Graphical and Computational Representation of Groups

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Combinatorial group theory concerns itself with the theory of group presentations, that is, it deals with groups specified by a set of generators and corresponding defining relations. The theory has its origins and roots in topology and complex analysis and in particular in the theory of the fundamental groups of combinatorial cell complexes. Because of its nature and origins, combinatorial group theory comes into contact with, and uses, many different areas of mathematics: clearly algebra and topology as mentioned above, but also hyperbolic geometry via the study of the Cayley graph; pure mathematical logic through the study of various decision problems; and computer science through the study of computational aspects of group theory and rewriting systems. These last three areas have been the subject of increased interest and intense study during the 1990s. The ties with hyperbolic geometry have evolved into geometric group theory—especially the theory of the hyperbolic groups—by considering the groups themselves as geometric objects via the Cayley graph. The ties with computation and computer science have evolved into the theory of automatic groups, where rewriting and computability properties are paramount. Finally, there have been extensive studies on the nature of solutions of the various decision problems and on the logical underpinnings of the the whole theory. Relative to the latter there has been a great deal of work on the Tarski conjectures (see below) and their ties to purely group theoretical properties.KeywordsFree GroupCayley GraphFree ProductHyperbolic GroupMaximal Abelian SubgroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A class of uni-directional Cayley graphs based on alternating groups is proposed in this paper. It is shown that this class of graphs is strongly connected and recursively scalable. The analysis of the shortest distance between any pair of nodes in a graph of this class is also given. Based on the analysis, we develop a polynomial time routing algorithm which yields a path distance at most one more than the theoretic lower bound. Furthermore, comparisons among uni-directional hypercubes, uni-directional star graphs, and uni-directional alternating group graphs are given. These observations validate the superiority of uni-directional alternating group graphs among known uni-directional topologies.

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).

The work is devoted to the consideration of applications of mathematics, in particular, graph theory and group theory in chemistry and physics. Chemistry was one of the first to adopt the ideas of graph theory. Mathematical chemistry is a section of theoretical chemistry that studies the application of mathematics to chemical problems. The possibility of formalizing molecules in the language of structural chemistry using graphs allows one to determine the number of different isomers using graph theory methods. The main tool in mathematical chemistry is mathematical modeling using computer technology. The methods of graph theory are actively used in three areas of chemistry: structural chemistry, chemical kinetics, and chemical physics of polymers. Group theory has found application in many areas of physics. Group theory serves to explain the most important characteristics of atomic spectra. Group theory provides an opportunity for a uniform approach to a large number of problems in solid state physics. Matrix groups and unitary group representations are actively used in these studies. Irreducible group representations are used to classify vibrational and electronic states of a crystal. Methods of group theory are applied to models of problems of classifying electronic states of a crystal. In crystallography, much attention is paid to applications of group theory methods to the study of vibrations of atoms that make up a molecule relative to their equilibrium positions. The results can be useful in encoding information in space communications sessions.

We address the problem of the construction of quantum walks on Cayley graphs. Our main motivation is the relationship between quantum algorithms and quantum walks. In particular, we discuss the choice of the dimension of the local Hilbert space and consider various classes of graphs on which the structure of quantum walks may differ. We completely characterise quantum walks on free groups and present partial results on more general cases. Some examples are given, including a family of quantum walks on the hypercube involving a Clifford Algebra.

The Cayley graph generated by a set A is the graph ΓA(V) on a set of positive integers V such that a pair (u, v) ∈ V×V is an edge of the graph if and only if |u−v| ∈ A or u+v ∈ A. We denote by (n,m) the class of graphs G = (V, E) such that G is a union of chains and cycles and |V| = n, |E| = m. In this paper, we present an upper bound for the number of independent sets in Cayley graphs ΓA(V) such that A = {⌈n/2⌉ − t, ⌈n/2⌉ − ƒ}, V ⊆ [⌊n/2⌋ + 1, ⌊n/2⌋+t] ∪ [n − t + 1, n], where n, t, ƒ ∈ N and ƒ < t < n/4. We also describe the graph with the maximum number of independent sets in the family (n, m).

One frequently studied problem in the context of information dissemination in communication networks is the broadcasting problem. In this paper, we study the following randomized broadcasting protocol: At some time t an information r is placed at one of the nodes of a graph G. In the succeeding steps, each informed node chooses one neighbor, independently and uniformly at random, and informs this neighbor by sending a copy of r to it. First, we consider the relationship between randomized broadcasting and random walks on graphs. In particular, we prove that the runtime of the algorithm described above is upper bounded by the corresponding mixing time, up to a logarithmic factor. One key ingredient of our proofs is the analysis of a continuous-type version of the afore mentioned algorithm, which might be of independent interest. Then, we introduce a general class of Cayley graphs, including (among others) Star graphs, Transposition graphs, and Pancake graphs. We show that randomized broadcasting has optimal runtime on all graphs belonging to this class. Finally, we develop a new proof technique by combining martingale tail estimates with combinatorial methods. Using this approach, we show the optimality of our algorithm on another Cayley graph and obtain new knowledge about the runtime distribution on several Cayley graphs.

