A characteristic polynomial for rooted graphs and rooted digraphs

We consider the one-variable characteristic polynomial p(G; ) in two settings. When G is a rooted digraph, we show that this polynomial essentially counts the number of sinks in G. When G is a rooted graph, we give combinatorial interpretations of several coe/cients and the degree of p(G; ). In particular, |p(G; 0)| is the number of acyclic orientations of G, while the degree of p(G; ) gives the size of the minimum tree cover (every edge of G is adjacent to some edge of T ), and the leading coe/cient gives the number of such covers. Finally, we consider the class of rooted fans in detail; here p(G; ) shows cyclotomic behavior. c © 2001 Elsevier Science B.V. All rights reserved.