Graphs Of Valency Research Articles

Semisymmetric Graphs of Order 2 p3 with Valency p2

A simple undirected regular graph is said to be semisymmetric if it is edge-transitive but not vertex-transitive. For a semisymmetric graph [Formula: see text] of order [Formula: see text], [Formula: see text] a prime, it is well known that [Formula: see text] is bipartite with two biparts having equal size. The complete classification of such graphs has been given for the full automorphism group [Formula: see text] acting unfaithfully on at least one bipart of [Formula: see text], which shows that there is only one infinite family of such graphs with valency [Formula: see text]. The graphs of this kind have been determined when [Formula: see text] acts faithfully and primitively on at least one bipart of [Formula: see text], and thus there is only one remaining case for classifying such graphs of valency [Formula: see text], [Formula: see text] acting faithfully and imprimitively on both biparts of [Formula: see text], which is dealt with in this paper. As a result, there is only one infinite family of semisymmetric graphs of order [Formula: see text] with valency [Formula: see text].

Nowhere-zero 3-flows in Cayley graphs on supersolvable groups

Tutte's 3-flow conjecture asserts that every 4-edge-connected graph admits a nowhere-zero 3-flow. We prove that this conjecture is true for every Cayley graph of valency at least four on any supersolvable group with a noncyclic Sylow 2-subgroup and every Cayley graph of valency at least four on any group whose derived subgroup is of square-free order.

Automorphisms of the double cover of a circulant graph of valency at most 7

Automorphisms of the double cover of a circulant graph of valency at most 7

Prime-valent symmetric graphs with a quasi-semiregular automorphism

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 problem regarding existence of semiregular automorphisms in vertex-transitive graphs. Symmetric graphs of valency three or four, admitting a quasi-semiregular automorphism, have been classified in recent two papers (Feng et al., 2019 [11]) and (Yin and Feng, 2021 [42]).Let Γ be a connected symmetric graph of prime valency p≥5 admitting a quasi-semiregular automorphism. In this paper, it is proved that either Γ is a connected Cayley graph Cay(M,S) such that M is a 2-group admitting a fixed-point-free automorphism of order p with S as an orbit of involutions, or Γ is a normal N-cover of a T-arc-transitive graph of valency p admitting a quasi-semiregular automorphism, where T is a non-abelian simple group and N is a nilpotent group. Further, for p=5 a complete classification of graphs Γ such that either Aut(Γ) has a solvable arc-transitive subgroup or Γ is T-arc-transitive with T a non-abelian simple group is given. Finally, a construction of an infinite family of symmetric graphs admitting a quasi-semiregular automorphism and having nonsolvable automorphism group is given.

NOWHERE-ZERO -FLOWS IN TWO FAMILIES OF VERTEX-TRANSITIVE GRAPHS

AbstractLet $\Gamma $ be a graph of valency at least four whose automorphism group contains a minimally vertex-transitive subgroup G. It is proved that $\Gamma $ admits a nowhere-zero $3$ -flow if one of the following two conditions holds: (i) $\Gamma $ is of order twice an odd number and G contains a central involution; (ii) G is a direct product of a $2$ -subgroup and a subgroup of odd order.

On locally primitive symmetric graph of valency twice prime

In this paper, all connected locally primitive symmetric graphs of valency 2r admitting a transitive alternating automorphism group are determined, where r is odd prime.

ON A CLASS OF EDGE-TRANSITIVE DISTANCE-REGULAR ANTIPODAL COVERS OF COMPLETE GRAPHS

The paper is devoted to the problem of classification of edge-transitive distance-regular antipodal covers of complete graphs. This extends the classification of those covers that are arc-transitive, which has been settled except for some tricky cases that remain to be considered, including the case of covers satisfying condition \(c_2=1\) (which means that every two vertices at distance 2 have exactly one common neighbour).Here it is shown that an edge-transitive distance-regular antipodal cover of a complete graph with \(c_2=1\) is either the second neighbourhood of a vertex in a Moore graph of valency 3 or 7, or a Mathon graph, or a half-transitive graph whose automorphism group induces an affine \(2\)-homogeneous group on the set of its fibres. Moreover, distance-regular antipodal covers of complete graphs with \(c_2=1\) that admit an automorphism group acting \(2\)-homogeneously on the set of fibres (which turns out to be an approximation of the property of edge-transitivity of such cover), are described. A well-known correspondence between distance-regular antipodal covers of complete graphs with \(c_2=1\) and geodetic graphs of diameter two that can be viewed as underlying graphs of certain Moore geometries, allows us to effectively restrict admissible automorphism groups of covers under consideration by combining Kantor's classification of involutory automorphisms of these geometries together with the classification of finite 2-homogeneous permutation groups.

