A generalization of Pappus graph

Algebra Colloquium, 2013

A Cayley graph Cay (G,S) on a group G with respect to a Cayley subset S is said to be normal if the right regular representation R(G) of G is normal in the full automorphism group of Cay (G,S). For a positive integer n, let Γn be a graph with vertex set {xi,yi|i ∈ ℤ2n} and edge set {{xi,xi+1}, {yi,yi+1}, {x2i,y2i+1}, {y2i,x2i+1}|i ∈ ℤ2n}. In this paper, it is shown that Γn is a Cayley graph and its full automorphism group is isomorphic to [Formula: see text] for n=2, and to [Formula: see text] for n > 2. Furthermore, we determine all pairs of G and S such that Γn= Cay (G,S) is non-normal for G. Using this, all connected cubic non-normal Cayley graphs of order 8p are constructed explicitly for each prime p.

Discrete Mathematics

It was proved in [Y.-Q. Feng, C. H. Li and J.-X. Zhou, Symmetric cubic graphs with solvable automorphism groups, European J. Combin. 45 (2015), 1-11] that a cubic symmetric graph with a solvable automorphism group is either a Cayley graph or a 2-regular graph of type 2 2 , that is, a graph with no automorphism of order 2 interchanging two adjacent vertices. In this paper an infinite family of non-Cayley cubic 2-regular graphs of type 2 2 with a solvable automorphism group is constructed. The smallest graph in this family has order 6174.

Periodica Mathematica Hungarica, 1976

It is well-knot~n tha~ its automorphism group A(X o H) must contain the regular subgroup L G corresponding to the set of left multiplication~ by elements of G. This paper is concerned with minimizing the index [A(Xo, t/):L a] for given G, in particular when this index is always greater than 1. If G is a.beli~n but not one of seven exceptional groups, then a Cayley graph of G exists for which this index is at most 2. Nearly complete results for the generalized dicyclic groups are also obtained.

Filomat

In this paper, generalized Cayley graphs are studied. It is proved that every generalized Cayley graph of order 2p is a Cayley graph, where p is a prime. Special attention is given to generalized Cayley graphs on Abelian groups. It is proved that every generalized Cayley graph on an Abelian group with respect to an automorphism which acts as inversion is a Cayley graph if and only if the group is elementary Abelian 2-group, or its Sylow 2-subgroup is cyclic. Necessary and sufficient conditions for a generalized Cayley graph to be unworthy are given.

2000

A graph is symmetric if its automorphism group acts transitively on the arcs of , and s-regular if its automorphism group acts regularly on the set of s-arcs of . Tutte (1947, 1959) showed that every cubic finite symmetric cubic graph is s-regular for some s ≤ 5. Djokoviÿc and Miller (1980) proved that there are seven types of arc-transitive

This paper introduces the basic definitions and properties of simple graphs which are mainly covered in [1] and [2]. Each definition and property is supported by examples and diagrams. There are also some basic facts used in this paper which have been demonstrated by other researchers such as [3] and [4]. The main concern and the focus in this paper are on the automorphism groups of some graphs. The final part of this work have been on cubic graphs and the Boolian graph B n. To achieve the main points, the group automorphisms have been applied on the automorphisms of some graphs. The permutation groups played the principle role in the case. This was used to study the nature of the graph automorphisms.

Graphs and Combinatorics, 1994

In this paper we review the characterization of point-color symmetric (PCS) graphs based on the color preserving automorphisms given in [3"1. In particular, we consider PCS pictures, arriving at another characterization theorem. We summarize a few results and give some examples.

2014

An algebraic approach to graph theory can be useful in numerous ways. There is a relatively natural intersection between the fields of algebra and graph theory, specifically between group theory and graphs. Perhaps the most natural connection between group theory and graph theory lies in finding the automorphism group of a given graph. However, by studying the opposite connection, that is, finding a graph of a given group, we can define an extremely important family of vertex-transitive graphs. This paper explores the structure of these graphs and the ways in which we can use groups to explore their properties.

Journal of the Indonesian Mathematical Society, 2010

A graph is said to be semisymmetric if its full automorphism group actstransitively on its edge set but not on its vertex set. In this paper, we prove thatthere is only one semisymmetric cubic graph of order 28p2, where p is a prime.DOI : http://dx.doi.org/10.22342/jims.16.2.38.139-143

Acta Universitatis Apulensis, 2014

A graph is called edge-transitive, if its automorphisms group acts transitively on the set of its edges. In this paper, we classify all connected cubic edge-transitive graphs of order 46p 2 , where p is a prime.