Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Matrix-Forest Theorems   The Laplacian matrix of a graph G is L(G)=D(G)-A(G), where A(G) is the
adjacency matrix and D(G) is the diagonal matrix of vertex degrees. According
to the Matrix-Tree Theorem, the number of spanning trees in G is equal to any
cofactor of an entry of L(G). A rooted forest is a union of disjoint rooted
trees. We consider the matrix W(G)=I+L(G) and prove that the (i,j)-cofactor of
W(G) is equal to the number of spanning rooted forests of G, in which the
vertices i and j belong to the same tree rooted at i. The determinant of W(G)
equals the total number of spanning rooted forests, therefore the (i,j)-entry
of the matrix W^{-1}(G) can be considered as a measure of relative
''forest-accessibility'' of the vertex i from j (or j from i). These results
follow from somewhat more general theorems we prove, which concern weighted
multigraphs. The analogous theorems for (multi)digraphs are established. These
results provide a graph-theoretic interpretation for the adjugate to the
Laplacian characteristic matrix.

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The categories of this paper are: hep-lat
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The categories of this paper are: alg-geom
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The categories of this paper are: econ
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The categories of this paper are: plasm-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: dg-ga
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The categories of this paper are: eess
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The categories of this paper are: chem-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: ao-sci
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The categories of this paper are: q-alg
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The categories of this paper are: adap-org
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The categories of this paper are: hep-ph
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The categories of this paper are: q-bio
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The categories of this paper are: chao-dyn
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The categories of this paper are: nlin