Application of Some Graph Invariants to the Analysis of Multiprocessor Interconnection Networks∗

Let G be a graph with diameter D, maximum vertex degree Δ, the largest eigenvalue λ1 and m distinct eigenvalues. The products mΔ and (D+1) λ1 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. Tightness values for graphs of some standard interconnection networks are determined. We also present some facts showing that the tightness of the second type is a relevant graph invariant. We prove that the number of connected graphs with a bounded tightness is finite.