Find, read and cite all the research you need on researchgate. A schur ring s over a group h is said to be primitive if the subgroups k 1 and k h are the only subgroups of h for which k. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Its definition is suggested by cayley s theorem and uses a specified, usually finite, set of generators for the group. Return to article details on minimal distance regular cayley graphs of generalized dihedral groups download download pdf. Section 4 discusses the special case, borel cayley graphs. Oct 15, 2018 sufficient conditions for 3 regular cayley graphs to have disjoint efficient k distance dominating sets are given. Cayley graph, coxeter group, distance spectrum, distance polyno mial, weak order, absolute order 1 introduction let w. On minimal distanceregular cayley graphs of generalized dihedral. It is proved that every nontrivial distance regular cayley graph on a dihedral group is bipartite, nonantipodal, has diameter 3 and arises either from a cyclic. Distanceregular cayley graphs with least eigenvalue tilburg. An arbitrary graph gis said to be a cayley graph if there exists a group band a generating set xsuch that gis isomorphic to the cayley graph for band x.
It belongs to the class np of computational complexity. Morris extended this result in 2 to show that a cayley graph on the cyclic group zz pn can be represented as a cayley graph. A distance regular graph is called antipodal if its distance d graph is a disjoint union of complete graphs. Distanceregular cayley graphs with small valency tilburg. At this point, the following general problem arises naturally. Efficient k distance dominating set in cayley graphs. May 01, 2009 as for the examples of strongly regular tri cayley graphs over noncyclic groups, observe that the paley graph p 3 n, n an even integer, is strongly regular tri cayley graph over the group z 3 n. A linear algorithm for the distance in cayley graph hl,p matematica. We determine the distanceregular cayley graphs with valency at most 4, the cayley graphs among the distance regular graphs with known putative intersection arrays for valency 5, and the cayley graphs among all distance regular graphs with girth 3 and valency 6 or 7. On distanceregular cayley graphs on abelian groups.
Kantor mathematics department, university of oregon, eugene, or 97403, usa received 9 january 1989 1. Cayley graphs are dependent on a speci c set of generators. As observed in, perfect codes in cayley graphs are another generalization of perfect codes in the classical setting. Remark figure 3 illustrates that a nonminimal generating set for a group can be used in a cayley graph speci cation of a graph.
We determine the distance regular cayley graphs with valency at most 4, the cayley graphs among the distance regular graphs with known putative intersection arrays for valency 5, and the cayley graphs among all distance regular graphs with girth 3 and valency 6 or 7. Regular permutation groups and cayley graphs cheryl e. Distanceregular cayley graphs on dihedral groups request pdf. Fn q fq be a nonsingular quadratic form, where n is even and q is a power of an odd prime p, and let d x. Introduction let 93 be a finite connected regular graph of degree k2 having n vertices and diameter d. The distance spectra of cayley graphs of coxeter groups cayley graphs distance spectrum i let u. Most of the information found here is taken from john meiers, groups graphs and trees, 1. Examples will be provided to show that the same group can have a di erent cayley graph. On the cayley graph of a generic finitely presented. Pdf on jan 1, 2007, stefko miklavic and others published distanceregular cayley graphs on dihedral groups. An asymmetric graph is an undirected graph with only the trivial automorphism. Here you will find distance regular graphs available for download in a variety of formats.
Note that the identity element 1 is excluded from g. Partial results have been obtained by abdollahi and the authors. This shows that every cayley graph is vertex transitive. Cayley graph distanceregular graph is a regular graph such that for any two vertices v and w, the number of vertices at distance j from v and at distance k from w depends only upon j, k, and i dv, w. A distance regular graph with diameter 2 is called strongly regular. We now look at some examples to help illustrate this theorem. Similar to the graph isomorphism problem, it is unknown whether it has a polynomial time algorithm or it is npcomplete. Final remarks as mentioned before, devroye and sbihi 1990 and biggs 1993 showed that if g is a distance regular graph, then the expected hitting times between any pair of vertices in the graph depend only on the distance between these vertices. Distanceregular cayley graphs on dihedral groups stefko. We also show that there are exactly 109 connected cubic corefree symmetric bi cayley graphs, 48 of which are bi cayley graphs over nonabelian simple groups, and that there are 1, 6, 81, 462 and. The graphs were classified by their properties and related eigenvalues and multiplicities.
