SGT Toolbox – A Playground for Spectral Graph Theory

We present an interactive toolbox application we have developed to enable users to gain an intuitive understanding of the “meaning” of eigenvalues and eigenvectors of graph-related matrices. In addition, we explore spectral embeddings and the effects of graph altering algorithms on the eigenvalues of a graph. Furthermore, we examine Luby’s fast distributed randomized algorithm for finding a maximal independent set (MIS) of a network of nodes. We compare different graph classes with respect to how many rounds the algorithm takes to complete on them on average. We suggest that certain graph classes exhibit a logarithmic lower bound for this number of rounds. Lastly, we prove a logarithmic upper bound for zmax, the largest distance from any vertex to its nearest MIS vertex after one round of Luby’s algorithm, once for path graphs and then for all graphs in general.