  Main results of the paper:
  (1) For any finite metric space $M$ the Lipschitz free space on $M$ contains
a large well-complemented subspace which is close to $\ell_1^n$.
  (2) Lipschitz free spaces on large classes of recursively defined sequences
of graphs are not uniformly isomorphic to $\ell_1^n$ of the corresponding
dimensions. These classes contain well-known families of diamond graphs and
Laakso graphs.
  Interesting features of our approach are: (a) We consider averages over
groups of cycle-preserving bijections of graphs which are not necessarily graph
automorphisms; (b) In the case of such recursive families of graphs as Laakso
graphs we use the well-known approach of Gr\"unbaum (1960) and Rudin (1962) for
estimating projection constants in the case where invariant projections are not
unique.
