL}_{n}(q)$ ($n=4, 6, 8$), $\mathrm{PSp}_{n}(q)$ ($n=6, 8$), $\mathrm{P}\Omega_{n}^{+}(q)$ ($n=8, 10, 12$) and $\mathrm{P}\Omega_{n}^{-}(q)$ ($n=8, 10, 12$), where $q = 2^f$ with $f \ge 1$. We prove that for each of these groups, with probability tending to $1$ as $q \rightarrow \infty$, any element $x$ of odd prime order dividing $2^{ef}-1$ but not $2^{i}-1$ for each $1 \le i < ef$ together with a random involution $y$ gives rise to a cubic GRR, where $e=n-2$ for $\mathrm{P}\Omega_{n}^{+}(q)$ and $e=n$ for other groups. Moreover, for sufficiently large $q$, there are elements $x$ satisfying these conditions, and for each of them there exists an involution $y$ such that $\{x,x^{-1},y\}$ produces a cubic GRR. This result together with certain known results in the literature implies that except for $\mathrm{PSL}_2(q)$, $\mathrm{PSL}_3(q)$, $\mathrm{PSU}_3(q)$ and a finite number of other cases, every finite non-abelian simple group contains an element $x$ and an involution $y$ such that $\{x,x^{-1},y\}$ produces a GRR, showing that a modified version of a conjecture by Spiga is true. Our results and several known results together also confirm a conjecture by Fang and Xia which asserts that except for a finite number of cases every finite non-abelian simple group has a cubic GRR.

A graphical regular representation (GRR) of a group $G$ is a Cayley graph of $G$ whose full automorphism group is equal to the right regular permutation representation of $G$. Towards a proof of the conjecture that only finitely many finite simple groups have no cubic GRR, this paper shows that $\mathrm{PSU}_3(q)$ has a cubic GRR if and only if $q\geq4$. Moreover, a cubic GRR of $\mathrm{PSU}_3(q)$ is constructed for each of these $q$.

Determining an upper bound on $s$ for finite vertex-primitive $s$-arc-transitive digraphs has received considerable attention dating back to a question of Praeger in 1990. It was shown by Giudici and Xia that the smallest upper bound on $s$ is attained for some digraph admitting an almost simple $s$-arc-transitive group. In this paper, based on the work of Pan, Wu and Yin, we prove that $s\leqslant 2$ in the case where the group is an alternating or symmetric group.

This is the fourth one in a series of papers classifying the factorizations of almost simple groups with nonsolvable factors. In this paper we deal with orthogonal groups of minus type., Comment: arXiv admin note: substantial text overlap with arXiv:2107.09630, arXiv:2106.03109, arXiv:2106.04278

This is the third one in a series of papers classifying the factorizations of almost simple groups with nonsolvable factors. In this paper we deal with orthogonal groups in odd dimension., Comment: arXiv admin note: substantial text overlap with arXiv:2106.04278, arXiv:2106.03109

This is the second one in a series of papers classifying the factorizations of almost simple groups with nonsolvable factors. In this paper we deal with almost simple unitary groups., Comment: arXiv admin note: substantial text overlap with arXiv:2106.03109

This is the first one in a series of papers classifying the factorizations of almost simple groups with nonsolvable factors. In this paper we deal with almost simple linear groups.

Let $\Gamma$ be a graph with vertex set $V$, and let $a$ and $b$ be nonnegative integers. A subset $C$ of $V$ is called an $(a,b)$-regular set in $\Gamma$ if every vertex in $C$ has exactly $a$ neighbors in $C$ and every vertex in $V\setminus C$ has exactly $b$ neighbors in $C$. In particular, $(0, 1)$-regular sets and $(1, 1)$-regular sets in $\Ga$ are called perfect codes and total perfect codes in $\Ga$, respectively. A subset $C$ of a group $G$ is said to be an $(a,b)$-regular set of $G$ if there exists a Cayley graph of $G$ which admits $C$ as an $(a,b)$-regular set. In this paper we prove that, for any generalized dihedral group $G$ or any group $G$ of order $4p$ or $pq$ for some primes $p$ and $q$, if a nontrivial subgroup $H$ of $G$ is a $(0, 1)$-regular set of $G$, then it must also be an $(a,b)$-regular set of $G$ for any $0\leqslant a\leqslant|H|-1$ and $0\leqslant b\leqslant |H|$ such that $a$ is even when $|H|$ is odd. A similar result involving $(1, 1)$-regular sets of such groups is also obtained in the paper., Comment: 9 pages

