Gossiping and routing in undirected triple-loop networks

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.

“…From this and (28), (29), (31) and (32), we obtain (a,x)∈G l(a, x) ≥ 2 n−1 r 2 /2 + 4q 2 1 − 12q 3/2 log(2q)…”

Section: Case Dr ≥ 2nmentioning

confidence: 75%

“…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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“…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

“…This has been achieved in [33] and [36] in the case of valency 4 and 6, respectively. (See also [34,35,41] for related results.) In this paper we classify first kind Frobenius circulants of valency 2p for any odd prime p (Theorem 5.3), and prove that all of them are p th cyclotomic graphs (Theorem 5.5).…”

Section: Introductionmentioning

confidence: 99%

Cyclotomic graphs and perfect codes

Journal of Pure and Applied Algebra

2019

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We study two families of cyclotomic graphs and perfect codes in them. They are Cayley graphs on the additive group ofis an mth primitive root of unity, A a nonzero ideal of Z[ζ m ], and φ Euler's totient function. We call them the mth cyclotomic graph and the second kind mth cyclotomic graph, and denote them by G m (A) and G * m (A), respectively. We give a necessary and sufficient condition for D/A to be a perfect t-code in G * m (A) and a necessary condition for D/A to be such a code in G m (A), where t ≥ 1 is an integer and D an ideal of Z[ζ m ] containing A. In the case when m = 3, 4, G m ((α)) is known as an Eisenstein-Jacobi and Gaussian networks, respectively, and we obtain necessary conditions for (β)/(α) to be a perfect t-code in G m ((α)), where 0 = α, β ∈ Z[ζ m ] with β dividing α. In the literature such conditions are known to be sufficient when m = 4 and m = 3 under an additional condition. We give a classification of all first kind Frobenius circulants of valency 2p and prove that they are all pth cyclotomic graphs, where p is an odd prime. Such graphs belong to a large family of Cayley graphs that are efficient for routing and gossiping.