What does an isometry corresponding to an integer n do to the graph. The classification of distance regular cayley graphs is an open problem in the area of algebraic graph theory 28, problem 71ii. Cayley graphs are regular, in some cases have logarithmic diameter, and can be used. 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. Cayley graphs are associated with groups and their generators 16.
Note that the above infinite families include the complete graphs k 2, k 3 and k 4, and that the above graphs are distance regular 2, section 9. View of on minimal distanceregular cayley graphs of generalized. On some spectral and combinatorial properties of distance. Cayley graphs definition 2 a gmph c v, g is a cayley gmph with uertez set v if two vertices ul, vz e v are adjacent w v2 g for some g e g where v, is a jinite group and g c v \ 1. We consider the problem of which distance regular graphs with small valency are cayley graphs. In the mathematical field of graph theory, a vertextransitive graph is a graph g in which, given. It is the antipodal distance regular cayley graph on the alternating group a 5, with diameter 3 and valency 29. Cayley graphs and the discrete fourier transform alan mackey advisor.
I is a basis for v i every root 2 can be written p i2 c i iwhere all c i 0 or all c i 0 i thesimple re ectionsare s. A cayley map is a cayley graph 2cell embedded in some orientable surface so that the local rotations at every vertex are identical. To examine this thoroughly, we will discuss the necessary background information on group theory and graph theory. Cayley graphs geometrically display the actions of a group.
We determine the distance regular cayley graphs with valency at most 4, the cayley graphs among the distance regular graphs with known putative intersection arrays for valency 5, and the cayley graphs among all distance regular graphs with girth 3. Oct 01, 1979 journal of combinatorial theory, series b 27, 180189 1979 spectra of cayley graphs l. Every distance transitive graph is distance regular. Moreover, as there are no nontrivial strongly regular graphs on 18, 24 or 30 vertices, it is clear that a nontrivial strongly regular tri. Jan 01, 2007 the heawood graph and its bipartite complement are distanceregular cayley graphs on the dihedral group of order 14, and as it was pointed out in 10, the shrikhande graph can be represented as a cayley graph on three nonisomorphic nonabelian groups of order 16, as well as a cayley graph on z 4.
This is the smallest example of a connected regular graph which is not cayley. The distance between two vertices xand yis the least number of edges in a path connecting xto y. Pdf distanceregular cayley graphs on dihedral groups. Vertextransitivity and routing for cayley graphs in gcr. The distinct cubic distance regular graphs are k 4 or tetrahedral graph, k 3,3, the petersen graph, the cubical graph, the heawood graph, the pappus graph, the coxeter graph, the tuttecoxeter graph, the dodecahedral graph, the desargues graph, tutte 12cage, the biggssmith graph, and the foster graph. Then the matrix representation of lwith respect to the basis elements w2 wis precisely d. Distanceregular cayley graphs on dihedral groups stefko miklavi. Pdf this paper establishes connections between the structure of a semigroup and the minimum spans of distance labellings of its cayley. More surprising is the fact that not every vertex transitive graph is a cayley graph. Distanceintegral cayley graphs over abelian groups and. Aug 29, 2019 welcome to, an online repository of distance regular graphs. The maximum distance over all pairs of vertices xand yis called the diameter of the graph, and the girth of the graph is the length of its smallest circuit. The graph is called k regular if each vertex has degree k. On the spectrum of a class of distancetransitive graphs neliti.
Without the labelings the cayley graphs ca, s and cb, t are isomorphic, but they are distinguished via the labels. The graph automorphism problem is the problem of testing whether a graph has a nontrivial automorphism. In section 3, we discuss the vertextransitive property of cayley graphs in gcr representations. The study of groups by their cayley graphs is a main topic of algebraic graph. Oct 01, 2003 let us remark that in the case of distance regular cayley graphs the distance module, viewed as a zalgebra, is isomorphic to the bosemesner algebra of the distance regular cayley graph. Distanceregular cayley graphs with small valency van. Pdf distance labellings of cayley graphs of semigroups. Given a group with a, typically nite, generating set, we can form a cayley graph for that group with respect to that generating set. Draw part of the cayley graph for these generators. The main result of this article is a classification of distanceregular cayley graphs on dihedral groups. Balanced cayley maps, discrete mathematics io9 1992 265276.
