Home archives vol 49 no 3 2018 papers normal edgetransitive and \frac12arctransitive cayley graphs on nonabelian groups of odd order 3pq, p and q are primes. On normal 2geodesic transitive cayley graphs, journal of. Finite 2arctransitive abelian cayley graphs sciencedirect. A group g is said to have a normal cayley di graph if g has a subset s such that the cayley di graph x cay g, s is normal. Moreover, a characterization of valencies of normal edgetransitive dihedrants is obtained, and some classes of examples with certain valences are constructed.
A cayley graph of a group is called normal edgetransitive if the. We complete the classification of normal edgetransitive cayley graphs of order a product of two primes by dealing with cayley graphs for frobenius groups of such. In this paper, we study some properties of normal edgetransitive cayley graphs. Finite normal edgetransitive cayley graphs bulletin of the.
On edgetransitive cayley graphs of valency four sciencedirect. Finite edgetransitive cayley graphs and rotary cayley maps cai heng li abstract. It is shown in that 3arc transitive cayley graphs of any given valency are rare. Every cayley graph of a finite group g can be expressed as an edge disjoint union of edgetransitive cayley graphs for the same group g, and moreover we may. Normal edgetransitive cayley graphs on nonabelian groups of. It is a central tool in combinatorial and geometric group theory. Abstractfor a finite group g with a normal subgroup h, the normal subgroup h based power graph of g, denoted by. In this paper, we determine all connected directed cayley graphs of finite abelian groups with valency at most 3 which are normal edge transitive but not normal.
Read on normal 2geodesic transitive cayley graphs, journal of algebraic combinatorics on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. The reasons for this are i that each cayley graph relative to a group g, is the edgedisjoint union of certain normal edgetransitive cayley graphs for the same group g, and ii the quotients of a normal edgetransitive cayley graph for g relative to certain. The completen i graps ahn k undirecte d cayley graph for. However, cayley graphs can be defined from other sets of generators as well. Bulletin of the australian mathematical society, vol. Pdf finite edgetransitive oriented graphs of valency four. On isomorphisms of connected cayley graphs, iii bulletin. Pdf nonnormal edgetransitive cubic cayley graphs of. In this paper, we determine all connected normal edgetransitive cayley graphs. Most of the information found here is taken from john meiers, groups graphs and trees, 1. S, let x c a yg,s be the corresponding cayley graph. The family of cayley graphs is one of the most important families of vertextransitive graphs, and provide a rich source of many interesting graphs.
Normal cayley graphs of finite groups springerlink. This paper aims to develop a theory for studying cayley graphs, especially for those with a high degree of symmetry. Available formats pdf please select a format to send. A cayley graph of a group is called normal edgetransitive if the normalizer of the right representation of the group in the automorphism of the cayley graph acts transitively on the set of edges of the graph. Normal edgetransitive cayley graphs of frobenius groups. This can be seen by considering the complete graphn os kn n vertices. In this paper, we prove that connected tetravalent edgetransitive cayley graphs on pgl2, p are either normal or bi normal when p. Recently, substantial progress has been made on the study of this problem, many longstanding open problems have been solved, and many new research problems have arisen. Cubic nonnormal cayley graphs of order 4p2 sciencedirect.
Potocnik, spiga and verret have constructed a census. It is proved that every finite group g has a normal cayley graph unlessg 4 2 org. Xuon finite edgetransitive cayley graphs of valency 4. In the case of a finite cyclic group, with its single generator, the cayley graph is a cycle graph, and for an infinite cyclic group with its generator the cayley graph is a doubly infinite path graph. Nonnormal edgetransitive cubic cayley graphs of dihedral groups. An approach to analysing the family of cayley graphs for a finite group g is given which identifies normal edgetransitive cayley graphs as a subfamily of central importance. We determine all connected normal edgetransitive cayley graphs on nonabelian groups with order 4 p, where p is a prime number. On normal edgetransitive cayley graphs of some abelian groups. The theory consists of analysing several types of basic cayley graphs normal, bi normal. Finite normal edgetransitive cayley graphs cambridge university. The wellknown hypercubes h n, 2 form one family of vertex biprimitive affine 2arctransitive graphs.
Various interesting families of normal 2arctransitive cayley graphs and bi normal 2arctransitive cayley graphs of elementary abelian 2groups appear in the classification of affine 2arctransitive graphs given by ivanov and praeger. These are the cayley graphs for g for which a subgroup of automorphisms exists which both normalises g and acts transitively on edges. However, it still seems quite difficult to give a nice characterization of general edgetransitive dihedrants. Normal edgetransitive cayley graphs were initiated by praeger and have some nice properties. Normal edgetransitive cayley graphs of the group hindawi. In this paper all connected cubic non normal cayley graphs of order 4 p 2 are constructed explicitly for each odd prime p. For convenience, we sometimes call a cayley graph on a dihedral group a dihedrant.
