We classify non-complete prime valency graphs satisfying the property that their automorphism group is transitive on both the set of arcs and the set of $2$-geodesics. We prove that either $Γ$ is 2-arc transitive or the valency $p$ satisfies $p\equiv 1\pmod 4$, and for each such prime there is a unique graph with this property: it is a non-bipartite antipodal double cover of the complete graph $K_{p+1}$ with automorphism group $PSL(2,p)\times Z_2$ and diameter 3.
One version of the polycirculant conjecture states that every vertex-transitive graph has a semiregular automorphism. We give a proof of the conjecture in the arc-transitive case for graphs of valency 8, which was the smallest open case.
We prove that every finite arc-transitive graph of valency twice a prime admits a nontrivial semiregular automorphism, that is, a non-identity automorphism whose cycles all have the same length. This is a special case of the Polycirculant Conjecture, which states that all finite vertex-transitive digraphs admit such automorphisms.
A graph is said to be {\em half-arc-transitive} if its automorphism group acts transitively on the set of its vertices and edges but not on the set of its arcs. With each half-arc-transitive graph of valency 4 a collection of the so called {\em alternating cycles} is associated, all of which have the same even length. Half of this length is called the {\em radius} of the graph in question. Moreover, any two adjacent alternating cycles have the same number of common vertices. If this number, the so called {\em attachment number}, coincides with the radius, we say that the graph is {\em tightly …
In this paper, we study arc-transitive Cayley graphs on non-abelian simple groups with soluble stabilizers and valency seven. Let $\Ga$ be such a Cayley graph on a non-abelian simple group $T$. It is proved that either $\Ga$ is a normal Cayley graph or $\Ga$ is $S$-arc-transitive, with $(S,T)=(\A_n,\A_{n-1})$ and $n=7,21,63$ or $84$; and, for each of these four values of $n$, there really exists arc-transitive $7$-valent non-normal Cayley graphs on $\A_{n-1}$ and specific examples are constructed.
A half-arc-transitive graph is a regular graph that is both vertex- and edge-transitive, but is not arc-transitive. If such a graph has finite valency, then its valency is even, and greater than $2$. In 1970, Bouwer proved that there exists a half-arc-transitive graph of every even valency greater than 2, by giving a construction for a family of graphs now known as $B(k,m,n)$, defined for every triple $(k,m,n)$ of integers greater than $1$ with $2^m \equiv 1 \mod n$. In each case, $B(k,m,n)$ is a $2k$-valent vertex- and edge-transitive graph of order $mn^{k-1}$, and Bouwer showed that $B(k,6,9…
A graph is edge-transitive if its automorphism group acts transitively on the edge set. In this paper, we investigate the automorphism groups of edge-transitive graphs of odd order and twice prime valency. Let $Γ$ be a connected graph of odd order and twice prime valency, and let $G$ be a subgroup of the automorphism group of $\Ga$. In the case where $G$ acts transitively on the edges and quasiprimitively on the vertices of $\Ga$, we prove that either $G$ is almost simple or $G$ is a primitive group of affine type. If further $G$ is an almost simple primitive group then, with two exceptions, t…
Given integers $k$ and $m$, we construct a $G$-arc-transitive graph of valency $k$ and an $L$-arc-transitive oriented digraph of out-valency $k$ such that $G$ and $L$ both admit blocks of imprimitivity of size $m$.
Let $G$ be a group and let $S$ be an inverse-closed and identity-free generating set of $G$. The \emph{Cayley graph} $\Cay(G,S)$ has vertex-set $G$ and two vertices $u$ and $v$ are adjacent if and only if $uv^{-1}\in S$. Let $CAY_d(n)$ be the number of isomorphism classes of $d$-valent Cayley graphs of order at most $n$. We show that $\log(CAY_d(n))\inΘ(d(\log n)^2)$, as $n\to\infty$. We also obtain some stronger results in the case $d=3$.
It is shown that there are infinitely many connected vertex-transitive graphs that have no Hamilton decomposition, including infinitely many Cayley graphs of valency 6, and including Cayley graphs of arbitrarily large valency.
