Journal of Algebraic Combinatorics 5 (1996), 5-11 1996 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. Summary: Journal of Algebraic Combinatorics 5 (1996), 5-11 © 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, l-walk, 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 Collections: Mathematics