Asymptotic Behavior of the Random 3-Regular Bipartite Graph

In 2001, two numerical experiments were performed to observe whether or not the second largest eigenvalue of the adjacency matrix for the random cubic bipartite graph approaches 2 √ 2 as the size of the graph increases. In the first experiment, by Kevin Chang, the graphs were chosen using an algorithm that constructed entirely new graphs at each step using three random permutations, in contrast to the second experiment, by Peter Richter, which used a random walk in the space of simple cubic connected bipartite graphs. Although the walk in Richter’s experiment was random, in that two randomly chosen edges were swapped, the eigenvalues of the graphs from two consecutive steps of the walk are shown here to be correlated. A walk in which the eigenvalues are uncorrelated is used here in a similar experiment. In addition, an experiment similar to Kevin Chang’s experiment is performed in which graphs are constructed using an algorithm that is proven to choose uniformly at random from the space of simple cubic connected bipartite graphs. The distributions of the eigenvalues, after being normalized to have mean zero and standard deviation one, appear to be stable but not symmetric, similar to the Tracy-Widom distributions. The mean and standard deviation appear to approach 2 √ 2 and zero, respectively, according to power laws with the mean approachig quicker than the standard deviation. ∗E-mail: [email protected]