Cayley Graphs Of Abelian Groups Research Articles

The Hamilton Compression of Highly Symmetric Graphs

We say that a Hamilton cycle C=(x1,…,xn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C=(x_1,\\ldots ,x_n)$$\\end{document} in a graph G is k-symmetric, if the mapping xi↦xi+n/k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_i\\mapsto x_{i+n/k}$$\\end{document} for all i=1,…,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$i=1,\\ldots ,n$$\\end{document}, where indices are considered modulo n, is an automorphism of G. In other words, if we lay out the vertices x1,…,xn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_1,\\ldots ,x_n$$\\end{document} equidistantly on a circle and draw the edges of G as straight lines, then the drawing of G has k-fold rotational symmetry, i.e., all information about the graph is compressed into a 360∘/k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$360^\\circ /k$$\\end{document} wedge of the drawing. The maximum k for which there exists a k-symmetric Hamilton cycle in G is referred to as the Hamilton compression of G. We investigate the Hamilton compression of four different families of vertex-transitive graphs, namely hypercubes, Johnson graphs, permutahedra and Cayley graphs of abelian groups. In several cases, we determine their Hamilton compression exactly, and in other cases, we provide close lower and upper bounds. The constructed cycles have a much higher compression than several classical Gray codes known from the literature. Our constructions also yield Gray codes for bitstrings, combinations and permutations that have few tracks and/or that are balanced.

Geometry of random Cayley graphs of Abelian groups

Geometry of random Cayley graphs of Abelian groups

Quantum symmetries of Cayley graphs of abelian groups

AbstractWe study Cayley graphs of abelian groups from the perspective of quantum symmetries. We develop a general strategy for determining the quantum automorphism groups of such graphs. Applying this procedure, we find the quantum symmetries of the halved cube graph, the folded cube graph, and the Hamming graphs.

Chromatic numbers of Cayley graphs of abelian groups: A matrix method

In this paper, we take a modest first step towards a systematic study of chromatic numbers of Cayley graphs on abelian groups. We lose little when we consider these graphs only when they are connected and of finite degree. As in the work of Heuberger and others, in such cases the graph can be represented by an m×r integer matrix, where we call m the dimension and r the rank. Adding or subtracting rows produces a graph homomorphism to a graph with a matrix of smaller dimension, thereby giving an upper bound on the chromatic number of the original graph. In this article we develop the foundations of this method. As a demonstration of its utility, we provide an alternate proof of Payan's theorem, which states that a cubelike graph (i.e., a Cayley graph on the product Z2×⋯×Z2 of the integers modulo 2 with itself finitely many times) cannot have chromatic number 3. In a series of follow-up articles using the method of Heuberger matrices, we completely determine the chromatic number in cases with small dimension and rank, as well as prove a generalization of Zhu's theorem on the chromatic number of 6-valent integer distance graphs.

On the inducibility problem for random Cayley graphs of abelian groups with a few deleted vertices

AbstractGiven a k‐vertex graph H and an integer n, what are the n‐vertex graphs with the maximum number of induced copies of H? This question is closely related to the inducibility problem introduced by Pippenger and Golumbic in 1975, which asks for the maximum possible fraction of k‐vertex subsets of an n‐vertex graph that induce a copy of H. Huang, Lee, and the first author proved that for a random k‐vertex graph H, almost surely the n‐vertex graphs maximizing the number of induced copies of H are the balanced iterated blow‐ups of H. In this article, we consider the case where the graph H is obtained by deleting a small number of vertices from a random Cayley graph of an abelian group. We prove that in this case, almost surely all n‐vertex graphs maximizing the number of induced copies of H are balanced iterated blow‐ups of .

Diameters of random Cayley graphs of finite nilpotent groups

Abstract We prove the existence of a limiting distribution for the appropriately rescaled diameters of random undirected Cayley graphs of finite nilpotent groups of bounded rank and nilpotency class, thus extending a result of Shapira and Zuck which dealt with the case of abelian groups. The limiting distribution is defined on a space of unimodular lattices, as in the case of random Cayley graphs of abelian groups. Our result, when specialised to a certain family of unitriangular groups, establishes a very recent conjecture of Hermon and Thomas. We derive this as a consequence of a general inequality, showing that the diameter of a Cayley graph of a nilpotent group is governed by the diameter of its abelianisation.

On Reconstruction of Normal Edge-Transitive Cayley Graphs

The main idea of this paper is to provide an algebraic algorithm for constructing symmetric graphs with optimal fault tolerance. For this purpose, we use normal edge-transitive Cayley graphs and the idea of reconstruction question posed by Praeger to present a special factorization of groups which induces a graphical decomposition of normal edge-transitive Cayley graphs to simpler normal edge-transitive Cayley graphs. Then as a consequence of our results, we continue the study of normal edge-transitive Cayley graphs of abelian groups and we show that knowing normal edge-transitive Cayley graphs of abelian p-groups, we can determine all normal edge-transitive Cayley graphs of abelian groups.

