Finite edge-transitive Cayley graphs and rotary Cayley maps

Regular maps from Cayley graphs II antibalanced Cayley maps

In this paper we apply Polya's Theorem to the problem of enumerating Cayley graphs on permutation groups up to isomorphisms induced by conjugacy in the symmetric group. We report the results of a search of all three-regular Cayley graphs on permutation groups of degree at most nine for small diameter graphs. We explore several methods of constructing covering graphs of these Cayley graphs. Examples of large graphs with small diameter are obtained.

The workshop Permutation groups organised by Robert Guralnick (Southern California), Cheryl Praeger (Western Australia), Jan Saxl (Cambridge) and Katrin Tent (Bielefeld) was held August 5th–11th, 2007. The focus was recent developments in permutation group theory and their influence on, and from, group representation theory, algebraic graph theory and other areas of geometry and combinatorics, algebraic geometry, model theory and infinite symmetrical structures. It was well attended with 50 participants. Especially valuable was the program to support young researchers which gave the opportunity for a considerable number of young mathematicians to participate in the workshop. They benefited from the ‘Oberwolfach experience’ and also enlivened and contributed enormously to the success of the meeting. The workshop had a friendly and vibrant atmosphere, aided by the excellent facilities of the Institute. All participants appreciated the beautiful lecture by John G. Thompson entitled ‘The divisor matrix and {\rm SL}(2,\Z) ’. (In fact, this meeting was the first Oberwolfach workshop in over twenty years that Thompson had attended.) There were 26 other talks, and the program featured in each of the first five sessions at least one talk by a student or young researcher at the beginning of their career. This facilitated interaction between younger and more established researchers. The number of talks was restricted to give plenty of time for discussion and cooperation among the participants. Other communications were presented as posters. Highlights of the workshop addressed both fundamental permutation group theory, and also significant outcomes from applying permutation group theory and methods in combinatorics, model theory and other areas. Some topics arose in more than one context, forming new connections. A common feature of many contributions was application of the finite simple group classification, or new results on the structure of simple groups. As examples, we mention two highlights reported by early career participants. The question of base size of permutation actions is of importance in computational group theory as well as in the study of the graph isomorphism problem. Recent research has thrown much light on the base sizes of actions of almost simple groups in particular. Burness reported on the very recent proof of a conjecture of Cameron and Kantor dating from 1993 concerning non-standard permutation actions of almost simple groups: these are all primitive actions apart from actions of alternating and symmetric groups on subsets or partitions, and of classical groups on subspaces or pairs of complementary subspaces. In all the latter families the minimum base size is unbounded as the group order increases. The somewhat surprising conjecture, now proved, is that the minimum base size of non-standard permutation groups is at most 7, with the unique example where this bound is attained being the Mathieu group M_{24} in its natural action on 24 points. Expander graphs play an important role in computer science and combinatorics, and, for example, are significant in modelling and analysing communication networks. In particular, their diameters grow only logarithmically with the numbers of vertices. This property on diameters was proved in 1989 by Babai, Kantor and Lubotzky to hold for suitable Cayley graphs of simple groups: specifically for each finite nonabelian simple group G , they constructed a valency 7 Cayley graph for G with diameter at most a constant times \log(|G|) . Although their graphs were not expanders, they conjectured that there exists an absolute constant k such that all infinite families of simple groups give rise to families of valency k Cayley graphs that are expanders. Kassabov lectured on amazing progress towards proving this conjecture – it is now proved for all simple groups apart from the Suzuki groups. It was a very happy and successful workshop. Indeed, several of the most distinguished participants commented that this was one of the best Oberwolfach meetings they had attended, with plenty of time for collaboration, and outstanding talks. Extended abstracts of the talks mentioned, as well as all the others, are given below (in the order in which they were presented).

We shall consider problems in two broad areas of mathematics, namely the area of the degree diameter problem and the area of regular maps. <br><br> In the degree diameter problem we investigate finding graphs as large as possible with a given degree and diameter. Further, we may consider additional properties of such extremal graphs, for example restrictions on the kinds of symmetries that the graph in question exhibits. <br><br> We provide two pieces of research relating to the degree diameter problem. First, we provide a new derivation of the Hoffman-Singleton graph and show that this derivation may be used with minor modification to derive the Bosák graph. Ultimately we show that no further natural modification of the construction we use can derive any other Moore or mixed-Moore graphs. Second, we answer the previously open question of whether the Gómez graphs, which are known to be vertex-transitive, are in addition also Cayley. In doing this, we also generalise the construction of the Gómez graphs and show that the Gómez graphs are the largest graphs for given degree and diameter following the generalised construction. <br><br> We also provide two pieces of research relating to regular maps. We aim to address the related questions of for which triples of parameters <i>k</i>, <i>l</i> and <i>m</i> there exist finite regular maps of face length <i>k</i>, vertex order <i>l</i> and Petrie walk length <i>m</i>. We then address the related question of determining for which <i>n</i> there exist regular maps which are self dual and self Petrie dual which have face length, vertex order and Petrie dual walk length n. We address both questions by constructions of regular maps in fractional linear groups, necessarily leading us to study some interesting related number theoretic questions.

