A Note on the Poincaré and Cheeger Inequalities for Simple Random Walk on a Connected Graph

In 1991, Persi Diaconis and Daniel Stroock obtained two canonical path bounds on the second largest eigenvalue for simple random walk on a connected graph, the Poincaré and Cheeger bounds, and they raised the question as to whether the Poincaré bound is always superior. In this paper, we present some background on these issues, provide an example where Cheeger beats Poincaré, establish some sufficient conditions on the canonical paths for the Poincaré bound to triumph, and show that there is always a choice of paths for which this happens. 1. Background and Notation Let G = G(X,E) be any simple, connected, undirected, and unweighted graph with edge set E and finite vertex set X. For each (x, y) ∈ X ×X with x 6= y, choose a unique oriented path γx,y from x to y. Let Γ denote this collection of canonical paths (also known as a routing) and define n = |X| , d = max x∈X deg(x), γ∗ = max γ∈Γ |γ| , and b = max − →e ∈ − → E |{γ ∈ Γ : γ 3 − →e }| . Note that in the definition of b, the bottleneck number of (G,Γ), the maximum is taken over directed edges and so is to be distinguished from the related concept of the edge-forwarding index of (G,Γ) (see [7]). We distinguish directed and undirected edges by adorning the former with an arrow, and if e is an undirected edge connecting vertices x and y, we write e = {x, y}. Throughout this paper, the term path refers to a sequence of vertices where successive terms are connected by an edge. Repeated vertices are allowed, but no edge may appear more than once. (Some authors refer to such a path as a trail.) If − →e is an edge from x to y, we write γ 3 e, γ 3 − →e if the vertex sequence defining γ contains x and y as consecutive terms, x preceding y in the latter case. The length of a path is defined as the number of edges it contains: |γ| = |{e ∈ E : γ 3 e}|. A simple random walk on G begins at some vertex x0 and then proceeds by moving to a neighboring vertex chosen uniformly at random. This defines a Markov process {Xk} with state space X, transition probabilities K(x, y) =  1 deg(x) , {x, y} ∈ E 0, otherwise , and stationary distribution π(x) = deg(x) 2 |E| . 2010 Mathematics Subject Classification. Primary: 05C81; Secondary: 15A42.