Two Basic Results Concerning Random Walks on Graphs

This article describes my investigation into several basic problems regarding random walks on graphs. On several occasions, I asked myself questions which my intuition failed to answer. I guessed at an answer, and spent some time in a fruitless attempt at proving that it was correct. Out of frustration I turned to computer simulations, only to discover that my guesses were faulty. Once I had the correct answer, I was able to supply the proofs. As every mathematician knows, it is much easier to solve a problem when you know the right answer ahead of time. This presentation is deliberately informal, as it represents the record of an actual investigation that took place, rather than a crafted paper. In fact, the notebook that I used to run my experiments has become the paper, with explanatory text added and unnecessary debris removed.