Strongly regular cayley graphs in rank two abelian groups ken w. Let g denote a finite abelian group with identity 1 and let s denote an inverseclosed subset of g. Implementing brouwers database of strongly regular graphs. As for general distanceregular cayley graphs, distanceregular cayley graphs over cyclic and dihedral groups were classified in 7, 8. This paper is a survey of distance regular cayley graphs of diameter two and three, primarily focusing on diameter three due to the extensive amount of research already done in diameter two. In this paper, cim abelian groups are characterized. We prove that every connected intransitive onematching semi cayley graph, with maximum degree three, over a finite abelian group is normal and characterize all such nonnormal graphs. Pdf on 3regular and 4regular cayley graphs of abelian groups.
On the other hand, mcmurray price 3 has shown that cayley graphs of abelian groups do have monotonic normalized heat di usion. In this paper a wide family of identifying codes over regular cayley graphs of degree four which are built over finite abelian groups is presented. Most of the information found here is taken from john meiers, groups graphs and trees, 1. This gives antipodal covers of some distance regular and distance transitive graphs. In 5, regev and shinkar also give an example, based on an idea of cheeger, of. A subset c of v is called a perfect code in if c is an independent set of and every vertex in v nc is. A group g is called an abelian or commutative if a b b a for all a, b, g. Alireza abdollahi, groups with maximal irredundant covers and minimal blocking sets, ars combinatoria 1 2014 337339. However, something much stronger rst appeared in 1979. Dec 06, 2005 strongly regular cayley graphs over the group z p n. Graphical regular representation a drr which is a graph is called agrrfor graphical regular representation. Which finitely generated abelian groups admit isomorphic. Strongly regular cayley graphs over the group z p n deepdyve. Cayley graph conjecture in the rst edition of lov aszs celebrated book combinatorial problems and exercises, there is an exercise asking for a proof of the assertion that connected cayley graphs on abelian groups are hamiltonian.
Di erence sets and partial di erence sets in abelian groups. International workshop on symmetries of graphs and networks. The cayley graph of the direct product of groups with the cartesian product of generating sets as a generating set is the cartesian product of the corresponding cayley graphs. In this paper all connected cubic nonnormal cayley graphs of order 4 p 2 are constructed explicitly for each odd prime p. Formally, for a given choice of generators, one has the word metric the natural distance on the cayley graph, which determines a metric space. 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. Reichard, schur rings over the elementary abelian group of order 64. The most celebrated theorem relating cayley graphs and h connectivity is the following theorem of chen and quimpo 7. Distanceregular cayley graphs on dihedral groups sciencedirect. In this paper, we study some properties of normal edgetransitive cayley graphs.
Let be a finitely generated group, and be its cayley graph with respect to some finite set of generators. To check that a group g is abelian, it su ces to only check that ab ba for all pairs ofgeneratorsof g. In section 2 we summarize some basic facts about distance regular graphs. Here we are interested in abelian groups with high exponent, in particular, g c pn c pn. In this paper, some basic properties and the automorphisms of bi cayley graphs are explored. Elena konstantinova lecture notes on some problems on cayley. Regev and shinkar is that there are cayley graphs which do not have monotonic normalized heat di usion. We classify the distanceregular cayley graphs with least. Pdf distanceregular cayley graphs with small valency. Suppose that m is another minimal normal subgroup of g, so n. Z p n strongly regular cayley graphs over the group z p n. The family of connected cayley graphs of valency three or more on abelian groups is an h connected family. On the other hand, the structure of cayley graphs of.
Theorem 6 let lbe the cayley table of an elementary abelian 2group g of order n. Group theory in the bedroom and other mathematical. Strongly regular cayley graphs are also partial di erence sets which are. Group theory introduction to higher mathematics lecture 16. View the article pdf and any associated supplements and figures for a period of 48 hours. Most examples so far are in abelian groups, however, this is not necessary condition for existence. Integral cayley graphs over abelian groups rims, kyoto university. Cayley and schreier graphs let gbe a group, and h gits subgroup. Properties of cayley digraphs there are 4 properties of cayley digraphs.