Finite edgetransitive dihedrant graphs request pdf. By the definition, every normal edgetransitive cayley graph is edgetransitive, but not every edgetransitive cayley graph is normal edgetransitive. Tetravalent edgetransitive cayley graphs of pgl 2, p. Newest cayleygraphs questions feed subscribe to rss.
The theory consists of analysing several types of basic cayley graphs normal, binormal, and corefree, and analysing several operations of cayley graphs. Let be a finite group and let be an inverse closed subset of that does not. Find all edgetransitive cayley graphs on cyclic group of valency 3. Normal cayley graphs have some very nice properties. On connected tetravalent normal edgetransitive cayley. In mathematics, a cayley graph, also known as a cayley colour graph, cayley diagram, group diagram, or colour group is a graph that encodes the abstract structure of a group. Then x is said to be normal edge transitive if n a u tx g is transitive on edges. Ams transactions of the american mathematical society. In this paper, we determine all connected normal edgetransitive cayley graphs of the group 1. S is either transitive on s or has two orbits in s in the form of t and t 1, where t is a nonempty subset of s and s t t 1. These three classes of cayley graphs are important for studying sarc transitive cayley graphs. Read tetravalent edgetransitive cayley graphs with odd number of vertices, journal of combinatorial theory, series b on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. A cayley graph cay g, s on a group g 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.
On finite edge transitive graphs and rotary maps on finite edge transitive graphs and rotary maps li, cai heng 20080901 00. Speci cally, we are going to develop cayley graphs and schreier diagrams, use them to study various kinds of groups, and from there prove some very deep and surprising theorems from abstract algebra. Several nice properties of bi normal cayley graphs are given in lemma 2. Moreover, every chiral or regular map has a normal cover that is a balanced chiral or regular cayley map, respectively. Finite edgetransitive dihedrant graphs the classification is then used to classify vertexquasiprimitive and vertexbiquasiprimitive edgetransitive dihedrants. Cayley graphs of abelian groups which are not normal edge. Normal edgetransitive circulant graphs request pdf. Moreover, given a normal edgetransitive cayley graph. In this paper we determine all connected, undirected edgetransitive cayley graphs of. Finite vertextransitive graphs include the symmetric graphs such as the petersen graph, the heawood graph and the vertices and edges of the platonic solids.
Nonnormal edgetransitive cubic cayley graphs of dihedral. Finite normal edgetransitive cayley graphs volume 60 issue 2 cheryl e. Pdf a cayley graph of a finite group is called normal edge transitive if its automorphism group has a subgroup which both normalizes and acts. Tetravalent halfedgetransitive graphs and non normal cayley graphs. The theory consists of analysing several types of basic cayley graphs normal, bi normal, and corefree, and analysing several operations of cayley graphs core quotient, normal quotient, and. Independently for our investigation, and as another attempt to study the structure of finite cayley graphs, xu 2 defined a cayley graph cay, to be normal. Nonnormal edgetransitive directed cayley graphs of. Throughout this paper, groups are assumed to be finite, and graphs are. In the mathematical field of graph theory, an edgetransitive graph is a graph g such that, given any two edges e 1 and e 2 of g, there is an automorphism of g that maps e 1 to e 2 in other words, a graph is edgetransitive if its automorphism group acts transitively upon its edges. Newest cayleygraphs questions mathematics stack exchange. Finite edgetransitive dihedrant graphs sciencedirect. The isomorphism problem for cayley graphs has been extensively investigated over the past 30 years. Citeseerx document details isaac councill, lee giles, pradeep teregowda.
If the degree is 4 or less, or the graph is also edgetransitive, or the graph is a minimal cayley graph, then the vertexconnectivity will also be equal to d. Tetravalent edgetransitive cayley graphs with odd number of vertices. Normal subgroup based power graphs of a finite group. It is thus natural to consider the following problem. Quotients of normal edgetransitive cayley graphs kastellorizo. It gives a characterization of a class of edgetransitive cayley graphs of finite. Preliminary results one of the principle theorems that helps us to connect the group properties of a group and normal edgetransitivity of. Finite edgetransitive cayley graphs and rotary cayley maps. Pdf we develop a new framework for analysing finite connected, oriented graphs of valency. H for a quotient group gh, necessary and sufficient conditions are obtained for the existence of a normal edgetransitive cayley graph.
Cayley graphs week 5 mathcamp 2014 today and tomorrows classes are focused nthe interplay of graph theory and algebra. A special type of normal cayley graph satisfies x1. On finite edge transitive graphs and rotary maps request pdf. Research article normal edgetransitive cayley graphs of. The theory consists of analysing several types of basic cayley graphs normal, bi normal, and corefree, and analysing several operations of cayley graphs core quotient, normal quotient, and imprimitive quotient. The theory consists of analysing several types of basic cayley graphs normal, bi normal, and corefree, and analysing several operations of. The finite cayley graphs such as cubeconnected cycles are also vertextransitive, as are the vertices and edges of the archimedean solids though only two of these are symmetric. A graph is said to be halftransitive if it is vertex and edgetransitive but not.
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