Symmetric graphs of valency seven and their basic normal quotient graphs

Abstract We characterize seven valent symmetric graphs of order 2 p q n 2p{q}^{n} with p < q p\lt q odd primes, extending a few previous results. Moreover, a consequence partially generalizes the result of Conder, Li and Potočnik [On the orders of arc-transitive graphs, J. Algebra 421 (2015), 167–186].

On Symmetric Cayley Graphs of Valency Eleven

A Cayley graph [Formula: see text] is said to be normal if [Formula: see text] is normal in [Formula: see text]. In this paper, we investigate the normality problem of the connected 11-valent symmetric Cayley graphs [Formula: see text] of finite nonabelian simple groups [Formula: see text], where the vertex stabilizer [Formula: see text] is soluble for [Formula: see text] and [Formula: see text]. We prove that either [Formula: see text] is normal or [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text]. Further, 11-valent symmetric nonnormal Cayley graphs of [Formula: see text], [Formula: see text] and [Formula: see text] are constructed. This provides some more examples of nonnormal 11-valent symmetric Cayley graphs of finite nonabelian simple groups after the first graph of this kind (of valency 11) was constructed by Fang, Ma and Wang in 2011.

Half-Arc-Transitive Graphs and the Fano Plane

A subgroup G of the automorphism group of a graph \(\Gamma\) acts half-arc-transitively on \(\Gamma\) if the natural actions of G on the vertex-set and edge-set of \(\Gamma\) are both transitive, but the natural action of G on the arc-set of \(\Gamma\) is not transitive. When \(G ={\text{Aut}}(\Gamma )\) the graph \(\Gamma\) is said to be half-arc-transitive. Given a bipartite cubic graph with a certain degree of symmetry two covering constructions that provide infinitely many tetravalent graphs admitting half-arc-transitive groups of automorphisms are introduced. Symmetry properties of constructed graphs are investigated. In the second part of the paper the two constructions are applied to the Heawood graph, the well-known incidence graph of the Fano plane. It is proved that the members of the infinite family resulting from one of the two constructions are all half-arc-transitive, and that the infinite family resulting from the second construction contains a mysterious family of arc-transitive graphs that emerged within the classification of tightly attached half-arc-transitive graphs of valence 4 back in 1998 and 2008.

Tetravalent edge-transitive Cayley graphs of Frobenius groups

In this paper, we give a characterisation for a class of edge-transitive Cayley graphs and provide a method for constructing edge-transitive graphs of valency 4 with arbitrarily large vertex stabiliser. In particular, in the last section, we obtain certain extensions of the results of Li et al. (Tetravalent edge-transitive Cayley graphs with odd number of vertices, J Comb Theory Ser B 96:164–181, 2006) on half-transitive graphs.

Symmetric graphs of valency 4 having a quasi-semiregular automorphism

Feng et al. (2019) characterized connected G-symmetric graphs of valency 4 having a quasi-semiregular automorphism, namely, a graph automorphism fixing a unique vertex in the vertex set of the graph and keeping the lengths of all other orbits equal, when G is soluble or the vertex stabilizer in G is not a 2-group. In this paper we prove that a connected symmetric graph with valency 4 having a quasi-semiregular automorphism is a Cayley graph on a group G with respect to S, where G is abelian of odd order and S is an orbit of a group of automorphisms of the group G.

CLASSIFICATION OF TETRAVALENT -TRANSITIVE NONNORMAL CAYLEY GRAPHS OF FINITE SIMPLE GROUPS

AbstractA graph $\Gamma $ is called $(G, s)$ -arc-transitive if $G \le \text{Aut} (\Gamma )$ is transitive on the set of vertices of $\Gamma $ and the set of s-arcs of $\Gamma $ , where for an integer $s \ge 1$ an s-arc of $\Gamma $ is a sequence of $s+1$ vertices $(v_0,v_1,\ldots ,v_s)$ of $\Gamma $ such that $v_{i-1}$ and $v_i$ are adjacent for $1 \le i \le s$ and $v_{i-1}\ne v_{i+1}$ for $1 \le i \le s-1$ . A graph $\Gamma $ is called 2-transitive if it is $(\text{Aut} (\Gamma ), 2)$ -arc-transitive but not $(\text{Aut} (\Gamma ), 3)$ -arc-transitive. A Cayley graph $\Gamma $ of a group G is called normal if G is normal in $\text{Aut} (\Gamma )$ and nonnormal otherwise. Fang et al. [‘On edge transitive Cayley graphs of valency four’, European J. Combin.25 (2004), 1103–1116] proved that if $\Gamma $ is a tetravalent 2-transitive Cayley graph of a finite simple group G, then either $\Gamma $ is normal or G is one of the groups $\text{PSL}_2(11)$ , ${\rm M} _{11}$ , $\text{M} _{23}$ and $A_{11}$ . However, it was unknown whether $\Gamma $ is normal when G is one of these four groups. We answer this question by proving that among these four groups only $\text{M} _{11}$ produces connected tetravalent 2-transitive nonnormal Cayley graphs. We prove further that there are exactly two such graphs which are nonisomorphic and both are determined in the paper. As a consequence, the automorphism group of any connected tetravalent 2-transitive Cayley graph of any finite simple group is determined.