An interesting fact is that most of the known connected $2$-arc-transitive nonnormal Cayley graphs of small valency on finite simple groups are $(\mathrm{A}_{n+1},2)$-arc-transitive Cayley graphs on $\mathrm{A}_n$. This motivates the study of $2$-arc-transitive Cayley graphs on $\mathrm{A}_n$ for arbitrary valency. In this paper, we characterize the automorphism groups of such graphs. In particular, we show that for a non-complete $(G,2)$-arc-transitive Cayley graph on $\mathrm{A}_n$ with $G$ almost simple, the socle of $G$ is either $\mathrm{A}_{n+1}$ or $\mathrm{A}_{n+2}$. We also construct the first infinite family of $(\mathrm{A}_{n+2},2)$-arc-transitive Cayley graphs on $\mathrm{A}_n$.

In this paper, we show that every finite simple group of order at least $5$ admits an oriented regular representation of out-valency $2$.

This paper addresses the factorization problem of almost simple groups, studies regular subgroups of quasiprimitive permutation groups, and characterizes $s$-arc transitive Cayley graphs of finite simple groups. The main result of the paper presents a classification of exact factorizations of almost simple groups, which has been a long-standing open problem initiated around 1980 by the work of Wiegold-Williamson (1980), and significantly progressed by Liebeck, Praeger and Saxl in 2010. A by-product theorem of classifying exact factorizations of almost simple groups shows that, if $G=HK$ is a classical group of Lie type and $H,K$ are nonsolvable, then $H$ or $K$ is `almost maximal', with only a couple of exceptions. As applications of the classification result, (1) a classification is given of quasiprimitive groups containing a regular simple group; (2) a characterization is obtained for $s$-arc transitive Cayley graphs of simple groups, showing that $3$-arc transitive Cayley graphs of simple groups are surprisingly rare; (3) a question regarding biperfect Hopf algebras is confirmed.

We start up the study of the stability of general graph pairs. This notion is a generalization of the concept of the stability of graphs. We say that a pair of graphs $(\Gamma,\Sigma)$ is stable if $Aut(\Gamma\times\Sigma) \cong Aut(\Gamma)\times Aut(\Sigma)$ and unstable otherwise, where $\Gamma\times\Sigma$ is the direct product of $\Gamma$ and $\Sigma$. An unstable graph pair $(\Gamma,\Sigma)$ is said to be a nontrivially unstable graph pair if $\Gamma$ and $\Sigma$ are connected coprime graphs, at least one of them is non-bipartite, and each of them has the property that different vertices have distinct neighbourhoods. We obtain necessary conditions for a pair of graphs to be stable. We also give a characterization of a pair of graphs $(\Gamma, \Sigma)$ to be nontrivially unstable in the case when both graphs are connected and regular with coprime valencies and $\Sigma$ is vertex-transitive. This characterization is given in terms of the $\Sigma$-automorphisms of $\Gamma$, which are a new concept introduced in this paper as a generalization of both automorphisms and two-fold automorphisms of a graph., Comment: 22 pages

