Diameter and Broadcast Time of the Knödel graph

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 W_(d, n) is a regular graph, of an even order n and degree d, 1 ≤ d ≤ floor(log 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 W_(d, 2^d). Recently, the general upper and lower bounds on diameter and broadcast time of the Knödel graph have been presented. In this thesis, our motivation is to find the diameter, the number of vertices at a particular distance and the broadcast time of the Knödel graph. Theoretically, we succeed to prove the diameter and the broadcast time of the Knödel graph W_(3, n). We also claim that the Knödel graph W_(3, n) for n = 4 mod 4 and n > 16, is a diametral broadcast graph. We present that W_(3, 22) is a broadcast graph. Experimentally, however, we obtain the following results; (a) the diameter of some specific Knödel graphs, and (b) the propositions on the number of vertices at a particular distance. We also construct a new graph, denoted as HW_(d,2^d), by connecting Knödel graph W_(d-1,2^(d-1)) to hypercube H_(d-1) and experimentally show that HW_(d,2^d) has even a smaller diameter than Knödel graph W_(d,2^d)

Diameter of the Knödel Graph

. Diameter of the 2 k -node Knodel graph is d(k + 2)=2e: 1 Introduction Recently, diverse properties and invariants of interconnection networks, not only those of parallel machines, have been studied and a number of interesting results has been found, see e.g. [1]. One of the important features of an interconnection network is its message passing ability. The quality of an interconnection network depends mainly on the time delay of the communication between the nodes, which can be either processors or computers or other type of terminals. Let us suppose that a node knows a piece of information and needs to transmit it to every other node in the network. This task is usually called broadcasting. When each node knows a piece of information that has to be transmitted to every other node, we are speaking about gossiping. We refer to [6, 8, 9] for surveys on broadcasting and gossiping. The Knodel graph was introduced 25 years ago by Knodel [11], as an interconnection network where gossip..