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.
The main result of this paper is that, if $Γ$ is a finite connected $4$-valent arc-transitive graph, then either $Γ$ is part of a well-understood family of graphs, or every non-identity automorphism of $Γ$ fixes at most $1/3$ of the vertices. As a corollary, we get a similar result for $3$-valent vertex-transitive graphs. Based on these results we propose a conjecture on the number of fixed points of non-identity automorphisms of vertex-transitive graphs of bounded valency.
We give certain properties which are satisfied by the descendant set of a vertex in an infinite, primitive, distance transitive digraph of finite out-valency and provide a strong structure theory for digraphs satisfying these properties. In particular, we show that there are only countably many possibilities for the isomorphism type of such a descendant set, thereby confirming a conjecture of the first Author. As a partial converse, we show that certain related conditions on a countable digraph are sufficient for it to occur as the descendant set of a primitive, distance transitive digraph.
We study finite four-valent graphs Gamma admitting an edge-transitive group G of automorphisms such that G determines and preserves an edge-orientation on Gamma, and such that at least one G-normal quotient is a cycle (a quotient modulo the orbits of a normal subgroup of G). We show on the one hand that the number of distinct cyclic G-normal quotients can be unboundedly large. On the other hand existence of independent cyclic G-normal quotients (that is, they are not extendable to a common cyclic G-normal quotient) places severe restrictions on the graph Gamma and we classify all examples. We …
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 …
We investigate the class of infinite distance-transitive digraphs $D$ of finite out-valency. We show that if $D$ is a weakly descendant-homogeneous in such a class then either (1) $D$ has property $Z$ and the reachability relation is not universal; or (2) $D$ does not have property $Z$, the reachability relation is universal and $D$ has infinite in-valency. Also, we show that earlier results, proved in the context of highly-arc-transitive digraphs, hold under the weaker condition of distance-transitivity. Finally, we give a description of a subclass of the class distance-transitive weakly desc…
A graph $\Ga=(V,E)$ is called a Cayley graph of some group $T$ if the automorphism group $\Aut(\Ga)$ contains a subgroup $T$ which acts on regularly on $V$. If the subgroup $T$ is normal in $\Aut(\Ga)$ then $\Ga$ is called a normal Cayley graph of $T$. Let $r$ be an odd prime. Fang et al. \cite{FMW} proved that, with a finite number of exceptions for finite simple group $T$, every connected symmetric Cayley graph of $T$ of valency $r$ is normal. In this paper, employing maximal factorizations of finite almost simple groups, we work out a possible list of those exceptions for $T$.
We develop a new framework for analysing finite connected, oriented graphs of valency 4, which admit a vertex-transitive and edge-transitive group of automorphisms preserving the edge orientation. We identify a sub-family of "basic" graphs such that each graph of this type is a normal cover of at least one basic graph. The basic graphs either admit an edge-transitive group of automorphisms that is quasiprimitive or biquasiprimitive on vertices, or admit an (oriented or unoriented) cycle as a normal quotient. We anticipate that each of these additional properties will facilitate effective furth…
We prove that every regular graph of valency at least four whose automorphism group contains a nilpotent subgroup acting transitively on the vertex set admits a nowhere-zero 3-flow.
