  The main aim of the paper is Fredholm properties of a class of bounded linear
operators acting on weighted Lebesgue spaces on an infinite metric graph
$\Gamma$ which is periodic with respect to the action of the group ${\mathbb
Z}^n$. The operators under consideration are distinguished by their local
behavior: they act as (Fourier) pseudodifferential operators in the class
$OPS^0$ on every open edge of the graph, and they can be represented as a
matrix Mellin pseudodifferential operator on a neighborhood of every vertex of
$\Gamma$. We apply these results to study the Fredholm property of a class of
singular integral operators and of certain locally compact operators on graphs.
