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Exploring the diameter and broadcast time of general Knödel graphs using extensive simulations

Efficient dissemination of information remains a central challenge for all types of networks. There are two ways to handle this issue. One way is to compress the amount of data being transferred and the second way is to minimize the delay of information distribution. Well-received approaches used in the second way either design efficient algorithms or implement reliable network architectures with optimal dissemination time. Among the well-known network architectures, the Knödel graph can be considered a suitable candidate for the problem of information dissemination. The Knödel graph Wd,n is a regular graph, of an even order n and degree d, 1 ≤ d ≤ ⌊log2 n⌋. The Knödel graph was introduced by W. Knödel almost four decades ago as network architecture with good properties in terms of broadcasting and gossiping in interconnected networks. Although the Knödel graph has a highly symmetric structure, its diameter is only known for, Wd2d. Recently, the general upper and lower bounds on diameter and broadcast time of the Knödel graph have been presented. In this paper, our motivation is to explore the communication properties of Knödel graph in terms of the diameter, the number of vertices at a particular distance and the broadcast time. Experimentally, we obtain the following results; (a) the diameter of some specific Knödel graphs, (b) the propositions on the number of vertices at a particular distance, and (c) the upper bound on the broadcast time of Knödel graph. We also construct a new graph, denoted as HWd2d by connecting Knödel graph Wd-1,2d-1 to hypercube Hd-1 and experimentally show that HWd2d have even a smaller diameter than Knödel graph Wd2d.