[Paper Review] Acyclic Digraphs and Eigenvalues of (0,1)-Matrices

Brendan D. McKay, Frédérique Oggier+4 more|arXiv (Cornell University)|Oct 28, 2003

Graph theory and applications4 references21 citations

TL;DR

This paper proves Weisstein's conjecture that the number of acyclic directed graphs with $n$ labeled vertices equals the number of $n \times n$ $(0,1)$-matrices with all positive real eigenvalues. The proof establishes a bijection via the transformation $B = I + A$, where $A$ is the adjacency matrix of an acyclic digraph, showing that such matrices have all eigenvalues equal to 1, and conversely, any $(0,1)$-matrix with positive real eigenvalues must be of this form.

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.

Motivation & Objective

• To resolve Weisstein's conjecture that the count of acyclic digraphs with $n$ labeled vertices matches the count of $n \times n$ $(0,1)$-matrices with all positive real eigenvalues.
• To establish a structural and spectral correspondence between acyclic digraphs and $(0,1)$-matrices with positive eigenvalues.
• To characterize the eigenvalue behavior of $(0,1)$-matrices derived from directed graphs, particularly focusing on positivity and reality of eigenvalues.
• To explore the implications of this correspondence for matrix theory, graph theory, and combinatorics, including symmetry and equivalence classes.

Proposed method

• Define $A$ as the adjacency matrix of an acyclic digraph $G$, ensuring $A$ has zero diagonal (no loops), so $B = I + A$ is a $(0,1)$-matrix with 1s on the diagonal.
• Use topological sorting to relabel vertices so $A$ becomes strictly upper triangular, making $B = I + A$ upper triangular with 1s on the diagonal, hence all eigenvalues of $B$ are 1.
• Apply the arithmetic-geometric mean inequality to show that if all eigenvalues of a $(0,1)$-matrix $B$ are positive reals, then they must all be equal to 1.
• Use the trace condition $\mathrm{Trace}(B^k) = n$ for all $k$ to show that all closed walks of length $k$ in the digraph of $B$ must be loops, implying no non-loop edges exist.
• Define $A = B - I$, so $A$ is a $(0,1)$-matrix with zero diagonal and no cycles in its associated digraph, proving $A$ corresponds to an acyclic digraph.
• Establish equivalence classes under permutation similarity, showing that the number of such matrices up to permutation is equal to the number of acyclic digraphs on $n$ unlabeled vertices.

Experimental results

Research questions

• RQ1 Are the sequences counting acyclic digraphs with $n$ labeled vertices and $(0,1)$-matrices with positive real eigenvalues identical?
• RQ2 What spectral properties characterize $(0,1)$-matrices that arise from acyclic digraphs via $B = I + A$?
• RQ3 Can every $(0,1)$-matrix with positive real eigenvalues be shown to be of the form $I + A$ for some acyclic digraph’s adjacency matrix $A$?
• RQ4 What constraints do the trace and determinant of a $(0,1)$-matrix impose on its eigenvalues when they are required to be positive reals?
• RQ5 How does the structure of the digraph (e.g., presence of cycles) affect the nature of the eigenvalues of its adjacency matrix?

Key findings

• The number of acyclic digraphs with $n$ labeled vertices is exactly equal to the number of $n \times n$ $(0,1)$-matrices with all positive real eigenvalues, confirming Weisstein’s conjecture.
• Any $(0,1)$-matrix with all positive real eigenvalues must have all eigenvalues equal to 1, and its diagonal entries must all be 1.
• The only symmetric $(0,1)$-matrix with positive eigenvalues is the identity matrix.
• A matrix $B$ with integer entries and $\mathrm{Trace}(B) \leq n$ has all eigenvalues real and positive if and only if $B = I + N$ for some nilpotent matrix $N$.
• If a digraph contains a cycle, its adjacency matrix has at least one eigenvalue that is zero, negative, or strictly complex; if the shortest cycle has length at least 3, there is a strictly complex eigenvalue.
• The number of equivalence classes of $n \times n$ $(0,1)$-matrices with positive eigenvalues under permutation similarity equals the number of acyclic digraphs on $n$ unlabeled vertices, corresponding to OEIS sequence A003087.