  This paper revisits the problem of multi-agent consensus from a graph signal
processing perspective. Describing a consensus protocol as a graph spectrum
filter, we present an effective new approach to the analysis and design of
consensus protocols in the graph spectrum domain for the uncertain networks,
which are difficult to handle by the existing time-domain methods. This novel
approach has led to the following new results in this paper: 1) New necessary
and sufficient conditions for both finite-time and asymptotic average consensus
of multi-agent systems. 2) Direct link between the consensus convergence rate
and the periodic consensus protocols. 3) Conversion of the fast consensus
problem to the problem of polynomial design of graph spectrum filter. 4) A
Lagrange polynomial interpolation method and a worst-case optimal interpolation
method for the design of periodic consensus protocols for the MASs on uncertain
graphs. 5) Explicit formulas for the convergence rate of the designed
protocols. Several numerical examples are given to demonstrate the validity,
effectiveness and advantages of these results.

Systems and Control