Gauss-Bonnet for multi-linear valuations (1601.04533v1)

Abstract: 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.

• The paper extends the Gauss-Bonnet theorem to multi-linear valuations on finite simple graphs by defining curvature on vertices.
• It introduces Wu's higher-order characteristics, generalizing the Euler characteristic via quadratic and cubic forms linked to graph f-vectors.
• The work also establishes a Poincaré-Hopf formula for graphs, offering a method to compute combinatorial invariants through local vertex indices.

An Overview of Gauss-Bonnet for Multi-linear Valuations on Finite Simple Graphs

The paper "Gauss-Bonnet for multi-linear valuations" by Oliver Knill addresses mathematical concepts related to graph theory, particularly with regard to valuative properties of finite simple graphs. The work extends classical results in differential geometry, such as the Gauss-Bonnet theorem, into the field of discrete mathematics by leveraging graph-theoretic constructs.

Core Contributions

The main contribution of the paper is the extension of the Gauss-Bonnet theorem to multi-linear valuations on finite simple graphs. A valuation in this context is a function that assigns a real number to subgraphs under certain structural constraints. The paper explores multiple facets of valuation theory, illustrating how Euler-type characteristics can be represented via these valuations.

1. Multi-linear Valuations and Graph Extensions: The notion of a multi-linear valuation generalizes the well-known Euler characteristic to more complex objects than simple graphs, allowing interactions among tuples of subgraphs. This builds upon the classical valuation theory that historically dealt with convex bodies and polytopes.
2. Wu's Higher-order Characteristics: A central highlight is the focus on Wu's characteristic, denoted as $\omega(G)$, which generalizes the Euler characteristic to include interactions between different dimensional simplices in a graph. This leads to notions of higher-order Wu characteristics that can be represented by quadratic and cubic forms related to the $f$-vectors and $f$-matrices of graphs.
3. Gauss-Bonnet Theorems for Graphs: The Gauss-Bonnet theorem is extended to multi-linear valuations on graphs. It is shown that for any given multi-linear valuation, there exists a curvature function defined on vertices of the graph such that the valuation can be realized as the sum of this curvature function over all vertices.
4. Poincaré-Hopf Formula: The paper establishes a Poincaré-Hopf index formula for graphs extending the concept to the field of multi-linear valuations, thus providing a way to compute these characteristics via local index functions at vertices.

Theoretical Implications

The results have implications for both combinatorial topology and the algebraic aspects of graph theory:

• Cobordism Invariants: The paper discusses the implications for cobordism invariants, particularly focusing on closed manifolds and their boundary properties in the discrete setting. The Wu characteristic $\omega$ provides a combinatorial invariant associated with the graph boundary, similar to known continuum results.
• Barycentric Refinement: An important property claimed by the paper is the invariance of the Wu characteristic under barycentric refinement. This is significant as it suggests that these higher-order valuations retain their value under this form of graph transformation, hinting at potential applications in discrete geometry simplifications and topological equivalence operations.
• Product and Multiplicative Properties: The fact that Wu characteristics are invariant under graph Cartesian products extends the algebraic symmetry seen in traditional Euler characteristics to complex higher dimensions and sheds light on structural product properties inherent in graphs.
• Grünbaum's Question Affirmation: It answers the longstanding question posed by Grünbaum regarding the existence of multi-linear Dehn-Sommerville valuations that vanish for specific $d$-graphs, affirming the solution through a constructive method involving combinatorial curvatures.

Potential Applications and Future Directions

The paper lays a foundational theory that can potentially impact numerous areas, including discrete geometry, computational topology, and perhaps even graph-theoretical aspects of network analysis:

• Computational Aspects: With efficient computation playing a significant role in graph theoretic contexts, potential algorithms emerging from such theoretical embeddings might see applications in computer science, especially in large-scale network topology processing.
• Algebraic and Homotopy Theoretical Approaches: The intersection number interpretations provide new ways to look at algebraic intersections and path-related computations in networks that can be pivotal in pathways and connectivity analysis within computational domains.

Future research could explore extending these combinatorial structures to broader classes of topological spaces and manifolds, possibly uncovering further links between discrete graph invariants and classical topological invariants seen in algebraic topology or differential geometry. Additionally, continued exploration of computational efficiency and algorithmic development based on these theoretical insights could yield powerful tools for network analysis in varied applications ranging from biology to social networks.