  In the past decade, significant progress has been made to generalize
classical tools from Fourier analysis to analyze and process signals defined on
networks. In this paper, we propose a new framework for constructing Gabor-type
frames for signals on graphs. Our approach uses general and flexible families
of linear operators acting as translations. Compared to previous work in the
literature, our methods yield the sharp bounds for the associated frames, in a
broad setting that generalizes several existing constructions. We also examine
how Gabor-type frames behave for signals defined on Cayley graphs by exploiting
the representation theory of the underlying group. We explore how natural
classes of translations can be constructed for Cayley graphs, and how the
choice of an eigenbasis can significantly impact the properties of the
resulting translation operators and frames on the graph.
