  We investigate the bottom of the spectra of infinite quantum graphs, i.e.,
Laplace operators on metric graphs having infinitely many edges and vertices.
We introduce a new definition of the isoperimetric constant for quantum graphs
and then prove the Cheeger-type estimate. Our definition of the isoperimetric
constant is purely combinatorial and thus it establishes connections with the
combinatorial isoperimetric constant, one of the central objects in spectral
graph theory and in the theory of simple random walks on graphs. The latter
enables us to prove a number of criteria for quantum graphs to be uniformly
positive or to have purely discrete spectrum. We demonstrate our findings by
considering trees, antitrees and Cayley graphs of finitely generated groups.
