bottleneck constants of finite graphs

In this note we consider an example to test three lower bounds presented in Diaconis– Stroock [4] (one by Diaconis and Stroock (DS), another by Sinclair (S) [9], and the third by Jerrum–Sinclair (JS) [8]) of the first positive eigenvalue of a finite graph. The lower bounds are quite sharp in many of the examples studied in [4], and seem to measure the bottlenecks or complexity of the graphs studied. Our example here is quite simple, and is motivated by differential geometric considerations — the Cheeger–Calabi dumbbell [3] and the lower bounds for the bottom of the spectrum of a Riemann surface of finite volume [6, 7] — from which we construct a family of examples in which we can make all the above estimates as poor as we like (the precise sense to be explicated below). The example is quite straightforward. One considers a graph all of whose points have one edge connecting each point to the other. Distinguish two points of the graph, take another copy of the graph with the same distinguished two points, connect the graphs with one edge connecting each of the two distinguished points to its copy in the second graph. Now fiddle with distinct weights for these two edges. The details follow. Before proceeding, however, we note that these examples show that the upper bound for λ1 in terms of the discrete Cheeger constant is sharp.