Lecture 1: Random Walks

In this note we study random walks and mixing time in weighted undirected non-regular graphs G. The notation that we define here will be used later to study high dimensional random walks on simplicial complexes. Given an undirected (weighted) graph G = (V,E) with weight function w : E → R≥0, let A ∈ RV×V ≥0 be the adjacency matrix of G. Consider a natural probability distribution on edges of G where for any edge {u, v}