Weighted multi-connected loop networks

Abstract Given relatively prime integers N , a 1 ,…, a k , a multi-connected loop network is defined as the directed graph with vertex set Z NZ = 0, 1, …, N − 1 , and directed edges i → r ≡ i + a j (mod N ). If each edge i → i + a j is given a positive real weight w j for j = I ,…, k , then we have a weighted multi-connected loop network. The weight of a path is the sum of weights on its edges. The distance from a vertex to another is the minimum weight of all paths from the first vertex to the second. The diameter of the network is the maximum distance, and the average diameter is the average distance in the network. In this paper we study the diameter and the average diameter of a weighted multi-connected loop network. We give a unified and generalized presentation of several results in the literature, and also some new results are obtained.

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