Frobenius Circulant Graphs of Valency Four

2009

Networks

Self Cite

Given integers n ≥ 7 and a, b, c with 1 ≤ a, b, c ≤ n − 1 such that a, n − a, b, n − b, c, n − c are pairwise distinct, the (undirected) triple-loop network TL n (a, b, c) is the degree-six graph with vertices 0, 1, 2, . . . , n − 1 such that each vertex x is adjacent to x ± a, x ± b, and x ± c, where the operation is modulo n. It is known that the maximum order of a connected triple-loop network of the form TL n (a, b, n − (a + b)) with given diameter d ≥ 2 is n d = 3d 2 + 3d + 1, which is achieved by2 −1). In this article, we study the routing and gossiping problems for such optimal tripleloop networks under the store-and-forward, all-port, and full-duplex model, and prove that they admit "perfect" gossiping and routing schemes which exhibit many interesting features. Using a group-theoretic approach we develop for TL n d a method for systematically producing such optimal gossiping and routing schemes. Moreover, we determine the minimum gossip time, the edge-and arc-forwarding indices, and the minimal edge-and arcforwarding indices of TL n d , and prove that our routing schemes are optimal with respect to these four indices simultaneously. As a key step towards these results, we prove that TL n d is a Frobenius graph on a Frobenius group with Frobenius kernel Z n d , and that TL n d is arc-transitive with respect to this Frobenius group. In addition, we show that TL n d admits complete rotations.

“…(a) H(d) and S(d) are identical as sets and H(d) is generated by a complete rotation. This phenomenon occurs for another family[17] of circulant graphs with degree four.…”

mentioning

confidence: 82%

Gossiping and routing in undirected triple-loop networks

Thomson

2009

Networks

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“…In [23] it was proved that orbital regular graphs (which are essentially Frobenius graphs [10] except cycles and stars) achieve the smallest possible edge-forwarding index. In [25,26,27], Thomson and Zhou gave formulas for the edge-forwarding and arc-forwarding indices of two interesting families of Frobenius circulant graphs. The exact value of edge-forwarding index of some other graphs have also been computed, including Knödel graphs [11] and recursive circulant graphs [14].…”

Section: Literature Reviewmentioning

confidence: 99%

Forwarding and optical indices of 4-regular circulant networks

Gan

Mokhtar

Journal of Discrete Algorithms

2015

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An all-to-all routing in a graph $G$ is a set of oriented paths of $G$, with exactly one path for each ordered pair of vertices. The load of an edge under an all-to-all routing $R$ is the number of times it is used (in either direction) by paths of $R$, and the maximum load of an edge is denoted by $\pi(G,R)$. The edge-forwarding index $\pi(G)$ is the minimum of $\pi(G,R)$ over all possible all-to-all routings $R$, and the arc-forwarding index $\overrightarrow{\pi}(G)$ is defined similarly by taking direction into consideration, where an arc is an ordered pair of adjacent vertices. Denote by $w(G,R)$ the minimum number of colours required to colour the paths of $R$ such that any two paths having an edge in common receive distinct colours. The optical index $w(G)$ is defined to be the minimum of $w(G,R)$ over all possible $R$, and the directed optical index $\overrightarrow{w}(G)$ is defined similarly by requiring that any two paths having an arc in common receive distinct colours. In this paper we obtain lower and upper bounds on these four invariants for $4$-regular circulant graphs with connection set $\{\pm 1,\pm s\}$, $1

“…For example, networks with smaller diameters will lead to shorter data transmission delay. The forwarding indices [5,12] and bisection width are also well-known measures of performance of interconnection networks [10,13,25,[28][29][30]32].…”

Section: Introductionmentioning

confidence: 99%

Recursive cubes of rings as models for interconnection networks

Mokhtar

Discrete Applied Mathematics

2017

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We study recursive cubes of rings as models for interconnection networks. We first redefine each of them as a Cayley graph on the semidirect product of an elementary abelian group by a cyclic group in order to facilitate the study of them by using algebraic tools. We give an algorithm for computing shortest paths and the distance between any two vertices in recursive cubes of rings, and obtain the exact value of their diameters. We obtain sharp bounds on the Wiener index, vertex-forwarding index, edge-forwarding index and bisection width of recursive cubes of rings. The cube-connected cycles and cube-of-rings are special recursive cubes of rings, and hence all results obtained in the paper apply to these well-known networks.