Multiprocessor Interconnection Networks

Dragoš Cvetković, Tatjana Davidović

Homogeneous multiprocessor systems are usually modelled by undirected graphs. Vertices of these graphs represent the processors, while edges denote the connection links between adjacent processors. Let $G$ be a graph with diameter $D$, maximum vertex degree $\Delta$, the largest eigenvalue $\lambda_1$ and $m$ distinct eigenvalues. The products $m\Delta$ and $(D+1)\lambda_1$ are called the tightness of $G$ of the first and second type, respectively. In the recent literature it was suggested that graphs with a small tightness of the first type are good models for the multiprocessor interconnection networks. We study these and some other types of tightness and some related graph invariants and demonstrate their usefulness in the analysis of multiprocessor interconnection networks. A survey of frequently used interconnection networks is given. Load balancing problem is presented. We prove that the number of connected graphs with a bounded tightness is finite and we determine explicitly graphs with tightness values not exceeding 9. There are 69 such graphs and they contain up to 10 vertices. In addition we identify graphs with minimal tightness values when the number of vertices is $n=2,\ldots,10$.