Consider the (p,q) simple connected graph . The sum absolute values of the spectrum of quotient matrix of a graph make up the graph's quotient energy. The objective of this study is to examine the quotient energy of identity graphs and zero-divisor graphs of commutative rings using group theory, graph theory, and applications. In this study, the identity graphs derived from the group and a few classes of zero-divisor graphs of the commutative ring R are examined.

This chapter considers a new type of graph coloring known as edge coloring. It begins with a discussion of an idea by Scottish physicist Peter Guthrie Tait that led to edge coloring. Tait proved that the regions of every 3-regular bridgeless planar graph could be colored with four or fewer colors if and only if the edges of such a graph could be colored with three colors so that every two adjacent edges are colored differently. Tait thought that he had found a new way to solve the Four Color Problem. The chapter also examines the chromatic index of a graph, Vizing's Theorem, applications of edge colorings, and a class of numbers in graph theory called Ramsey numbers. Finally, it describes the Road Coloring Theorem which deals with traffic systems consisting only of one-way streets in which the same number of roads leave each location.

It was proved by Glover and Marusic (J. Eur. Math. Soc. 9:775---787, 2007), that cubic Cayley graphs arising from groups G={a,x|a2=x s=(ax)3=1,...} having a (2,s,3)-presentation, that is, from groups generated by an involution a and an element x of order s such that their product ax has order 3, have a Hamiltonian cycle when |G| (and thus also s) is congruent to 2 modulo 4, and have a Hamiltonian path when |G| is congruent to 0 modulo 4. In this article the existence of a Hamiltonian cycle is proved when apart from |G| also s is congruent to 0 modulo 4, thus leaving |G| congruent to 0 modulo 4 with s either odd or congruent to 2 modulo 4 as the only remaining cases to be dealt with in order to establish existence of Hamiltonian cycles for this particular class of cubic Cayley graphs.

In this paper, we give a necessary and sufficient condition for the integrality of Cayley graphs over the dihedral group $$D_n=\langle a,b\mid a^n=b^2=1,bab=a^{-1}\rangle $$ . Moreover, we also obtain some simple sufficient conditions for the integrality of Cayley graphs over $$D_n$$ in terms of the Boolean algebra of $$\langle a\rangle $$ , from which we find infinite classes of integral Cayley graphs over $$D_n$$ . In particular, we completely determine all integral Cayley graphs over the dihedral group $$D_p$$ for a prime p.

A Cayley graph for a group G is called normal edge-transitive if it admits an edge-transitive action of some subgroup of the holomorph of G [the normaliser of a regular copy of G in $${{\mathrm{Sym}}}(G)$$Sym(G)]. We complete the classification of normal edge-transitive Cayley graphs of order a product of two primes by dealing with Cayley graphs for Frobenius groups of such orders. We determine the automorphism groups of these graphs, proving in particular that there is a unique vertex-primitive example, namely the flag graph of the Fano plane.

For a group $G$ generated by $S\doteq \{g_1,\ldots ,g_n\}$, one can construct the Cayley graph $\mathrm {Cayley}({G},{S})$. Given a distance set $D\subset \mathbb Z _{\geq 0}$ and $\mathrm{Cayley}{G}{S}$, one can construct a $D$-neighborhood complex. This neighborhood complex is a simplicial complex to which we can associate a chain complex. Group $G$ acts on this chain complex, and this leads to an action on the homology of the chain complex. These group actions decompose into several representations of $G$. This paper uses tools from group theory, representation theory and homological algebra to further our understanding of the interplay between generated groups, corresponding representations on their associated $D$-neighborhood complexes and the homology of the $D$-neighborhood complexes. This paper is an exposition of the results in my dissertation focusing on the case of two generators.

We propose new techniques to address low-level image segmentation problem under clustering theory. The goal of this paper is to provide a compromised solution between methods that produce two different kinds of segmentation results: one generates coherent regions but views disjoint regions as totally different objects, and the others do not consider the spatial relationship at all. For our approach, spatial geometries are partially preserved and disjoint regions are also allowed to be grouped into a single cluster. The approach is built on the feature space clustering algorithm called K-MSVC, but constrained by the graph to maintain the capability of partially preserving the spatial coherence. A new type of graph called Random Grouping Graph (RGG) is introduced then, to overcome the high computational cost on the grid-graph based image representation. It's fast to construct, greatly reduce the graph size and can speedup other graph-based segmentation algorithms. Though with less vertices, the segmentation on RGG works better than on downsampled version of the image. Nontrivial experimental results on the Berkeley Segmentation Dataset demonstrate that our method outperforms the existing algorithms and yields more satisfactory results.