Distanceregular cayley graphs with least eigenvalue. At this point, the following general problem arises naturally. What proves so interesting about this problem is that no one can yet identify what the integral trees are or which 5 regular graphs are integral. As a natural generalization of distanceregular graphs see 3, 4 for the theory of distanceregular graphs, wang and suzuki 6 introduced the concept of weakly distance regular. A vertextransitive graph is an undirected graph in which every vertex may be mapped by an automorphism into any other vertex. Firstly, to classify all distanceregular cayley graphs on dihedral groups, and thus to contin ue the pro ject of classi. Distanceregular cayley graphs with small valency arxiv. On distanceregular cayley graphs on abelian groups sciencedirect. Pdf on jan 1, 2007, stefko miklavic and others published distanceregular cayley graphs on dihedral groups. It is a central tool in combinatorial and geometric group theory.
Then the matrix representation of lwith respect to the basis elements w2. This interplay of concepts proves to be especially intimate when a subclass of distance regular cayley graphs is considered see section 2 for formal definitions of distance regular graphs and cayley graphs. Several topics on the integral cayley graphs are presented. Strongly regular semicayley graphs rims, kyoto university. Cayley graph on three nonisomorphic nonabelian groups of order 16, as well as a cayley. The distance spectra of cayley graphs of coxeter groups the distance spectra of cayley graphs of coxeter groups paul renteln 1department of physics california state university san bernardino 2department of mathematics. The main result of this article is a classification of distanceregular cayley graphs on dihedral groups. Upper bound in terms of order and maximum degree of an independent set s of vertices in a graph \gv,e\ called an efficient k distance dominating set if every vertex in \vs\ be distance k from. Among distance regular cayley graphs, those of diameter 2 also called strongly regular cayley graphs have been investigated most. Such cayley graphs arise on the symmetric group sym n and the hyperoctahedral group b n of signed permutations when transpositions or reversals are considered as generators for these graphs where reversals are the operations reversing the order of a substring of. Regular graphs of degree at most 2 are easy to classify. Distanceregular cayley graphs with small valency nasaads. Normal cayley graphs of finite groups springerlink. In mathematics, a 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 every distance transitive graph is distance regular.
Most of the information found here is taken from john meiers, groups graphs and trees, 1. Cayley graphs week 5 mathcamp 2014 today and tomorrows classes are focused nthe interplay of graph theory and algebra. There exist four obvious families of such graphs, which are called trivial. Cayley graphs abstract there are frequent occasions for which graphs with a lot of symmetry are required. In addition, we classify the possible connection sets for the lattice graphs and obtain some results on the structure of distance. Regular maps from cayley graphs ii antibalanced cayley maps. In this thesis, integral cayley graphs are studied. 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. We obtain asymptotically exact bounds for minimal resistances in cay s n, t n. It is the antipodal distance regular cayley graph on the alternating group a 5, with diameter 3 and valency 29. The graphs were classified by their properties and related eigenvalues and multiplicities. The pdf file you selected should load here if your web browser has a.
Distanceregular cayley graphs on dihedral groups stefko. Distanceregular cayley graphs on dihedral groups fmf. Efficient k distance dominating set in cayley graphs. A 0 regular graph consists of disconnected vertices, a 1 regular graph consists of disconnected edges, and a 2 regular graph consists of a disjoint union of cycles and infinite chains. Distanceregular cayley graphs on dihedral groups distanceregular cayley graphs on dihedral groups miklavic stefko. Maximal and average resistances in cayley graphs on symmetric groups are given in terms of degree, girth and spectral gap. The distance spectra of cayley graphs of coxeter groups paul renteln. This problem is much more complicated when a cayley graph is not distance regular. Pdf regular maps from cayley graphs ii antibalanced. 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.
Citeseerx document details isaac councill, lee giles, pradeep teregowda. Distanceregular graphs are graphs with a lot of combinatorial symmetry, in the sense that given an arbitrary ordered pair of vertices at distance h, the number of vertices that are at distance ifrom the rst vertex and distance jfrom the second is a constant i. 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. Pdf distanceregular cayley graphs on dihedral groups. An asymmetric graph is an undirected graph with only the trivial automorphism. Pdf distanceregular cayley graphs with least eigenvalue. 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.
Several families of graphs are defined by having certain types of automorphisms. In particular, it follows that a distance regular graph with least eigenvalue. Pdf distanceregular cayley graphs with small valency. Finally, we show that these characterizations are effective for the endregular graphs of finite degree which are the graphs finitely decomposable by distance.
We classify the distanceregular cayley graphs with least. The present paper is devoted to estimating effective resistances in large weighted undirected cayley graph cay s n, t n on symmetric group s n. 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. We gave background material on groups and graphs and gave nu. We consider the problem of which distance regular graphs with small valency are cayley graphs. Cayley graph, coxeter group, distance spectrum, distance polynomial, weak order, absolute order. Indeed, distance regular graphs were introduced as a combinatorial generalization of distance transitive graphs, having the.
Distanceregular cayley graphs with small valency van. Elena konstantinova lecture notes on some problems on cayley. In mathematics, a 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. Vertex reconstruction in cayley graphs sciencedirect.
Pdf on distanceregular cayley graphs on abelian groups. If the degree of the vertices in is and the degree of the vertices in is, then the graph is said to be,biregular. Distanceregular graphs form an important class of graphs in the area of algebraic graph theory. Strongly related to strongly regular graphs are regular two graphs, and chapter 10 mainly discusses seidels work on sets of equiangular lines. For s t, both the incidence graph and its line graph are distance regular. 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. This interplay of concepts proves to be especially intimate when a subclass of distanceregular cayley graphs is considered see section 2 for formal definitions. Its definition is suggested by cayley s theorem named after arthur cayley and uses a specified, usually finite, set of generators for the group. We obtain that the incidence graphs of desarguesian a.
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. I the constructions we will talk about use nite elds, semi elds, quadratic forms over nite elds or more generally, pary bent functions, and cyclotomy. The purpose of this study was to examine multiple examples of cayley graphs through group theory, graph theory, and applications. Cayley graphs are graphs associated to a group and a set of generators for that group there is also an associated directed graph. Regular permutation groups and cayley graphs cheryl e. The distance spectra of cayley graphs of coxeter groups. Resistance distances in cayley graphs on symmetric groups. Distanceregular cayley graphs on dihedral groups, journal. 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. Introduction i this talk is about strongly regular cayley graphs, or equivalently, partial di erence sets. In graph theoretic mathematics, a biregular graph or semiregular bipartite graph is a bipartite graph, for which every two vertices on the same side of the given bipartition have the same degree as each other.