Uniform Mixing on Cayley Graphs

This thesis investigates uniform mixing on Cayley graphs over Z3. We apply Mullin’s results on Hamming quotients, and characterize the 2(d+2)-regular connected Cayley graphs over Z3 that admit uniform mixing at time 2π/9. We generalize Chan’s construction on the Hamming scheme H(d, 2) to the schemeH(d, 3), and find some distance graphs of the Hamming graph H(d, 3) that admit uniform mixing at time 2π/3 for any k ≥ 2. To restrict the mixing time, we derive a sufficient and necessary condition for uniform mixing to occur on a Cayley graph over Z3 at a given time. Using this, we obtain three results. First, we give a lower bound of the valency of a Cayley graph over Z3 that could admit uniform mixing at some time. Next, we prove that no Hamming quotient H(d, 3)/〈1〉 admits uniform mixing at time earlier than 2π/9. Finally, we explore the connected Cayley graphs over Z3 with connected complements, and show that five complementary graphs admit uniform mixing with earliest mixing time 2π/9.