Multi-iterative techniques of Multigrid type for solving large linear systems with structure of graph

We consider multi-iterative techniques of multigrid type for the numerical solution of large linear systems with (weighted) structure of graph. We combine proper coarser-grid operators with auxiliary techniques. We show that the most effective smoothers have to be of Krylov type with spanning tree preconditioners, while the projectors have to be designed for maintaining as much as possible the structure of graph matrix at the inner levels. Some necessary and sufficient conditions are proved; in this framework it is possible to explain why the classical projectors inherited from differential equations are good in the differential context and why they behave unsatisfactorily for unstructured graphs. Several numerical experiments have been conducted showing that our approach is effective even in very difficult cases where the known approaches are rather slow.