In a graph $\Gamma$ with vertex set $V$, a subset $C$ of $V$ is called an $(a,b)$-perfect set if every vertex in $C$ has exactly $a$ neighbors in $C$ and every vertex in $V\setminus C$ has exactly $b$ neighbors in $C$, where $a$ and $b$ are nonnegative integers. In the literature $(0,1)$-perfect sets are known as perfect codes and $(1,1)$-perfect sets are known as total perfect codes. In this paper we prove that, for any finite group $G$, if a non-trivial normal subgroup $H$ of $G$ is a perfect code in some Cayley graph of $G$, then it is also an $(a,b)$-perfect set in some Cayley graph of $G$ for any pair of integers $a$ and $b$ with $0\leqslant a\leqslant|H|-1$ and $0\leqslant b\leqslant |H|$ such that $\gcd(2,|H|-1)$ divides $a$. A similar result involving total perfect codes is also proved in the paper., Comment: 12 pages

This paper is devoted to the classification of flag-transitive 2-(v,k,2) designs. We show that apart from two known symmetric 2-(16,6,2) designs, every flag-transitive subgroup G of the automorphism group of a nontrivial 2-(v,k,2) design is primitive of affine or almost simple type. Moreover, we classify the 2-(v,k,2) designs admitting a flag transitive almost simple group G with socle PSL(n,q) for some n \geq 3. Alongside this analysis, we give a construction for a flag-transitive 2-(v,k-1,k-2) design from a given flag-transitive 2-(v,k,1) design which induces a 2-transitive action on a line. Taking the design of points and lines of the projective space PG(n-1,3) as input to this construction yields a G-flag-transitive 2-(v,3,2) design where G has socle PSL(n,3) and v=(3^n-1)/2. Apart from these designs, our PSL-classification yields exactly one other example, namely the complement of the Fano plane.

We prove that, given a finite graph $\Sigma$ satisfying some mild conditions, there exist infinitely many tetravalent half-arc-transitive normal covers of $\Sigma$. Applying this result, we establish the existence of infinite families of finite tetravalent half-arc-transitive graphs with certain vertex stabilizers, and classify the vertex stabilizers up to order $2^8$ of finite connected tetravalent half-arc-transitive graphs. This sheds some new light on the longstanding problem of classifying the vertex stabilizers of finite tetravalent half-arc-transitive graphs., Comment: 12 pages

Half-arc-transitive graphs are a fascinating topic which connects graph theory, Riemann surfaces and group theory. Although fruitful results have been obtained over the last half a century, it is still challenging to construct half-arc-transitive graphs with prescribed vertex stabilizers. Until recently, there have been only six known connected tetravalent half-arc-transitive graphs with nonabelian vertex stabilizers, and the question whether there exists a connected tetravalent half-arc-transitive graph with nonabelian vertex stabilizer of order $2^s$ for every $s\geqslant3$ has been wide open. This question is answered in the affirmative in this paper via the construction of a connected tetravalent half-arc-transitive graph with vertex stabilizer $\mathrm{D}_8^2\times\mathrm{C}_2^m$ for each integer $m\geqslant1$, where $\mathrm{D}_8^2$ is the direct product of two copies of the dihedral group of order $8$ and $\mathrm{C}_2^m$ is the direct product of $m$ copies of the cyclic group of order $2$. The graphs constructed have surprisingly many significant properties in various contexts.

A subset $C$ of the vertex set of a graph $\Gamma$ is called a perfect code in $\Gamma$ if every vertex of $\Gamma$ is at distance no more than $1$ to exactly one vertex of $C$. A subset $C$ of a group $G$ is called a perfect code of $G$ if $C$ is a perfect code in some Cayley graph of $G$. In this paper we give sufficient and necessary conditions for a subgroup $H$ of a finite group $G$ to be a perfect code of $G$. Based on this, we determine the finite groups that have no nontrivial subgroup as a perfect code, which answers a question by Ma, Walls, Wang and Zhou.

A reductive characterization of arc-transitive circulants was given independently by Kovacs in 2004 and the first author in 2005. In this paper, we give an explicit characterization of arc-transitive circulants and their automorphism groups. Based on this, we give a proof of the fact that arc-transitive circulants are all CI-digraphs.

