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De Bruijn digraphs and shuffle-exchange graphs are useful models for interconnection networks. They can be represented as group action graphs of the wrapped butterfly graph and the cube-connected cycles, respectively. The Kautz digraph has the similar definitions and properties to de Bruijn digraphs. It is d-regular and strongly d-connected, thus it is a group action graph. In this paper, we use another representation of the Kautz digraph and settle the open problem posed by M.-C. Heydemann in [ G. Hahn (ed.) et al., Graph symmetry: algebraic methods and applications. Proceedings of the NATO Advanced Study Institute and séminaire de mathématiques supérieures, Montréal, Canada, July 1-12, 1996. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 497, 167–224 (1997; Zbl 0885.05075)].

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... At first, we give the following description derived from the vertex labeling of the Kautz digraph in [10]. ...

... Proposition 7 [10] ...

In this paper, we investigate isomorphic factorizations of the Kronecker product graphs. Using these relations, it is shown that (1) the Kronecker product of the d-out-regular digraph and the complete symmetric digraph is factorized into the line digraph, (2) the Kronecker product of the Kautz digraph and the de Bruijn digraph is factorized into the Kautz digraph, (3) the Kronecker product of binary generalized de Bruijn digraphs is factorized into the binary generalized de Bruijn digraph. (c) 2006 Published by Elsevier B.V.

... It is not in general straightforward to give explicit descriptions of these Cayley regular covers. Responding to a question in [8,Problem 47], such descriptions were given independently in [3,16] for the family of Kautz digraphs, a class of digraphs which is well behaved in terms of various useful parameters for communication network design. Also the Cayley regular covers of the de Bruijn digraphs are known [6,12]. ...

We study the family of \emph{derangement action digraphs}, which are a subfamily of the group action graphs introduced in [Fred Annexstein, Marc Baumslag, and Arnold L. Rosenberg, Group action graphs and parallel architectures, \emph{SIAM J. Comput.} 19 (1990), no. 3, 544--569]. For any non-empty set $X$ and a non-empty subset $S$ of $\Der(X)$, the set of derangments of $X$, we define the derangement action digraph $\rm\overrightarrow{DA}(X;S)$ to have vertex set $X$, and an arc from $x$ to $y$ if and only if $y=x^s$ for some $s\in S$. In common with Cayley graphs and digraphs, derangement action digraphs may be useful to model networks as the same routing and communication scheme can be implemented at each vertex. We determine necessary and sufficient conditions on $S$ under which $\rm\overrightarrow{DA}(X;S)$ may be viewed as a simple graph of valency $|S|$, and we call such graphs derangement action graphs. Also we investigate the structural and symmetry properties of these digraphs and graphs. Several open problems are posed and many examples are given.

This paper introduces a new class of interconnection scheme based on the Cayley graph of the alternating group. It is shown that this class of graphs are edge symmetric and 2-transitive. We then describe an algorithm for (a) packet routing based on the shortest path analysis, (b) finding a Hamiltonian cycle, (c) ranking and unranking along the chosen Hamiltonian cycle, (d) unit expansion and dilation three embedding of a class of two-dimensional grids, (e) unit dilation embedding of a variety of cycles, and (f) algorithm for broadcasting messages. The paper concludes with a short analysis of contention resulting from a typical communication scheme. Although this class of graphs does not possess many of the symmetry properties of the binary hypercube, with respect to the one source broadcasting, these graphs perform better than does a hypercube, and with respect to the contention problem, these graphs perform better than do the star graphs and are close to the hypercube. © 1993 by John Wiley & Sons, Inc.

The authors develop an algebraic framework that exposes the structural kinship among the deBruijn, shujCfle-exchange, butterjy, and cube-connected cycles networks and illustrate algorithmic benefits that ensue from the exposed relationships. The framework builds on two alge- braically specified genres of graphs: A group action graph (GAG, for short) is given by a set V of vertices and a set H of permutations of V: For each v E V and each r EII, there is an arc labeled r from vertex v to vertex v. A Cayley graph is a GAG (V, H), where V is the group Gr(H) generated by H and where each r 6 H acts on each g 6 Gr(H) by right multiplication. The graphs (Gr(H), H) and (V, H) are called associated graphs. It is shown that every GAG is a quotient graph of its associated Cayley graph. By applying such general results, the authors determine the following: The butterfly network (a Cayley graph) and the deBruijn network (a GAG) are associated graphs. The cube-connected cycles network (a Cayley graph) and the shuffle-exchange network (a GAG) are associated graphs. The order-n instance of both the butterfly and the cube-connected cycles share the same underlying group, but have slightly different generator sets II. By analyzing these algebraic results, it is delimited, for any Cayley graph G and associated GAG 7-/, a family of "leveled" algorithms which run as efficiently on T/ as they do on (the much larger) G. Further analysis of the results yields new, surprisingly efficient simulations by the shuffle-oriented networks (the shuffle-exchange and deBruijn networks) of like-sized butterfly-oriented networks (the butterfly and cube-connected cycles networks): An N-vertex butterfly-oriented network can be simulated by the smallest shuffle-oriented network that is big enough to hold it with slowdown O(log log N). This simulation is exponentially faster than the anticipated logarithmic slowdown. The mappings that underlie the simulation can be computed in linear time; and they afford one an algorithmic tech- nique for translating any program developed for a butterfly-oriented architecture into an equivalent program for a shuffle-oriented architecture, the latter program incurring only the indicated slowdown factor.

