 
Summary: Journal of Algebraic Combinatorics 5 (1996), 511
© 1996 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.
Spectra of Some Interesting Combinatorial
Matrices Related to Oriented Spanning Trees
on a Directed Graph
CHRISTOS A. ATHANASIADIS cathan@math.mit.edu
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139
Received October 19, 1994; Revised December 27, 1994
Abstract. The Laplacian of a directed graph G isthe matrix L(G) = 0(G) A(G), where A(G)isthe adjacency
matrix of G and O(G) the diagonal matrix of vertex outdegrees. The eigenvalues of G are the eigenvalues of
A(G). Given a directed graph G we construct a derived directed graph D(G) whose vertices are the oriented
spanning trees of G. Using a counting argument, we describe the eigenvalues of D(G) and their multiplicities
in terms of the eigenvalues of the induced subgraphs and the Laplacian matrix of G. Finally we compute the
eigenvalues of D(G) for some specific directed graphs G. A recent conjecture of Propp for D(Hn) follows, where
Hn stands for the complete directed graph on n vertices without loops.
Keywords: oriented spanning tree, lwalk, eigenvalue
1. Introduction
Consider a directed graph G = (V, E) on a set of n vertices V, with multipleedges and
loops allowed. An oriented rooted spanning tree on G, or simply an oriented spanning
tree on G, is a subgraph T of G containing all n vertices of G and having a distinguished
