  Graph signals arise in various applications, ranging from sensor networks to
social media data. The high-dimensional nature of these signals implies that
they often need to be compressed in order to be stored and transmitted. The
common framework for graph signal compression is based on sampling, resulting
in a set of continuous-amplitude samples, which in turn have to be quantized
into a finite bit representation. In this work we study the joint design of
graph signal sampling along with quantization, for graph signal compression. We
focus on bandlimited graph signals, and show that the compression problem can
be represented as a task-based quantization setup, in which the task is to
recover the spectrum of the signal. Based on this equivalence, we propose a
joint design of the sampling and recovery mechanisms for a fixed quantization
mapping, and present an iterative algorithm for dividing the available bit
budget among the discretized samples. Furthermore, we show how the proposed
approach can be realized using graph filters combining elements corresponding
the neighbouring nodes of the graph, thus facilitating distributed
implementation at reduced complexity. Our numerical evaluations on both
synthetic and real world data shows that the joint sampling and quantization
method yields a compact finite bit representation of high-dimensional graph
signals, which allows reconstruction of the original signal with accuracy
within a small gap of that achievable with infinite resolution quantizers.
