An Adjacency Matrix Perspective of Talented Monoids and Leavitt Path Algebras

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.