New Spectral Lower Bounds on the Bisection Width of Graphs

The communication overhead is a major bottleneck for the execution of a process graph on a parallel computer system. In the case of two processors, the minimization of the communication can be modeled by the graph bisection problem. The spectral lower bound of 2jV j 4 for the bisection width of a graph is well-known. The bisection width is equal to 2jV j 4 iff all vertices are incident to 2 2 cut edges in every optimal bisection. We discuss the case for which this fact is not satisfied and present a new method to get tighter lower bounds on the bisection width. This method makes use of the level structure defined by the bisection. Under certain conditions we get a lower bound depending on 2 jV j with 12 < 1. We also present examples of graphs for which our new bounds are tight up to a constant factor. As a by-product, we derive new lower bounds for the bisection widths of 3and 4-regular graphs. We use them to establish tighter lower bounds for the bisection width of 3and 4-regular Ramanujan graphs.