Arc-transitive graphs of valency twice a prime admit a semiregular automorphism

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.

Strong cliques in vertex‐transitive graphs

AbstractA clique (resp, independent set) in a graph is strong if it intersects every maximal independent set (resp, every maximal clique). A graph is clique intersect stable set (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 in a vertex‐transitive graph is strong if and only if for every maximal independent set of . On the basis of 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‐transitive graphs of valency at most 4 admitting a strong clique, and give a partial characterization of 5‐valent vertex‐transitive graphs admitting a strong clique. Our results imply that every vertex‐transitive graph of valency at most 5 that admits a strong clique is localizable. We answer an open question by providing an example of a vertex‐transitive CIS graph which is not localizable.

On Certain Edge-Transitive Bicirculants

A graph $\Gamma$ of even order is a bicirculant if it admits an automorphism with two orbits of equal length. Symmetry properties of bicirculants, for which at least one of the induced subgraphs on the two orbits of the corresponding semiregular automorphism is a cycle, have been studied, at least for the few smallest possible valences. For valences $3$, $4$ and $5$, where the corresponding bicirculants are called generalized Petersen graphs, Rose window graphs and Tabač jn graphs, respectively, all edge-transitive members have been classified. While there are only 7 edge-transitive generalized Petersen graphs and only 3 edge-transitive Tabač jn graphs, infinite families of edge-transitive Rose window graphs exist. The main theme of this paper is the question of the existence of such bicirculants for higher valences. It is proved that infinite families of edge-transitive examples of valence $6$ exist and among them infinitely many arc-transitive as well as infinitely many half-arc-transitive members are identified. Moreover, the classification of the ones of valence $6$ and girth $3$ is given. As a corollary, an infinite family of half-arc-transitive graphs of valence $6$ with universal reachability relation, which were thus far not known to exist, is obtained.

On arc-transitive Cayley digraphs of out-valency 3

In this paper, based on the calculation using GAP, we give a classification result on arc-transitive Cayley digraphs of finite simple groups. Let G be a finite simple group and S⊂G∖{1} with |S|=3, S≠S−1 and G=〈SS−1〉. If the Cayley digraph Γ=Cay(G,S) is arc-transitive, then Γ is either normal or isomorphic to one of 382 Cayley digraphs of the alternating group A47. Further, we consider the underlying graphs and standard double covers of these 382 Cayley digraphs, and then we get 172 (non-isomorphic) half-transitive Cayley graphs of valency 6, and 144 semisymmetric cubic graphs.

Arc-transitive digraphs of given out-valency and with blocks of given size

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.

Symmetric graphs of valency five and their basic normal quotients

A graph Γ is symmetric or arc-transitive if its automorphism group Aut(Γ) is transitive on the arc set of the graph, and Γ is basic if Aut(Γ) has no non-trivial normal subgroup N such that the quotient graph ΓN has the same valency as Γ. In this paper, we classify symmetric basic graphs of order 2qpn 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|(q−1), the complete graph K6 of order 6, the complete bipartite graph K5,5 of order 10, or one of the nine sporadic coset graphs associated with non-abelian simple groups. As an application, connected pentavalent symmetric graphs of order kpn for some small integers k and n are classified.

The validity of Tutte's 3-flow conjecture for some Cayley graphs

Tutte’s 3 -flow conjecture claims that every bridgeless graph with no 3 -edge-cut admits a nowhere-zero 3 -flow. In this paper we verify the validity of Tutte’s 3 -flow conjecture on Cayley graphs of certain classes of finite groups. In particular, we show that every Cayley graph of valency at least 4 on a generalized dicyclic group has a nowhere-zero 3 -flow. We also show that if G is a solvable group with a cyclic Sylow 2 -subgroup and the connection sequence S with | S | ≥ 4 contains a central generator element, then the corresponding Cayley graph Cay( G , S ) admits a nowhere-zero 3 -flow.