Semisymmetric Graphs 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].

A family of \(2\)-groups and an associated family of semisymmetric, locally \(2\)-arc-transitive graphs

A mixed dihedral group is a group \(H\) with two disjoint subgroups \(X\) and \(Y\), each elementary abelian of order \(2^n\), such that \(H\) is generated by \(X\cup Y\), and \(H/H'\cong X\times Y\). In this paper, for each \(n\geq 2\), we construct a mixed dihedral \(2\)-group \(H\) of nilpotency class \(3\) and order \(2^a\) where \(a=(n^3+n^2+4n)/2\), and a corresponding graph \(\Sigma\), which is the clique graph of a Cayley graph of \(H\). We prove that \(\Sigma\) is semisymmetric, that is, \({\mathop{\rm Aut}}(\Sigma)\) acts transitively on the edges but intransitively on the vertices of \(\Sigma\). These graphs are the first known semisymmetric graphs constructed from groups that are not \(2\)-generated (indeed \(H\) requires \(2n\) generators). Additionally, we prove that \(\Sigma\) is locally \(2\)-arc-transitive, and is a normal cover of the `basic' locally \(2\)-arc-transitive graph \({\rm\bf K}_{2^n,2^n}\). As such, the construction of this family of graphs contributes to the investigation of normal covers of prime-power order of basic locally \(2\)-arc-transitive graphs – the `local' analogue of a question posed by C. H. Li.

Resolution of a conjecture about linking ring structures

An LR-structure is a tetravalent vertex-transitive graph together with a special type of a decomposition of its edge-set into cycles. LR-structures were introduced in a paper by P. Potočnik and S. Wilson (2014) [12], as a tool to study tetravalent semisymmetric graphs of girth 4. In this paper, we use the methods of group amalgams to resolve some problems left open in the above-mentioned paper.

Semisymmetric graphs of order twice prime powers with the same prime valency

Semisymmetric graphs of order twice prime powers with the same prime valency

A Note on Two-Generator 2-Group Covers of Cubic Symmetric Graphs of Order 2p

Let [Formula: see text] be a prime. In this paper, a complete classification of edge-transitive [Formula: see text]-covers of a cubic symmetric graph of order [Formula: see text] is given for the case when [Formula: see text] is a two-generator 2-group whose derived subgroup is either isomorphic to [Formula: see text] or generated by at most two elements. As an application, it is shown that 11 is the smallest value of [Formula: see text] for which there exist infinitely many cubic semisymmetric graphs with order of the form [Formula: see text].

Semisymmetric Graphs Defined by Finite-Dimensional Generalized Kac–Moody Algebras

Semisymmetric Graphs Defined by Finite-Dimensional Generalized Kac–Moody Algebras

LDPC codes from cubic semisymmetric graphs

In this paper we study LDPC codes having cubic semisymmetric graphs as their Tanner graphs. We discuss the structure of the smallest absorbing sets of these LDPC codes. Further, we give an expression for the variance of the syndrome weight of the constructed codes, and present computational and simulation results.

Pentavalent semisymmetric graphs of square‐free order

AbstractA regular graph is semisymmetric if its automorphism group acts transitively on the edge set but not on the vertex set. In this article, we give a complete list of connected semisymmetric graphs of square‐free order and valency 5. The list consists of a single graph, the incidence graph of a generalized hexagon of order , and an infinite family arising from some groups with cyclic Fitting subgroup.

On semisymmetric cubic graphs of order 20p^2, p prime

On semisymmetric cubic graphs of order 20p^2, p prime

Classifying semisymmetric cubic graphs of order20p

A simple graph is called semisymmetric if it is regular and edge-transitive but not vertex-transitive. In this paper we classify all connected cubic semisymmetric graphs of order 20p, p prime.

Some semisymmetric graphs arising from finite vector spaces

A graph is worthy if no two vertices have the same neighborhood. In this paper, we characterize the automorphism groups of unworthy edge-transitive bipartite graphs, and present some worthy semisymmetric graphs arising from vector spaces over finite fields. We also determine the automorphism groups of these graphs.

