Average Distance in Interconnection Networks via Reduction Theorems for Vertex-Weighted Graphs

Abstract

Average distance is an important parameter for measuring the communication cost of computer networks. A popular approach for its computation is to first partition the edge set of a network into convex components using the transitive closure of the Djoković-Winkler's relation and then to compute the average distance from the respective invariants of the components. In this paper we refine this idea further by shrinking the quotient graphs into smaller weighted graph called reduced graph, so that the average distance of the original graph is obtained from the reduced graphs. We demonstrate the significance of this technique by computing the average distance of butterfly and hypertree architectures. Along the way a computational error from [European J. Combin. 36 (2014) 71-76] is corrected.

The Graph Theory General Position Problem on Some Interconnection Networks

“…Two nodes w, i and w , i are linked by an edge if i = i + 1 and either w and w are identical or w and w differ only in the bit in position i . We refer to [21,Section 11.4] for basic properties of butterfly networks and to [5,12] for a recent application and the average distance of these networks, respectively. Now, for r ≥ 1 the r-dim Beneš network BN (r) is constructed by merging two r-dim butterfly networks as shown in Fig.…”

A survey of the all-pairs shortest paths problem and its variants in graphs

“…S. Klavžar at el. [55], mentions that the average distance can be studied equivalently as the Wiener index (or the network distance)-hence the fundamental task would be the computation of the average distance or the Wiener index efficiently. For a general graph G, one way of computing W(G) or µ(G) algorithmically is to run the APSD problem; that is it can be computed in polynomial time, [58].…”

“…The average distance is one of the important parameter in metric graph theory. As mentioned in [55], it has numerous applications in many areas including computer science, cheminformatics, mathematics, and recent application in phylogenetics (see [88]). One of the fundamental parameter in computer science is to measure the cost of communication in a computer net-work.…”

The Wiener Polarity Index of Benzenoid Systems and Nanotubes

“…[23] Some recent applications of this method include investigations of various molecular descriptors on benzenoid systems, [24,25] partial cubes, [26] and interconnection networks. [27] …”