Its definition is suggested by cayleys theorem named after arthur cayley and uses a specified, usually finite, set of generators for the group. For investigation of distance regular cayley graphs,schur rings prove to be a useful tool. Of special interest were the graphs formed by the dihedral and cyclic groups. It is easy to see that paley graphs are distanceregular graphs of diameter 2. Some formula for spin models on distanceregular graphs. To acquire the book to read, as what your contacts do, you.
Isomorphic cayley graphs on nonisomorphic groups joy morris department of mathematics and statistics, simon fraser university, burnaby, bc. Hamiltonian decompositions of cayley graphs on abelian. 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. Graph 5 as a unitary cayley graph of even order is bipartite, while 6 and 7 are not bipartite. Graph 5 as a unitary cayley graph of even order is. A generalized dgon is a pointline incidence structure whose bipartite incidence graph has diameter d and girth 2d. Independent perfect domination sets in cayley graphs lee. Cayley graphs on abelian groups brian alspach university. The naive corresponding conjecture for graphs is false, for simple. The first isomorphism theorem is revised for groups and leads to the classical chinese remainder theorem. Thus the cayley graph of the abelian group with the set of generators consisting of four elements,, is the infinite grid on the plane, while for the direct.
Caya,s is said to have an automorphism group as small as possible if aut. Bipartite covers of linear cayley graphs are natural examples that possess abelian groups of automorphisms acting regularly on the colour classes. In this paper we obtain the complete classification of distanceregular. The quotient group of a group by normal subgroups are introduced. Pdf on the spectral gap and the diameter of cayley. Mina rajabiparsa, on normal edgetransitive cayley graphs sven reichard, schur rings over elementary abelian two groups grigory ryabov, on the isomorphism problem for cayley graphs over abelian p groups leonid shalaginov, on deza graphs with disconnected second neighbourhoods of vertices. Primorska, slovenia classification of cubic symmetric pentacirculants 26 jernej tonjek u. The smallest such graph is the graph of the icosahedron, which can be repre3. In addition, we classify the possible connection sets for the lattice graphs and obtain some results on the structure of distance. Such graphs may be calledbi abelian graphs, bidihedrants, and bimetacirculantsrespectively. Cayley graphs 3 group isomorphisms are nice since they mean two groups are the same except for the labeling of their elements. Families of nested completely regular codes and distance. It is the antipodal distance regular cayley graph on the alternating group a 5, with diameter 3 and v alency 29. However, many open problems for pds in abelian groups remain, and for the rest of the talk we will always assume g is abelian.
Walls2, kaishun wang3, sanming zhou4 abstract let be a graph with vertex set v. Cayley graph of an abelian group are isomorphic to one of the following graphs. In some cases, the constructed codes are also completely transitive and the corresponding coset graphs are distance transitive. Cayley graphs of abelian groups which are not normal edge. Miklavic and potocnik 20, 21 classified the distance regular cayley graphs over a cyclic or dihedral group. Groups of order n, 5 groups with orders 5 through 19, with their elements and some statistics. Every connected bi cayley graph over an abelian group. In the bi abelian case, for example, its easy to prove this theorem. Within each order, the abelian groups precede the nonabelian ones.
Standard topics on graph automorphisms are presented early on, while in later chapters more specialised topics are tackled, such as graphical regular representations and pseudosimilarity. On isomorphisms of connected cayley graphs, iii bulletin. The graphs considered include some wellknown graphs such as tori, twisted tori and kronecker products of two cycles. In this section we want to find all 3regular cayley graphs of abelian groups. The graphs were classified by their properties and related eigenvalues and multiplicities. Cayley graphs were first considered for finite groups by arthur cayley in 1878. As for general distanceregular cayley graphs, distanceregular cayley graphs over cyclic and dihedral groups were classified in. Cayley graphs give a way of encoding information about group in a graph. 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. Nontrivial distanceregular cayley graphs on nonabelian groups seem to be more di.
