PhD Thesis: Topology of Interconnection Networks with Given Degree and Diameter

In this thesis I deal 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 modeled by an undirected graph referred to as the “topology” of the network. In graph theory our interpretation of optimality is known as the degree/diameter problem [8]. Under these constraints, there is an upper bound on the maximum number of nodes that a network of maximum degree ∆ and diameter D can have, which is known as the Moore bound and is denoted by M∆,D [2, 8]. Considering bipartite network topology, the bipartite Moore bound, denoted by M b ∆,D, is defined in a similar way to the general case of the Moore bound [2, 8]. I address the degree/diameter problem for both general graphs and bipartite graphs. Interconnection networks modeled by (bipartite) Moore graphs—(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 such optimal graphs fall into two mainstreams. (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.