  In this article we establish relationships between Leavitt path algebras,
talented monoids and the adjacency matrices of the underlying graphs. We show
that indeed the adjacency matrix generates in some sense the group action on
the generators of the talented monoid. With the help of this we deduce a form
of the aperiodicity index of a graph via the talented monoid. We classify
hereditary and saturated subsets via the adjacency matrix. Moreover we give a
formula to compute all paths of a given length in a Leavitt path algebra based
on the adjacency matrix. In addition we discuss the number of cycles in a
graph. In particular we give an equivalent characterization of acylic graphs
via the adjacency matrix, the talented monoid and the Leavitt path algebra.