The cayley graph on the left is with respect to generating set s f12. Several families of graphs are defined by having certain types of automorphisms. Lovasz, the determination of the spectrum of any graph with transitive automorphism group easily reduces to that of. The distance dx, y of vertices x and y of a graph g is the length number of. A vertextransitive graph is an undirected graph in which every vertex may be mapped by an automorphism into any other vertex. University, department of algebra and number theory, h1088 budapest, museum krt 68, hungary communicated by alan j. I for any two adjacent vertices x,y, there are exactly. Cayley graphs give a way of encoding information about group in a graph. Regular permutation groups and cayley graphs cheryl e praeger. Distance regular cayley graphs with small valency 205 smallest ifor which c i 1.
The cayley graph of the free group on two generators a and b graph families defined by their automorphisms distance. The distance spectra of cayley graphs of coxeter groups. Finally, we show that these characterizations are effective for the endregular. On distanceregular cayley graphs on abelian groups request pdf. 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. Besides sporadic examples, these comprise of the lattice graphs, certain triangular graphs, and line graphs of incidence graphs of certain projective planes. Section 5 consists of a routing algorithm that exploits the vertextransitivity of cayley graphs in the gcr domain. Based on the representation theory of finite groups, we first show that an abelian cayley graph is integral if and only if it is distance integral, which naturally contains a main result obtained in electron. Draw the cayley graph of d 4 with generators l 1 and r 90. Its definition is suggested by cayley s theorem named after arthur cayley and uses a specified, usually finite, set of generators for the group. Distance polynomials of absolute order graphs for some coxeter types.
In addition, we classify the possible connection sets for the lattice graphs and obtain some results on the structure of distance. Characterization of the 3 regular connected circulant graphs that admit an efficient k distance dominating set for \k2,3\ has been obtained. Distanceregular cayley graphs on dihedral groups citeseerx. Pdf on distanceregular cayley graphs on abelian groups. In this paper, we show that every bipartite distance regular cayley graph with diameter 3 can be constructed on the. The corresponding strongly regular graph cayfnq,d is the socalled a ne polar graph. Distanceregular cayley graphs on dihedral groups sciencedirect. This proposition shows that not all connected regular graphs are cayley. Finally, all distance transitive cayley graphs on dihedral groups are determined.
There exist four obvious families of such graphs, which are called trivial. Request pdf on distanceregular cayley graphs on abelian groups let g denote a finite abelian group with identity 1 and let s denote an inverseclosed subset of g. Introduction recall that a cayley graph may be defined as a graph f v, e which admits an automorphism group h acting regularly on the vertex set v see. The finite cayley graphs such as cubeconnected cycles are also. For a cayley graph ch, s, left multiplication induces a simply transitive action of h on ch, s by graph automorphisms. More background information on distance regular graphs can be found in. Distanceregular cayley graphs with small valency van dam.
There is a polynomial time algorithm for solving the graph automorphism problem for graphs where vertex. S is distance regular if and only if it is isomorphic to one of the following graphs. Citeseerx distanceregular cayley graphs on dihedral groups. This paper is a survey of distanceregular cayley graphs of diameter two and three, primarily focusing on diameter three due to the extensive amount of research already done in diameter two. As for the examples of strongly regular tri cayley graphs over noncyclic groups, observe that the paley graph p 3 n, n an even integer, is strongly regular tri cayley graph over the group z 3 n.
Joseph proved in 1995 1 that a cayley graph on the cyclic group zz p2 can be represented as a cayley graph on the elementary abelian group of order p2 where p is prime, if and only if the graph is a wreath product. The maximum distance over all pairs of vertices xand yis called the diameter of the graph. The pdf file you selected should load here if your web browser has a. Sep 01, 2014 note that the above infinite families include the complete graphs k 2, k 3 and k 4, and that the above graphs are distance regular 2, section 9. The heawood graph and its bipartite complement are distanceregular cayley graphs on the dihedral group of order 14, and as it was pointed out in 10, the shrikhande graph can be represented as a cayley graph on three nonisomorphic nonabelian groups of order 16, as well as a cayley graph on z 4. Pdf random walks on edgetransitive graphs ii pedro. The distance spectra of cayley graphs of coxeter groups re ectioncoxeter groups simple systems i. Request pdf distanceregular cayley graphs on dihedral groups the main result of this article is a classification of distanceregular cayley graphs on dihedral groups. Using cayley graphs, researchers have discovered new regular graphs with.
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