In this paper, we classify finite quasiprimitive permutation groups with a metacyclic transitive subgroup, solving a problem initiated by Wielandt in 1949. It also involves the classification of factorizations of almost simple groups with a metacyclic factor., Comment: 25 pages

We study existence of Hamilton cycles in connected Cayley graphs on generalized dihedral groups, Comment: Sorry there is an error in the proof of the main theorem

The canonical double cover $\D(\Gamma)$ of a graph $\Gamma$ is the direct product of $\Gamma$ and $K_2$. If $\Aut(\D(\Gamma))\cong\Aut(\Gamma)\times\ZZ_2$ then $\Gamma$ is called stable; otherwise $\Gamma$ is called unstable. An unstable graph is said to be nontrivially unstable if it is connected, non-bipartite and no two vertices have the same neighborhood. In 2008 Wilson conjectured that, if the generalized Petersen graph $\GP(n,k)$ is nontrivially unstable, then both $n$ and $k$ are even, and either $n/2$ is odd and $k^2\equiv\pm 1 \pmod{n/2}$, or $n=4k$. In this note we prove that this conjecture is true. At the same time we determine all possible isomorphisms among the generalized Petersen graphs, the canonical double covers of the generalized Petersen graphs, and the double generalized Petersen graphs. Based on these we completely determine the full automorphism group of the canonical double cover of $\GP(n,k)$ for any pair of integers $n, k$ with $1 \leqslant k < n/2$., Comment: This is the final version that will appear in Journal of Graph Theory

The canonical double cover $\mathrm{D}(\Gamma)$ of a graph $\Gamma$ is the direct product of $\Gamma$ and $K_2$. If $\mathrm{Aut}(\mathrm{D}(\Gamma))=\mathrm{Aut}(\Gamma)\times\mathbb{Z}_2$ then $\Gamma$ is called stable; otherwise $\Gamma$ is called unstable. An unstable graph is nontrivially unstable if it is connected, non-bipartite and distinct vertices have different neighborhoods. In this paper we prove that every circulant graph of odd prime order is stable and there is no arc-transitive nontrivially unstable circulant graph. The latter answers a question of Wilson in 2008. We also give infinitely many counterexamples to a conjecture of Maru\v{s}i\v{c}, Scapellato and Zagaglia Salvi in 1989 by constructing a family of stable circulant graphs with compatible adjacency matrices.

A good knowledge of the Jordan canonical form (JCF) for a tensor product of `Jordan blocks' is key to understanding the actions of $p$-groups of matrices in characteristic $p$. The JCF corresponds to a certain partition which depends on the characteristic $p$, and the study of these partitions dates back to Aitken's work in 1934. Equivalently each JCF corresponds to a certain permutation $\pi$ introduced by Norman in 1995. These permutations $\pi = \pi(r,s,p)$ depend on the dimensions $r$, $s$ of the Jordan blocks, and on $p$. We give necessary and sufficient conditions for $\pi(r,s,p)$ to be trivial, building on work of M.J. Barry. We show that when $\pi(r,s,p)$ is nontrivial, it is an involution involving reversals. Finally, we prove that the group $G(r,p)$ generated by $\pi(r,s,p)$ for all $s$, `factors' as a wreath product corresponding to the factorisation $r=ab$ as a product of its $p'$-part $a$ and $p$-part $b$: precisely $G(r, p)={\sf S}_a\wr {\sf D}_b$ where ${\sf S}_a$ is a symmetric group of degree $a$, and ${\sf D}_b$ is a dihedral group of degree $b$., Comment: 27 pages, 5 tables, 1 figure Minor typos corrected

We study the covering radii of $2$-transitive permutation groups of Lie rank one, giving bounds and links to finite geometry.

This paper classifies the factorizations of almost simple groups with a factor having at least two nonsolvable composition factors. This together with a previous classification result of the authors reduces the factorization problem of almost simple groups to the case where both factors have a unique nonsolvable composition factor.