We show that a cayley graph x on a dihedral group is distancetransitive if and only if x is isomorphic to one of the following graphs. Then a minimal n 3perfect set in the latin square graph is a row, a column, a letter, or the set of n24 cells corresponding to a subgroup of gof index 2. Distanceregular cayley graphs with least eigenvalue. Hamilton paths in cayley graphs on generalized dihedral groups.
Pdf distanceregular cayley graphs on dihedral groups. In recent years, perfect codes in cayley graphs have received considerable attention 4,5,7,11,14,1618,20,26. The main result of this article is a classification of distance regular cayley graphs on dihedral groups. Topics in graph automorphisms and reconstruction by josef lauri. In particular, it follows that a distanceregular graph with least. Automorphism groups of cayley graphs on symmetric groups. We also give another prove for a wellknown theorem that g is 2dci if and only. Tetravalent 2arctransitive cayley graphs on nonabelian simple groups. Some of the codes in this construction are also perfect. In addition, we classify the possible connection sets for the lattice graphs and obtain some results on the structure of. S of gis the graph of the action of gon itself by left multiplication.
Symmetric cayley graphs on nonabelian simple groups with small valencies. Subgroup perfect codes in cayley graphs xuanlong ma1, gary l. We survey some of the known results on eigenvalues of cayley graphs and their applications, together with related results on eigenvalues of cayley digraphs and generalizations of cayley graphs. Let cayg, s be a connected cubic cayley graph on a non abelian simple group g. Diameter bounds on geometric distanceregular graphs. The issue of when two cayley digraphs on di erent abelian groups of prime power order can be isomorphic is examined. Let g be a finite group with identity element e and s\subseteq g\ setminus \e\ be.
Gabriel coutinho new examples of perfect state transfer. For example, any in nite cyclic group is isomorphic to the integers simply map the identity to 1. The group g is called a cim group if for any minimal generating subset s of g,s. Cayley graph on three nonisomorphic nonabelian groups of order 16, as well as a cayley. Alspach has conjectured that any 2kregular connected cayley graph caya,s on a finite abelian group a can be decomposed into k hamiltonian cycles. In this paper we find all 3regular and 4regular cayley graphs of abelian groups.
Read on distanceregular cayley graphs on abelian groups, 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. Codes over finite rings and finite abelian groupsa survey. For the rest of the paper we thus only consider graphs of valence at least 3. Given a group with a, typically nite, generating set, we can form a cayley graph for that group with respect to that generating set.
I will show that in contrast, it is possible for cayley graphs of given valency to have many more automorphisms. Sep 14, 2006 the main result of this article is a classification of distancetransitive cayley graphs on dihedral groups. There exist four obvious families of such graphs, which are called trivial. Automorphism groups of tetravalent cayley graphs on regular cayley graphs on abelian groups, in graph symmetry. Here we will restrict ourselves to the case of nite abelian groups. Implementing brouwers database of strongly regular graphs nathann cohen 0 1 dmitrii v. Nontrivial distanceregular cayley graphs on non abelian groups seem to be more di. This information will be needed in section 4, where we give representations of transitive graphs as cayley graphs. Since the smallest nonabelian simple group has order ja5j 60, one can show that theorem 2 implies theorem 1 see section 3. The group is then said to be hyperbolic if is a hyperbolic space in the. Distance regular cayley graphs over cyclic groups distance regular cayley graphs over dihedral groups \minimal distance regular cayley graphs on abelian groups a connected nite graph is distance regular if the cardinality of the intersection of two spheres depends only on their radiuses and the distance between their centres. Asa longterm perspectiveone might hopethat a thoroughunderstanding of the combinatorics of cayley graphs of abelian groups could lead to. Jazaeri, groups all of whose undirected cayley graphs are integral, european journal of combinatorics 38 2014 102109.