An improved Moore bound and some new optimal families of mixed Abelian Cayley graphs

We consider the case in which mixed graphs (with both directed and undirected edges) are Cayley graphs of Abelian groups. In this case, some Moore bounds were derived for the maximum number of vertices that such graphs can attain. We first show these bounds can be improved if we know more details about the order of some elements of the generating set. Based on these improvements, we present some new families of mixed graphs. For every fixed value of the degree, these families have an asymptotically large number of vertices as the diameter increases. In some cases, the results obtained are shown to be optimal.

New Moore-Like Bounds and Some Optimal Families of Abelian Cayley Mixed Graphs

Mixed graphs can be seen as digraphs that have both arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case where such graphs are Cayley graphs of abelian groups. Such groups can be constructed using a generalization to $$\mathbb {Z}^n$$ of the concept of congruence in $$\mathbb {Z}$$. Here we use this approach to present some families of mixed graphs, which, for every fixed value of the degree, have an asymptotically large number of vertices as the diameter increases. In some cases, the results obtained are shown to be optimal.

On the Isomorphism Problem for Cayley Graphs of Abelian Groups whose Sylow Subgroups are Elementary Abelian or Cyclic

We show that if certain arithmetic conditions hold, then the Cayley isomorphism problem for abelian groups, all of whose Sylow subgroups are elementary abelian or cyclic, reduces to the Cayley isomorphism problem for its Sylow subgroups. This yields a large number of results concerning the Cayley isomorphism problem, perhaps the most interesting of which is the following: if $p_1,\ldots, p_r$ are distinct primes satisfying certain arithmetic conditions, then two Cayley digraphs of $\mathbb{Z}_{p_1}^{a_1}\times\cdots\times\mathbb{Z}_{p_r}^{a_r}$, $a_i\le 5$, are isomorphic if and only if they are isomorphic by a group automorphism of $\mathbb{Z}_{p_1}^{a_1}\times\cdots\times\mathbb{Z}_{p_r}^{a_r}$. That is, that such groups are CI-groups with respect to digraphs.

Locality of percolation for Abelian Cayley graphs

We prove that the value of the critical probability for percolation on an Abelian Cayley graph is determined by its local structure. This is a partial positive answer to a conjecture of Schramm: the function pcpc defined on the set of Cayley graphs of Abelian groups of rank at least 22 is continuous for the Benjamini–Schramm topology. The proof involves group-theoretic tools and a new block argument.

Discrepancy and eigenvalues of Cayley graphs

We consider quasirandom properties for Cayley graphs of finite abelian groups. We show that having uniform edge-distribution (i.e., small discrepancy) and having large eigenvalue gap are equivalent properties for such Cayley graphs, even if they are sparse. This positively answers a question of Chung and Graham ["Sparse quasi-random graphs", Combinatorica 22 (2002), no. 2, 217-244] for the particular case of Cayley graphs of abelian groups, while in general the answer is negative.

Normally Regular Digraphs

A normally regular digraph with parameters $(v,k,\lambda,\mu)$ is a directed graph on $v$ vertices whose adjacency matrix $A$ satisfies the equation $AA^t=k I+\lambda (A+A^t)+\mu(J-I-A-A^t)$. This means that every vertex has out-degree $k$, a pair of non-adjacent vertices have $\mu$ common out-neighbours, a pair of vertices connected by an edge in one direction have $\lambda$ common out-neighbours and a pair of vertices connected by edges in both directions have $2\lambda-\mu$ common out-neighbours. We often assume that two vertices can not be connected in both directions. We prove that the adjacency matrix of a normally regular digraph is normal. A connected $k$-regular digraph with normal adjacency matrix is a normally regular digraph if and only if all eigenvalues other than $k$ are on one circle in the complex plane. We prove several non-existence results, structural characterizations, and constructions of normally regular digraphs. In many cases these graphs are Cayley graphs of abelian groups and the construction is then based on a generalization of difference sets.We also show connections to other combinatorial objects: strongly regular graphs, symmetric 2-designs and association schemes.

On the construction of all shortest vertex-disjoint paths in Cayley graphs of abelian groups

Cayley graphs provide a group-theoretic model for designing and analyzing symmetric interconnection networks. In this paper, we give a sufficient condition for the existence of m vertex-disjoint shortest paths from one source vertex to other m (not necessarily distinct) destination vertices in a Cayley graph of an abelian group, where m≤n and n is the cardinality of a (group) generator of the abelian group. In addition, when the condition holds, the m vertex-disjoint shortest paths can be constructed in O(mn) time, which is optimal in the worst case when O(n) ≤ the order of diameter. By applying our results, we can easily obtain the necessary and sufficient conditions, which can be verified in nearly optimal time, for the existence of all shortest vertex-disjoint paths in generalized hypercubes and odd tori. In the situation that all of the source vertex and destination vertices are mutually distinct, brute-force computations show that the probability of the existence of the m vertex-disjoint shortest paths in an r-dimensional generalized hypercube with r coordinates each of order k is greater than 94%, 70%, 96%, 99%, and 99% for (k,r,m)=(2,7,7),(3,4,8),(4,3,6),(5,3,6), and (6,3,6), respectively.

