Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Gauss-Bonnet for multi-linear valuations   We prove Gauss-Bonnet and Poincare-Hopf formulas for multi-linear valuations
on finite simple graphs G=(V,E) and answer affirmatively a conjecture of
Gruenbaum from 1970 by constructing higher order Dehn-Sommerville valuations
which vanish for all d-graphs without boundary. An example of a quadratic
valuation is the Wu characteristic w(G) which sums (-1)^(dim(x)+dim(y)) over
all intersecting pairs of complete subgraphs x,y of a G. More generally, an
intersection number w(A,B) sums (-1)^(dim(x)+dim(y)) over pairs x,y, where x is
in A and y is in B and x,y intersect. w(G) is a quadratic Euler characteristic
X(G), where X sums (-1)^dim(x) over all complete subgraphs x of G. We prove
that w is multiplicative, like Euler characteristic: w(G x H) = w(G) w(H) for
any two graphs and that w is invariant under Barycentric refinements. We
construct a curvature K satisfying Gauss-Bonnet w(G) = sum K(a). We also prove
w(G) = X(G)-X(dG) for Euler characteristic X which holds for any d-graph G with
boundary dG. We also show higher order Poincare-Hopf formulas: there is for
every multi-linear valuation X and function f an index i(a) such that sum
i(a)=X(G). For d-graphs G and X=w it agrees with the Euler curvature. For the
vanishing multi-valuations which were conjectured to exist, like for the
quadratic valuation X(G) = (V X) Y with X=(1,-1,1,-1,1),Y=(0,-2,3,-4,5) on
4-graphs, discrete 4 manifolds, where V_{ij}(G) is the f-matrix counting the
number of i and j-simplices in G intersecting, the curvature is constant zero.
For all graphs and multi-linear Dehn-Sommerville relations, the
Dehn-Sommerville curvature K(v) at a vertex is a Dehn-Sommerville valuation on
the unit sphere S(v). We show X V(G) Y = v(G) Y for any linear valuation Y of a
d-graph G with f-vector v(G). This provides examples for the Gruenbaum
conjecture.

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The categories of this paper are: adap-org
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The categories of this paper are: bayes-an
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The categories of this paper are: mtrl-th
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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: patt-sol
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-ph
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The categories of this paper are: nlin
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The categories of this paper are: quant-ph
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The categories of this paper are: stat
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The categories of this paper are: solv-int
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The categories of this paper are: funct-an
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The categories of this paper are: hep-ex
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The categories of this paper are: math-ph
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The categories of this paper are: supr-con