Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Euler characteristic of an even-dimensional graph   We write the Euler characteristic X(G) of a four dimensional finite simple
geometric graph G=(V,E) in terms of the Euler characteristic X(G(w)) of
two-dimensional geometric subgraphs G(w). The Euler curvature K(x) of a four
dimensional graph satisfying the Gauss-Bonnet relation sum_x K(x) = X(G) can so
be rewritten as an average 1-E[K(x,f)]/2 over a collection two dimensional
"sectional graph curvatures" K(x,f) through x. Since scalar curvature, the
average of all these two dimensional curvatures through a point, is the
integrand of the Hilbert action, the integer 2-2 X(G) becomes an
integral-geometrically defined Hilbert action functional. The result has an
interpretation in the continuum for compact 4-manifolds M: the Euler curvature
K(x), the integrand in the classical Gauss-Bonnet-Chern theorem, can be seen as
an average over a probability space W of 1-K(x,w)/2 with curvatures K(x,w) of
compact 2-manifolds M(w). Also here, the Euler characteristic has an
interpretation of an exotic Hilbert action, in which sectional curvatures are
replaced by surface curvatures of integral geometrically defined random
two-dimensional sub-manifolds M(w) of M.

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The categories of this paper are: q-fin
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The categories of this paper are: supr-con
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The categories of this paper are: dg-ga
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The categories of this paper are: acc-phys
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The categories of this paper are: patt-sol
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The categories of this paper are: cond-mat
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The categories of this paper are: nucl-ex
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The categories of this paper are: math-ph
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The categories of this paper are: econ
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The categories of this paper are: adap-org
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The categories of this paper are: nlin
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The categories of this paper are: stat
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-th