Topology of Interconnection Networks With Given Degree and Diameter

Abstract

This thesis deals with the design of optimal interconnection networks. Optimality is interpreted as the largest possible number of nodes in the network, under given constraints on the number of connections attached to a node (degree of the node), and on the length of shortest paths between any two nodes (diameter of the network). Any interconnection network with bidirectional channels can be modelled by an undirected graph referred to as the 'topology' of the network. In graph theory this interpretation of optimality is known as the degree/diameter problem [8]. Under these constraints, there is an upper bound M ,D on the maximum number of nodes that a network of maximum degree and diameter D can have, which is known as the Moore bound [2,8]. For bipartite network topology, the bipartite Moore bound M b ,D is defined in a similar way to the general Moore bound [2,8].I address the degree/diameter problem for both general graphs and bipartite graphs. Interconnection networks modeled by (bipartite) Moore graphs-namely, (bipartite) graphs attaining the (bipartite) Moore bound-are optimal.Studies on Moore graphs and bipartite Moore graphs proved their rarity [1,6,12]. Consequently, research efforts to search for such optimal graphs fall into two main streams. (i) Lowering the upper bound on the number of vertices of a graph of maximum degree and diameter D by proving the nonexistence or otherwise of graphs whose number of vertices is close to the corresponding Moore bounds. (ii) Increasing the lower bound on the number of vertices of a graph of maximum degree and diameter D by constructing largest known graphs of maximum degree and diameter D.