Let $G$ be a permutation group on a set $Ω$ and recall that a base for $G$ is a subset of $Ω$ such that its pointwise stabiliser is trivial. In a recent paper, Burness and Giudici introduced the Saxl graph of $G$, denoted $Σ(G)$, with vertex set $Ω$ and two vertices adjacent if they form a base. If $G$ is transitive, then $Σ(G)$ is vertex-transitive and it is natural to consider its valency (which we refer to as the valency of $G$). In this paper we present a general method for computing the valency of any finite transitive group and we use it to calculate the exact valency of every primitive …
A graph $\G$ is {\em symmetric} or {\em arc-transitive} if its automorphism group $\Aut(\G)$ is transitive on the arc set of the graph, and $\G$ is {\em basic} if $\Aut(\G)$ has no non-trivial normal subgroup $N$ such that the quotient graph $\G_N$ has the same valency with $\G$. In this paper, we classify symmetric basic graphs of order $2qp^n$ and valency 5, where $q<p$ are two primes and $n$ is a positive integer. It is shown that such a graph is isomorphic to a family of Cayley graphs on dihedral groups of order $2q$ with $5\di (q-1)$, the complete graph $K_6$ of order $6$, the complete bi…
In 2011, Fang et al. in (J. Combin. Theory A 118 (2011) 1039-1051) posed the following problem: Classify non-normal locally primitive Cayley graphs of finite simple groups of valency $d$, where either $d\leq 20$ or $d$ is a prime number. The only case for which the complete solution of this problem is known is of $d=3$. Except this, a lot of efforts have been made to attack this problem by considering the following problem: Characterize finite nonabelian simple groups which admit non-normal locally primitive Cayley graphs of certain valency $d\geq4$. Even for this problem, it was only solved f…
A clique (resp., independent set) in a graph is strong if it intersects every maximal independent sets (resp., every maximal cliques). A graph is CIS if all of its maximal cliques are strong and localizable if it admits a partition of its vertex set into strong cliques. In this paper we prove that a clique $C$ in a vertex-transitive graph $Γ$ is strong if and only if $|C||I|=|V(Γ)|$ for every maximal independent set $I$ of $Γ$. Based on this result we prove that a vertex-transitive graph is CIS if and only if it admits a strong clique and a strong independent set. We classify all vertex-transi…
A graph $Γ$ is basic if Aut$Γ$ has no normal subgroup $N\ne1$ such that $Γ$ is a normal cover of the normal quotient graph $Γ_N$. In this paper, we completely determine the basic normal quotient graphs of all connected 7-valent symmetric graphs of order $2pq^n$ with $p < q$ odd primes, which consist of an infinite family of dihedrants of order $2p$ with $p\equiv1$(mod 7), and 6 specific graphs with order at most 310. As a consequence, it shows that, for any given positive integer n, there are only finitely many connected 2-arc-transitive 7-valent graphs of order $2pq^n$ with $7\ne p<q$ primes,…
A detailed description of the structure of two-ended arc-transitive digraphs is given. It is also shown that several sets of conditions, involving such concepts as Property Z, local quasi-primitivity and prime out-valency, imply that an arc-transitive digraph must be highly-arc-transitive. These are then applied to give a complete classification of two-ended highly-arc-transitive digraphs with prime in- and out-valencies.
A graph is called {\em half-arc-transitive} if its full automorphism group acts transitively on vertices and edges, but not on arcs. It is well known that for any prime $p$ there is no tetravalent half-arc-transitive graph of order $p$ or $p^2$. Xu~[Half-transitive graphs of prime-cube order, J. Algebraic Combin. 1 (1992) 275-282] classified half-arc-transitive graphs of order $p^3$ and valency $4$. In this paper we classify half-arc-transitive graphs of order $p^3$ and valency $6$ or $8$. In particular, the first known infinite family of half-arc-transitive Cayley graphs on non-metacyclic $p$…
In this paper, we introduce a method for finding all edge-transitive graphs of small order, using faithful representations of transitive permutation groups of small degree, and we explain how we used this method to find all edge-transitive graphs of order up to $47$, and all bipartite edge-transitive graphs of order up to $63$. We also give an answer to a 1967 question of Folkman about semi-symmetric graphs of large valency; in fact we show that for semi-symmetric graphs of order $2n$ and valency $d$, the ratio $d/n$ can be arbitrarily close to $1$.
A connected graph $Γ=(V,E)$ of valency at least $3$ is called a basic $2$-arc-transitive graph if its full automorphism group has a subgroup $G$ with the following properties: (i) $G$ acts transitively on the set of $2$-arcs of $Γ$, and (ii) every minimal normal subgroup of $G$ has at most two orbits on $V$. In her papers [17,18], Praeger proved a connected $2$-arc-transitive graph of valency at least $3$ is a normal cover of some basic $2$-arc-transitive graph, and characterized the group-theoretic structures for basic $2$-arc-transitive graphs. Based on Praeger's theorems on $2$-arc-transiti…
Tutte's 3-flow conjecture asserts that every 4-edge-connected graph has a nowhere-zero 3-flow. In this note we prove that every regular graph of valency at least four admitting a solvable arc-transitive group of automorphisms admits a nowhere-zero 3-flow.