Representativity of Cayley maps

The motivating question of this paper is to ask when, given a group A and generating set X for A, one can find a Cayley map CM(A,X,ρ) with representativity at least two (also called a strong embedding of the graph; note that in such an embedding, each face is bounded by a cycle of the graph).We first investigate some general consequences of a Cayley map’s having representativity one, which means that there exist faces that are self-adjacent (and consequently these faces are not bounded by proper cycles of the graph). We find that there exist Cayley graphs of Abelian groups all of whose Cayley maps have representativity one. This indicates that if these graphs have strong embeddings (orientable or non-orientable), then these embeddings cannot be obtained by applying the symmetry of Cayley graphs in the most natural way. This also indicates that the Cycle Double Cover of these graphs, if it exists, cannot be constructed by applying the symmetry of these graphs in the most natural way.In addition, we investigate specific consequences in the case |X|=2 or 3; we provide a few classes of examples of A and X which may be forced to have representativity at least two; and we provide a combinatorial formalism for calculating the representativity of a Cayley map without appealing to the “lifted” embedding.

Non-normal edge-transitive directed Cayley graphs of abelian groups

For a group G ,a nd as ubsetS of G such that 1GS ,l etX = Cay(G, S) be the corresponding Cayley graph. Then X is said to be normal edge transitive if NAut(X)(G) is transitive on edges. In this paper, we determine all connected directed Cayley graphs of finite abelian groups with valency at most 3 which are normal edge transitive but not normal.

Hamilton cycle and Hamilton path extendability of Cayley graphs on abelian groups

AbstractIn this paper the concepts of Hamilton cycle (HC) and Hamilton path (HP) extendability are introduced. A connected graph Γ isn‐HC‐extendableif it contains a path of lengthnand if every such path is contained in some Hamilton cycle of Γ. Similarly, Γ isweakly n‐HP‐extendableif it contains a path of lengthnand if every such path is contained in some Hamilton path of Γ. Moreover, Γ isstrongly n‐HP‐extendableif it contains a path of lengthnand if for every such pathPthere is a Hamilton path of Γ starting withP. These concepts are then studied for the class of connected Cayley graphs on abelian groups. It is proved that every connected Cayley graph on an abelian group of order at least three is 2‐HC‐extendable and a complete classification of 3‐HC‐extendable connected Cayley graphs of abelian groups is obtained. Moreover, it is proved that every connected Cayley graph on an abelian group of order at least five is weakly 4‐HP‐extendable. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory

Group velocity of discrete-time quantum walks

We show that certain types of quantum walks can be modeled as waves that propagate in a medium with phase and group velocities that are explicitly calculable. Since the group and phase velocities indicate how fast wave packets can propagate causally, we propose the use of these wave velocities in our definition for the hitting time of quantum walks. Our definition of hitting time has the advantage that it requires neither the specification of a walker's initial condition nor of an arrival probability threshold. We give full details for the case of quantum walks on the Cayley graphs of Abelian groups. This includes the special cases of quantum walks on the line and on hypercubes.

Transitive permutation groups of prime-squared degree

We explicitly determine all of the transitive groups of degree p-squared, p a prime, whose Sylow p-subgroup is not the wreath product of two cyclic groups of order p. Furthermore, we provide a general description of the transitive groups of degree p-squared whose Sylow p-subgroup is such a wreath product, and explicitly determine most of them. As applications, we solve the Cayley Isomorphism problem for Cayley objects of an abelian group of order p-squared, explicitly determine the full automorphism group of Cayley graphs of abelian groups of order p-squared, and find all nonnormal Cayley graphs of order p-squared.

Hard Metrics from Cayley Graphs of Abelian Groups

Hard metrics are the class of extremal metrics with respect to embedding into Euclidean spaces; they incur $\Omega(\log n)$ multiplicative distortion, which is as large as it can possibly get for any metric space of size $n$. Besides being very interesting objects akin to expanders and good error-correcting codes, and having a rich structure, such metrics are important for obtaining lower bounds in combinatorial optimization, e.g., on the value of MinCut/MaxFlow ratio for multicommodity flows. For more than a decade, a single family of hard metrics was known (Linial, London, Rabinovich (Combinatorica 1995) and Aumann, Rabani (SICOMP 1998)). Recently, a different family was found by Khot and Naor (FOCS 2005). In this paper we present a general method of constructing hard metrics. Our results extend to embeddings into negative type metric spaces and into $\ell_1.$