Computation of the bisection width for random d-regular graphs

In this paper we provide an explicit way to compute asymptotically almost sure upper bounds on the bisection width of random d-regular graphs, for any value of d. We provide the bounds for 5 less than or equal to d less than or equal to 12. The upper bounds are obtained from the analysis of the performance of a randomized greedy algorithm to find bisections of d-regular graphs. We also give empirical values of the size of bisection found by the algorithm for some small values of d and compare it with numerical approximations of our theoretical bounds. Our analysis also gives asymptotic lower bounds for the size of the maximum bisection.