This paper initiates the investigation of the family of $(G,s)$-geodesic-transitive digraphs with $s\geq 2$. We first give a global analysis by providing a reduction result. Let $Γ$ be such a digraph and let $N$ be a normal subgroup of $G$ maximal with respect to having at least $3$ orbits. Then the quotient digraph $Γ_N$ is $(G/N,s')$-geodesic-transitive where $s'=\min\{s,\diam(Γ_N)\}$, $G/N$ is either quasiprimitive or bi-quasiprimitive on $V(Γ_N)$, and $Γ_N$ is either directed or an undirected complete graph. Moreover, it is further shown that if $Γ$ is not $(G,2)$-arc-transitive, then $G/N…
A transitive permutation group is called elusive if it contains no semiregular element. We show that no group of automorphisms of a connected graph of valency at most four is elusive and determine all the elusive groups of automorphisms of connected digraphs of out-valency at most three.
It is well-known that a simple $G$-arc-transitive graph can be represented as a coset graph for the group $G$. This representation is extended to a construction of $G$-arc-transitive coset graphs $\Cos(G,H,J)$ with finite valency and finite edge-multiplicity, where $H, J$ are stabilisers in $G$ of a vertex and incident edge, respectively. Given a group $G=ła,z\r$ with $|z|=2$ and $|a|$ finite, the coset graph $\Cos(G,ła\r,łz\r)$ is shown, under suitable finiteness assumptions, to have exactly two different arc-transitive embeddings as a $G$-arc-transitive map $(V,E,F)$, namely, a {\it $G$-rota…
A connected graph of order $n$ admitting a semiregular automorphism of order $n/k$ is called a $k$-multicirculant. Highly symmetric multicirculants of small valency have been extensively studied, and several classification results exist for cubic vertex- and arc-transitive multicirculants. In this paper we study the broader class of cubic vertex-transitive graphs of order $n$ admitting an automorphism of order $n/3$ or larger that may not be semiregular. In particular, we show that any such graph is either a $k$-multicirculant for some $k \leq 3$, or it belongs to an infinite family of graphs …
We classify all the $2$-arc-transitive strongly regular graphs, and use this classification to study the family of finite $(G,3)$-geodesic-transitive graphs of girth $4$ or $5$ for some group $G$ of automorphisms. For this application we first give a reduction result on the latter family of graphs: let $N$ be a normal subgroup of $G$ which has at least $3$ orbits on vertices. We show that $Γ$ is a cover of its quotient $Γ_N$ modulo the $N$-orbits, and that either $Γ_N$ is $(G/N,3)$-geodesic-transitive of the same girth as $Γ$, or $Γ_N$ is a $(G/N,2)$-arc-transitive strongly regular graph, or $…
For a non-complete graph $Γ$, a vertex triple $(u,v,w)$ with $v$ adjacent to both $u$ and $w$ is called a $2$-geodesic if $u\neq w$ and $u,w$ are not adjacent. Then $Γ$ is said to be $2$-geodesic transitive if its automorphism group is transitive on both arcs and 2-geodesics. In previous work the author showed that if a $2$-geodesic transitive graph $Γ$ is locally disconnected and its automorphism group $\Aut(Γ)$ has a non-trivial normal subgroup which is intransitive on the vertex set of $Γ$, then $Γ$ is a cover of a smaller 2-geodesic transitive graph. Thus the `basic' graphs to study are th…
The polycirculant conjecture asserts that every vertex-transitive digraph has a semiregular automorphism, that is, a nontrivial automorphism whose cycles all have the same length. In this paper we investigate the existence of semiregular automorphisms of edge-transitive graphs. In particular, we show that any regular edge-transitive graph of valency three or four has a semiregular automorphism.
