Enumerative combinatorics
Categories of acyclic graphs and automorphisms of free groups
Categories of acyclic graphs and automorphisms of free groups
A multicomplex of partially edge-rooted forests
European Journal of Combinatorics
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We define the @a-invariant of a finite graph G on n+1 vertices to be the alternating sum @a(G)@?f"n-f"n"-"1+...+(-1)^nf"0 where each f"i is the number of spanning forests in G with i edges. The cone G@^ on a graph G is obtained by adjoining a new vertex p and then joining each vertex of G to p by a single edge. The main result of this paper is a simple and elegant combinatorial interpretation of @a(G@^) as the cardinality of the set of all edge-rooted spanning forests in the base graph G. We apply this result to discuss several examples of @a(G@^) including the complete graphs.