An Adjacency Matrix Perspective of Talented Monoids and Leavitt Path Algebras

TL;DR

This paper explores the deep connections between Leavitt path algebras, talented monoids, and graph adjacency matrices, providing new methods to analyze graph properties and algebraic structures through matrix representations.

Contribution

It introduces a novel adjacency matrix approach to understand talented monoids and Leavitt path algebras, including classification, path counting, and acyclicity characterization.

Findings

Adjacency matrix generates the group action on talented monoid generators.

A formula for counting paths of a given length in Leavitt path algebras.

Characterization of acyclic graphs via adjacency matrix and algebraic structures.

Abstract

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.