Graphs for small multiprocessor interconnection networks

Let D be the diameter of a graph G and let λ1 be the largest eigenvalue of its (0, 1)-adjacency matrix. We give a proof of the fact that there are exactly 69 non-trivial connected graphs with (D + 1)λ1 ⩽ 9. These 69 graphs all have up to 10 vertices and were recently found to be suitable models for small multiprocessor interconnection networks. We also examine the suitability of integral graphs to model multiprocessor interconnection networks, especially with respect to the load balancing problem. In addition, we classify integral graphs with small values of (D + 1)λ1 in connection with the load balancing problem for multiprocessor systems.