Acyclic Digraphs and Eigenvalues of (0,1)-Matrices

TL;DR

This paper establishes a surprising equivalence between the count of acyclic directed graphs with labeled vertices and the number of (0,1)-matrices with positive real eigenvalues, revealing a deep combinatorial-matrix spectral connection.

Contribution

It introduces a novel correspondence between acyclic digraphs and (0,1)-matrices characterized by positive eigenvalues, bridging graph enumeration and matrix spectral properties.

Findings

Number of acyclic digraphs equals the number of (0,1)-matrices with positive eigenvalues

Provides a new combinatorial interpretation of matrix eigenvalues

Establishes a spectral characterization of acyclic digraphs

Abstract

We show that the number of acyclic directed graphs with n labeled vertices is equal to the number of n X n (0,1)-matrices whose eigenvalues are positive real numbers.