  In this paper, we discuss the convergence of an Algebraic MultiGrid (AMG)
method for general symmetric positive-definite matrices. The method relies on
an aggregation algorithm, named \emph{coarsening based on compatible weighted
matching}, which exploits the interplay between the principle of compatible
relaxation and the maximum product matching in undirected weighted graphs. The
results are based on a general convergence analysis theory applied to the class
of AMG methods employing unsmoothed aggregation and identifying a quality
measure for the coarsening; similar quality measures were originally introduced
and applied to other methods as tools to obtain good quality aggregates leading
to optimal convergence for M-matrices. The analysis, as well as the coarsening
procedure, is purely algebraic and, in our case, allows an \emph{a posteriori}
evaluation of the quality of the aggregation procedure which we apply to
analyze the impact of approximate algorithms for matching computation and the
definition of graph edge weights. We also explore the connection between the
choice of the aggregates and the compatible relaxation convergence, confirming
the consistency between theories for designing coarsening procedures in purely
algebraic multigrid methods and the effectiveness of the coarsening based on
compatible weighted matching. We discuss various completely automatic
algorithmic approaches to obtain aggregates for which good convergence
properties are achieved on various test cases.