Given a permutation group G, the derangement graph ΓG of G is the Cayley graph with connection set the set of all derangements of G. We prove that, when G is transitive of degree at least 3, ΓG contains a triangle.The motivation for this work is the question of how large can be the ratio of the independence number of ΓG to the size of the stabilizer of a point in G. We give examples of transitive groups where this ratio is maximum.

Gossiping and routing in second-kind Frobenius graphs

A generalized strongly regular graph of grade p, as a generalization of strongly regular graphs, is a regular graph such that the number of common neighbours of both any two adjacent vertices and any two non-adjacent vertices takes on p distinct values. In this paper, we study generalized strongly regular graphs of grade 2 and provide some inequalities for the eigenvalues of them. In particular, we investigate a special family of generalized strongly regular graphs of grade 2, i.e., semi-strongly regular graphs. We obtain a relation between the parameters and two inequalities for the eigenvalues of these graphs. We also present some constructions of generalized strongly regular graphs based on Cayley graphs, graph operations and association schemes, respectively.

Orthogonal graphs are natural extensions of the classical binary and b-ary hypercubes b=2/sup l/ and are abstractions of interconnection schemes used for conflict-free orthogonal memory access in multiprocessor design. Based on the type of connection mode, these graphs are classified into two categories: those with disjoint and those with nondisjoint sets of modes. The former class coincides with the class of b-ary b=2/sup l/ hypercubes, and the latter denotes a new class of interconnection. It is shown that orthogonal graphs are Cayley graphs, a certain subgroup of the symmetric (permutation) group. Consequently these graphs are vertex symmetric, but it turns out that they are not edge symmetric. For an interesting subclass of orthogonal graphs with minimally nondisjoint set of modes, the shortest path routing algorithm and an enumeration of node disjoint (parallel) paths are provided. It is shown that while the number of node disjoint paths is equal to the degree, the distribution is not uniform with respect to Hamming distance as in the binary hypercube.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Vertex primitive [formula omitted]-transitive Cayley graphs

Quotient-complete arc-transitive graphs

We give a unified approach to analyzing, for each positive integer s, a class of finite connected graphs that contains all the distance transitive graphs as well as the locally s‐arc transitive graphs of diameter at least s. A graph is in the class if it is connected and if, for each vertex v, the subgroup of automorphisms fixing v acts transitively on the set of vertices at distance i from v, for each i from 1 to s. We prove that this class is closed under forming normal quotients. Several graphs in the class are designated as degenerate, and a nondegenerate graph in the class is called basic if all its nontrivial normal quotients are degenerate. We prove that, for s≥2, a nondegenerate, nonbasic graph in the class is either a complete multipartite graph or a normal cover of a basic graph. We prove further that, apart from the complete bipartite graphs, each basic graph admits a faithful quasiprimitive action on each of its (1 or 2) vertex‐orbits or a biquasiprimitive action. These results invite detailed additional analysis of the basic graphs using the theory of quasiprimitive permutation groups. © 2011 Wiley Periodicals, Inc. J Graph Theory 69:176‐197, 2012

Classification of reflexible Cayley maps for dihedral groups

On solvable factors of almost simple groups

A group G is called a CI-group if, for any subsets S,T⊂ G, whenever two Cayley graphs Cay(G,S) and Cay(G,T) are isomorphic, there exists an element σ∊Aut(G) such that Sσ = T. The problem of seeking finite CI-groups is a long-standing open problem in the area of Cayley graphs. This paper contributes towards a complete classification of finite CI-groups. First it is shown that the Frobenius groups of order 4p and 6p, and the metacyclic groups of order 9p of which the centre has order 3 are not CI-groups, where p is an odd prime. Then a shorter explicit list is given of candidates for finite CI-groups. Finally, some new families of finite CI-groups are found, that is, the metacyclic groups of order 4p (with centre of order 2) and of order 8p (with centre of order 4) are CI-groups, and a proof is given for the Frobenius group of order 3p to be a CI-group, where p is a prime.

The normality of symmetry property of Cayley graphs of valencies 3 and 4 on the alternating group A5 is studied. We prove that all but four such graphs are normal; that A5 is not 5-CI. A complete classification of all arc-transitive Cayley graphs on A5 of valencies 3 and 4 as well as some examples of trivalent and tetravalent GRRs of A5 is given.