A digraph is k-arc transitive if it has a group of automorphisms which acts transitively on the set of k-arcs. Unlike the undirected case, in which the cycles are the only k-arc transitive finite graphs for k⩾8, there are k-arc transitive finite digraphs with arbitrary out-degree for every positive integer k. We show that every regular finite digraph admits a covering digraph which is k-arc transitive. The result provides a technique to construct k-arc transitive digraphs from arbitrary regular digraphs. Some examples are given from complete graphs with and without loops.

This survey provides a comprehensive and unified analysis of symmetry in a wide variety of Cayley graphs of permutation groups. These include the star graph, bubble-sort graph, modified bubble-sort graph, complete-transposition graph, prefix-reversal graph, alternating-group graph, binary and base-b (b ≥ 3) hypercube, cube connected cycles, bisectional graph, folded hypercube and binary orthogonal graph. In addition, we also define a variety of generalizations of the hypercube and orthogonal graphs. The types of symmetry analyzed include vertex and edge transitivity, distance regularity and distance transitivity. Since these notions of symmetry depend on the shortest paths, as a by product we also describe the shortest path routing algorithms for these graphs. We present a number of open problems related to the networks described in this paper.

We discuss Cayley graphs and show how they may be used as a tool for the design and analysis of network architectures for parallel computers. We then present a known powerful factoring technique in computational group theory to produce a fast efficient routing algorithm on the associated Cayley graph. This research can be regarded as a first attempt to find general purpose routing algorithms for interconnection networks. Further we present some evidence that average diameter of a network for a large scale MIMD machine is the predominant factor in determining network performance.

A number of researchers have proposed Cayley graphs and Schreier coset graphs as models for interconnection networks. New algorithms are presented for generating Cayley graphs in a more time-efficient manner than was previously possible. Alternatively, a second algorithm is provided for storing Cayley graphs in a space-efficient manner (log2(3) bits per node), so that copies could be cheaply stored at each node of an interconnection network. The second algorithm is especially useful for providing a compact encoding of an optimal routing table (for example, a 13 kilobyte optimal table for 64,000 nodes). The algorithm relies on using a compact encoding of group elements known from computational group theory. Generalizations of all of the above are presented for Schreier coset graphs.

In this paper, we study the problem of homomorphisms of a general class of line digraphs. We show that the homomorphisms can always be defined using a partial binary operation on the alphabet whose letters form labels of the vertices. We apply these results to de Bruijn and Kautz (in short B/K) digraphs to characterize their uniform homomorphisms. For d non-prime, we describe algorithms for constructing non-trivial uniform homomorphisms of d-ary B/K digraphs of diameter D onto d-ary B/K digraphs of diameter D − 1. Using the properties of the uniform homomorphisms and shortest-path spanning trees of B/K digraphs, we also describe optimal emulations of Divide&Conquer computations on B/K digraphs.

The authors develop a formal group-theoretic model, called the Cayley graph model, for designing, analyzing, and improving such networks. They show that this model is universal and demonstrate how interconnection networks can be concisely represented in this model. It is shown that this model enables the authors to design networks based on representations of finite groups. They can then analyze these networks by interpreting the group-theoretic structure graph theoretically, Using these ideas, and motivated by certain well-known combinatorial problems, they develop two classes of networks called star graphs and pancake graphs. These networks are shown to have better performance than previous networks

Given a colouring Δ of a d-regular digraph G and a colouring Π of the symmetric complete digraph on d vertices with loops, the uniformly induced colouring LΠΔ of the line digraph LG is defined. It is shown that the group of colour-preserving automorphisms of (LG, LΠΔ) is a subgroup of the group of colour-permuting automorphisms of (G, Δ). This result is then applied to prove that if (G, Δ) is a d-regular coloured digraph and (LG, LΠΔ) is a Cayley digraph, then (G, Δ) is itself a Cayley digraph Cay (Ω, Δ) and Π is a group of automorphisms of Ω. In particular, a characterization of those Kautz digraphs which are Cayley digraphs is given.If d=2, for every arc-transitive digraph G, LG is a Cayley digraph when the number k of orbits by the action of the so-called Rankin group is at most 5. If k ⩾ 3 the arc-transitive k-generalized cycles for which LG is a Cayley digraph are characterized.