For a graph $Γ$, a positive integer $s$ and a subgroup $G\leq \Aut(Γ)$, we prove that $G$ is transitive on the set of $s$-arcs of $Γ$ if and only if $Γ$ has girth at least $2(s-1)$ and $G$ is transitive on the set of $(s-1)$-geodesics of its line graph. As applications, we first prove that the only non-complete locally cyclic $2$-geodesic transitive graphs are the complete multipartite graph $K_{3[2]}$ and the icosahedron. Secondly we classify 2-geodesic transitive graphs of valency 4 and girth 3, and determine which of them are geodesic transitive.
A graph $Γ$ is a bi-Cayley graph over a group $G$ if $G$ is a semiregular group of automorphisms of $Γ$ having two orbits. Let $G$ be a non-abelian metacyclic $p$-group for an odd prime $p$, and let $Γ$ be a connected bipartite bi-Cayley graph over the group $G$. In this paper, we prove that $G$ is normal in the full automorphism group ${\rm Aut}(Γ)$ of $Γ$ when $G$ is a Sylow $p$-subgroup of ${\rm Aut}(Γ)$. As an application, we classify half-arc-transitive bipartite bi-Cayley graphs over the group $G$ of valency less than $2p$. Furthermore, it is shown that there are no semisymmetric and no …
A graph is called cubic and tetravalent if all of its vertices have valency 3 and 4, respectively. It is called vertex-transitive and arc-transitive if its automorphism group acts transitively on its vertex-set and on its arc- set, respectively. In this paper, we combine some new theoretical results with computer calculations to construct all cubic vertex-transitive graphs of order at most 1280. In the process, we also construct all tetravalent arc-transitive graphs of order at most 640.
A finite simple graph $Γ$ is called a Nest graph if it is regular of valency $6$ and admits an automorphism $ρ$ with two orbits of the same length such that at least one of the subgraphs induced by these orbits is a cycle. We say that $Γ$ is core-free if no non-trivial subgroup of the group generated by $ρ$ is normal in $\mathrm{Aut}(Γ)$. In this paper, we show that, if $Γ$ is edge-transitive and core-free, then it is isomorphic to one of the following graphs: the complement of the Petersen graph, the Hamming graph $H(2,4)$, the Shrikhande graph and a certain normal $2$-cover of $K_{3,3}$ by $…
We compare three transitivity properties of finite graphs, namely, for a positive integer $s$, $s$-distance transitivity, $s$-geodesic transitivity and $s$-arc transitivity. It is known that if a finite graph is $s$-arc transitive but not $(s+1)$-arc transitive then $s\leq 7$ and $s\neq 6$. We show that there are infinitely many geodesic transitive graphs with this property for each of these values of $s$, and that these graphs can have arbitrarily large diameter if and only if $1\leq s\leq 3$. Moreover, for a prime $p$ we prove that there exists a graph of valency $p$ that is 2-geodesic trans…
A vertex triple $(u,v,w)$ of a graph is called a $2$-geodesic if $v$ is adjacent to both $u$ and $w$ and $u$ is not adjacent to $w$. A graph is said to be $2$-geodesic transitive if its automorphism group is transitive on the set of $2$-geodesics. In this paper, a complete classification of $2$-geodesic transitive graphs of order $p^n$ is given for each prime $p$ and $n\leq 3$. It turns out that all such graphs consist of three small graphs: the complete bipartite graph $K_{4,4}$ of order $8$, the Schläfli graph of order $27$ and its complement, and fourteen infinite families: the cycles $C_p,…
In this paper, we discuss the structural information about 2-arc-transitive (non-bipartite and bipartite) graphs of product action type. It is proved that a 2-arc-transitive graph of product action type requires certain restrictions on either the vertex-stabilizers or the valency. Based on the existence of some equidistant linear codes, a construction is given for 2-arc-transitive graphs of non-diagonal product action type, which produces several families of such graphs. Besides, a nontrivial construction is given for 2-arc-transitive bipartite graphs of diagonal product action type
