Random Seidel Switching on Graphs

We consider the random process arising from a sequence of random Seidel switching operations on \( n \) vertices. We show that this process can be interpreted as a random walk on a Cayley graph of an abelian group, and use spectral methods to show that the random process converges to a stationary distribution in \( O(n \log(n)) \) steps. We then consider two generalizations: we allow multiple states for each edge, and restrict the process to a fixed host graph \( H \). We then analyze the general case and obtain convergence results for any graph \( H \).