Cubic Semisymmetric Graphs of Order 2qp2

A regular edge-transitive graph is said to be semisymmetric if it is not vertex-transitive. Let p be a prime. By Folkman [J. Combin. Theory 3 (1967), 215–232], there is no cubic semisymmetric graph of order 2p or 2p2, and by Hua et al. [Science in China A 54 (2011), 1937–1949], there is no cubic semisymmetric graph of order 4p2. Lu et al. [Science in China A 47 (2004), 11–17] classified connected cubic semisymmetric graphs of order 6p2. In this paper, for p > q ≥ 5 two distinct odd primes, it is shown that the sufficient and necessary conditions which a connected cubic edge transitive bipartite graph of order 2qp2 is semisymmetric.

Edge-transitive graphs of small order and the answer to a 1967 question by Folkman

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.

Cubic Edge-Transitive Bi-$p$-Metacirculants

A graph is said to be a bi-Cayley graph over a group $H$ if it admits $H$ as a group of automorphisms acting semiregularly on its vertices with two orbits. For a prime $p$, we call a bi-Cayley graph over a metacyclic $p$-group a bi-$p$-metacirculant. In this paper, the automorphism group of a connected cubic edge-transitive bi-$p$-metacirculant is characterized for an odd prime $p$, and the result reveals that a connected cubic edge-transitive bi-$p$-metacirculant exists only when $p=3$. Using this, a classification is given of connected cubic edge-transitive bi-Cayley graphs over an inner-abelian metacyclic $3$-group. As a result, we construct the first known infinite family of cubic semisymmetric graphs of order twice a $3$-power.

English

We study graphs whose vertices possess the same value of betweenness centrality (which is defined as the sum of relative numbers of shortest paths passing through a given vertex). Extending previously known results of S. Gago, J. Hurajova, T. Madaras (2013), we show that, apart of cycles, such graphs cannot contain 2-valent vertices and, moreover, are 3-connected if their diameter is 2. In addition, we prove that the betweenness uniformity is satisfied in a wide graph family of semi-symmetric graphs, which enables us to construct a variety of nontrivial cubic betweenness-uniform graphs.

Linking rings structures and semisymmetric graphs: Combinatorial constructions

This paper considers combinatorial methods of constructing LR structures: two isolated constructions, RC and SoP, two closely related constructions, CS(Γ, B , 0) and CS(Γ, B , 1) using cycle decompositions of tetravalent graphs, a generalization of those, CS(Γ, B , k) for k > 2 , and finally a construction LDCS relating to cycle decompositions of graphs of higher even valence. This last construction is used to classify all LR structures of types {3, *} or {4, *} .

The automorphisms of bi-Cayley graphs

A bi-Cayley graph Γ is a graph which admits a semiregular group H of automorphisms with two orbits. In this paper, the normalizer of H in the full automorphism group of Γ is determined. Applying this, a characterization of cubic edge-transitive graphs of order a 2-power is given. As byproducts, we answer a problem proposed in Godsil (1983) [16] regarding the existence of arc-regular non-normal Cayley graphs of order a 2-power, and construct the first known family of cubic semisymmetric graphs of order a 2-power.

Linking rings structures and semisymmetric graphs: Cayley constructions

An LR structure is a tetravalent vertex-transitive graph together with a special type of a decomposition of its edge-set into cycles. LR structures were introduced in Potočnik and Wilson (2014) as a tool to study tetravalent semisymmetric graphs of girth 4. In this paper, we consider algebraic methods of constructing LR structures, using number theory, Cayley graphs, affine groups, abelian groups and fields.

Semisymmetric graphs admitting primitive groups of degree 9p

Let Γ be a connected regular bipartite graph of order 18p, where p is a prime. Assume that Γ admits a group acting primitively on one of the bipartition subsets of Γ. Then, in this paper, it is shown that either Γ is arc-transitive, or Γ is isomorphic to one of 17 semisymmetric graphs which are constructed from primitive groups of degree 9p.

Primitive Permutation Representations of PSL(3, p) and Its Applications

The main result of this paper is a determination of the suborbits of primitive permutation representations of PSL(3, p) relative to a maximal subgroup PGL(2, p), where p ≡ 13(mod 24). As an application, a new family of -transitive graphs and semisymmetric graphs are obtained, respectively.