An automorphism of a graph is called quasi-semiregular if it fixes a unique vertex of the graph and its remaining cycles have the same length. This kind of symmetry of graphs was first investigated by Kutnar, Malnič, Martínez and Marušič in 2013, as a generalization of the well-known semiregular automorphism of a graph. Symmetric graphs of valency three or four, admitting a quasi-semiregular automorphism, have been classified in recent two papers. Let $p\geq 5$ be a prime and $Γ$ a connected symmetric graph of valency $p$ admitting a quasi-semiregular automorphism. In this paper, we first prov…
Let X be a coherent configuration associated with a transitive group G. In terms of the intersection numbers of X, a necessary condition for the point stabilizer of G to be a TI-subgroup, is established. Furthermore, under this condition, X is determined up to isomorphism by the intersection numbers. It is also proved that asymptotically, this condition is also sufficient. More precisely, an arbitrary homogeneous coherent configuration satisfying this condition is associated with a transitive group, the point stabilizer of which is a TI-subgroup. As a byproduct of the developed theory, recent …
A graph is called {\em arc-transitive} (or {\em symmetric}) if its automorphism group has a single orbit on ordered pairs of adjacent vertices, and 2-arc-transitive its automorphism group has a single orbit on ordered paths of length 2. In this paper we consider the orders of such graphs, for given valency. We prove that for any given positive integer $k$, there exist only finitely many connected 3-valent 2-arc-transitive graphs whose order is $kp$ for some prime $p$, and that if $d\ge 4$, then there exist only finitely many connected $d$-valent 2-arc-transitive graphs whose order is $kp$ or $…
A CIS graph is a graph in which every maximal stable set and every maximal clique intersect. A graph is well-covered if all its maximal stable sets are of the same size, co-well-covered if its complement is well-covered, and vertex-transitive if, for every pair of vertices, there exists an automorphism of the graph mapping one to the other. We show that a vertex-transitive graph is CIS if and only if it is well-covered, co-well-covered, and the product of its clique and stability numbers equals its order. A graph is irreducible if no two distinct vertices have the same neighborhood. We classif…
Let $X$ be a finite vertex-transitive graph of valency $d$, and let $A$ be the full automorphism group of $X$. Then the arc-type of $X$ is defined in terms of the sizes of the orbits of the action of the stabiliser $A_v$ of a given vertex $v$ on the set of arcs incident with $v$. Specifically, the arc-type is the partition of $d$ as the sum $$n_1 + n_2 + \dots + n_t + (m_1 + m_1) + (m_2 + m_2) + \dots + (m_s + m_s),$$ where $n_1, n_2, \dots, n_t$ are the sizes of the self-paired orbits, and $m_1,m_1, m_2,m_2, \dots, m_s,m_s$ are the sizes of the non-self-paired orbits, in descending order. In …
A Frobenius group is a transitive but not regular permutation group such that only the identity element can fix two points. A finite Frobenius group can be expressed as $G = K \rtimes H$ with $K$ a nilpotent normal subgroup. A first-kind $G$-Frobenius graph is a Cayley graph on $K$ with connection set $S$ an $H$-orbit on $K$ generating $K$, where $H$ is of even order or $S$ consists of involutions. We classify all 6-valent first-kind Frobenius circulant graphs such that the underlying kernel $K$ is cyclic. We give optimal gossiping and routing algorithms for such a circulant and compute its fo…
Let $Γ$ be a finite X-symmetric graph with a nontrivial X-invariant partition $\mathcal {B}$ on $V(Γ)$ such that $Γ_{\mathcal {B}}$ is a connected (X,2)-arc-transitive graph and $Γ$ is not a multicover of $Γ_{\mathcal {B}}$. This article aims to give a characterization of $(Γ, X, \mathcal {B})$ for the case where $|Γ(C) \cap B| = 3$ for $B\in \mathcal {B}$ and $C \in Γ_{\mathcal {B}}(B)$. This investigation requires a study on (X,2)-arc-transitive graphs of valency 4 or 7. We give a characterization of tetravalent (X,2)-arc-transitive graphs at first; and as a byproduct, we prove that every te…
