Abstract
The Möbius coinvariant μ⊥(G) of a graph G is defined to be the Möbius invariant of the dual of the cycle matroid of G. This invariant is known to equal the rank of the reduced homology of the cycle matroid complex of G. For a complete graph Km+1, W. Kook gave an interpretation of μ⊥(Km+1) as the number of edge-rooted forests in Km. In this paper, we obtain a new combinatorial interpretation of μ⊥(Km+1,n+1) as the number of B-edge-rooted forests in Km,n, which is a bipartite analogue of the previous result.Based on these interpretations, we will give new bijective proofs of the formulas for μ⊥(Km+1) and μ⊥(Km+1,n+1) given by I. Novik, A. Postnikov, and B. Sturmfels in terms of the Hermite polynomials. In addition, we will construct a homology basis for the cycle matroid complex of Km+1,n+1 indexed by the B-edge-rooted forests. Also we will discuss the Möbius coinvariant of bi-coned graphs which generalize complete bipartite graphs.
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