  We study the free product of rooted graphs and its various decompositions
using quantum probabilistic methods. We show that the free product of rooted
graphs is canonically associated with free independence, which completes the
proof of the conjecture that there exists a product of rooted graphs
canonically associated with each notion of noncommutative independence which
arises in the axiomatic theory. Using the `orthogonal product' of rooted
graphs, we decompose the branches of the free product of rooted graphs as
`alternating orthogonal products'. This leads to alternating decompositions of
the free product itself, with the star product or the comb product followed by
orthogonal products. These decompositions correspond to the recently studied
decompositions of the free additive convolution of probability measures in
terms boolean and orthogonal convolutions, or monotone and orthogonal
convolutions. We also introduce a new type of `quantum decomposition' of the
free product of graphs, where the distance partition of the set of vertices is
taken with respect to a set of vertices instead of a single vertex. We show
that even in the case of widely studied graphs this yields new and more
complete information on their spectral properties, like spectral measures of a
(usually infinite) set of cyclic vectors under the action of the adjacency
matrix.