Pasechnik 0 1 0 department of computer science, the university of oxford, oxford, uk 1 cnrs and universite parissud 11, orsay, france andries brouwer maintains a public database of existence results for strongly regular graphs on n. Normality of onematching semi cayley graphs over finite abelian groups with maximum degree three majid arezoomand and mohsen ghasemi abstract. On distance regular cayley graphs for abelian groups combinatorics seminar monday at 2. Strongly regular cayley graphs in rank two abelian groups. In this paper, the conjecture is shown to be true if s sl,sz, s3 is a minimal generating set of a with 1 al odd, or. Let g be a finite group with identity element e and s\subseteq g\setminus \e\ be. In this talk, i will present a geometric condition for a bipartite cover of a linear cayley graph to have only ve distinct eigenvalues. Cubic bicayley graphs over abelian groups sciencedirect. See also the monograph by konstantinova 21 for some basic facts and problems on cayley graphs and distanceregular graphs. In 3 the following result was obtained also, see, corollary 3. The coarse equivalence class of this space is an invariant of the group. Pdf on 3regular and 4regular cayley graphs of abelian. Let gbe the group of isometries of z2 with its natural action on r2. On distanceregular cayley graphs on abelian groups deepdyve.
On the isomorphisms of cayley graphs of abelian groups. The reader is referred to 11, section 1 for a brief account of results on perfect codes in cayley graphs and connections between such codes and factorizations and tilings of the underlying groups. Though many of the graphs could be formed in abelian groups, some were found to be genuinely non abelian. For more about cayley graph and graphs in general refer any standard book on graph theory. Normality of onematching semicayley graphs over finite. An abstract group h is said to be a dt group, if every distanceregular cayley graph of h is distancetransitive. Distanceregular cayley graphs on dihedral groups stefko miklavi. Pdf on distanceregular cayley graphs on abelian groups. Besides sporadic examples, these comprise of the lattice graphs, certain triangular graphs, and line graphs of incidence graphs of certain projective planes.
As an application, a classification of connected cubic vertextransitive bi cayley graphs over abelian groups is given, and using this, a problem posed in zhou and feng 2012 regarding the cayley property of a class of graphs is solved. Sep 01, 2014 read on distance regular cayley graphs on abelian groups, 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. In 229, miklavic andsparl classified all distanceregular cayley graphs on abelian groups with respect to maximal inverseclosed generating sets. Cubic nonnormal cayley graphs of order 4p2 sciencedirect. On distanceregular cayley graphs on abelian groups request pdf. In this paper, we show that a cayley graph for an abelian group has an independent perfect domination set if and only if it is a covering graph of a complete graph. In this paper, we show that almost all cayley graphs on abelian groups have automorphism group as small as possible, proving a conjecture of babai and godsil. Finally cayley s theorem which shows isomorphism between abstract group and group of permutations is established. Mozhey, torsion free affine connections on threedimensional homogeneous spaces s.
Distanceregular cayley graphs of abelian groups fmf. Praeger 17 proved that if n autcayg,s rg is transitive on edges then the cayley graph cayg, s is normal, and fang et al. Co 22 jan 2019 eigenvalues of cayley graphs xiaogangliu department of applied mathematics northwestern polytechnical university. Distanceregular cayley graphs on dihedral groups stefko. Let g be a finite group, s a subset of g\1, and let cay g,s denote the cayley digraph of g with respect to s. Regular permutation groups and cayley graphs cheryl e. This indepth coverage of important areas of graph theory maintains a focus on symmetry properties of graphs.
The set is endowed with its graph metric in which edges are of length one and the distance between two vertices is the minimal number of edges in a path connecting them which turns it into a length space. In fact, the study of paley type pds in non abelian groups is rich and interesting. Normal cayley graphs of finite groups springerlink. The pattern on the left never appears in the cayley graph for an abelian group, whereas the pattern on the right illustrates the relation ab ba. On distanceregular cayley graphs on abelian groups.
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. To show that the determined graphs are not isomorphic, we need only care for the graphs on 12 vertices, i. For small values of n, the vast majority of undirected vertextransitive graphs with n vertices are cayley graphs see 5, table 1. Rosemary and i decided to have a go at generalising this result.