A matrix for counting paths in acyclic colored digraphs

TL;DR

This paper introduces a matrix associated with a k-colored acyclic digraph whose determinant counts the number of paths, providing a novel algebraic tool for path enumeration in such graphs.

Contribution

The paper presents a new matrix construction for k-colored acyclic digraphs that directly encodes path counts through its determinant, offering a novel algebraic approach.

Findings

The determinant of the matrix equals the number of paths in the digraph.

The matrix construction applies specifically to acyclic, k-colored digraphs.

This method simplifies path enumeration in complex directed graphs.

Abstract

In this paper, we introduce a matrix $A_{\Gamma_{k, R}}$ associate with a $k$-colored acyclic digraph $\Gamma_{k, R}$ such that $\text{det}(A_{\Gamma_{k, R}})$ enumerates the paths in the digraph $\Gamma_{k, R}.$