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09/05/2023
The Three Tree Theorem
We prove that every 2-sphere graph different from a prism can be vertex ...
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05/22/2023
Cohomology of open sets
If G is a finite abstract simplicial complex and K is a subcomplex of G ...
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04/03/2023
Spectral monotonicity of the Hodge Laplacian
If K,G are finite abstract simplicial complexes and K is a subcomplex of...
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02/06/2023
Characteristic Topological Invariants
The higher characteristics w_m(G) for a finite abstract simplicial compl...
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01/13/2023
The Sphere Formula
The sphere formula states that in an arbitrary finite abstract simplicia...
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05/26/2022
The Babylonian Graph
The Babylonian graph B has the positive integers as vertices and connect...
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05/23/2022
The Tree-Forest Ratio
The number of rooted spanning forests divided by the number of spanning ...
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05/23/2022
Eigenvalue bounds of the Kirchhoff Laplacian
We prove that each eigenvalue l(k) of the Kirchhoff Laplacian K of a gra...
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01/24/2022
Analytic torsion for graphs
Analytic torsion is a functional on graphs which only needs linear algeb...
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08/11/2021
Shannon capacity, Chess, DNA and Umbrellas
A vexing open problem in information theory is to find the Shannon capac...
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06/18/2021
Remarks about the Arithmetic of Graphs
The arithmetic of N, Z, Q, R can be extended to a graph arithmetic where...
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01/18/2021
Graph complements of circular graphs
Graph complements G(n) of cyclic graphs are circulant, vertex-transitive...
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12/14/2020
Complexes, Graphs, Homotopy, Products and Shannon Capacity
A finite abstract simplicial complex G defines the Barycentric refinemen...
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10/19/2020
Green functions of Energized complexes
If h is a ring-valued function on a simplicial complex G we can define t...
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12/24/2019
Constant index expectation curvature for graphs or Riemannian manifolds
An integral geometric curvature is defined as the index expectation K(x)...
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12/02/2019
More on Poincare-Hopf and Gauss-Bonnet
We illustrate connections between differential geometry on finite simple...
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11/11/2019
Poincare Hopf for vector fields on graphs
We generalize the Poincare-Hopf theorem sum_v i(v) = X(G) to vector fiel...
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10/07/2019
A simple sphere theorem for graphs
A finite simple graph G is declared to have positive curvature if every ...
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08/19/2019
Energized simplicial complexes
For a simplicial complex with n sets, let W^-(x) be the set of sets in G...
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07/07/2019
The energy of a simplicial complex
A finite abstract simplicial complex G defines a matrix L, where L(x,y)=...
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06/15/2019
A parametrized Poincare-Hopf Theorem and Clique Cardinalities of graphs
Given a locally injective real function g on the vertex set V of a finit...
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05/31/2019
More on Numbers and Graphs
In this note we revisit a "ring of graphs" Q in which the set of finite ...
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05/13/2019
Dehn-Sommerville from Gauss-Bonnet
We give a zero curvature proof of Dehn-Sommerville for finite simple gra...
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05/06/2019
The average simplex cardinality of a finite abstract simplicial complex
We study the average simplex cardinality Dim^+(G) = sum_x |x|/(|G|+1) of...
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03/25/2019
A Reeb sphere theorem in graph theory
We prove a Reeb sphere theorem for finite simple graphs. The result brid...
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11/26/2018
Cartan's Magic Formula for Simplicial Complexes
Cartan's magic formula L_X = i_X d + d i_X = (d+i_X)^2=D_X^2 relates the...
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08/22/2018
Eulerian edge refinements, geodesics, billiards and sphere coloring
A finite simple graph is called a 2-graph if all of its unit spheres S(x...
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06/17/2018
Combinatorial manifolds are Hamiltonian
Extending a theorem of Whitney of 1931 we prove that all connected d-gra...
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03/19/2018
The Cohomology for Wu Characteristics
While Euler characteristic X(G)=sum_x w(x) super counts simplices, Wu ch...
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03/05/2018
The hydrogen identity for Laplacians
For any finite simple graph G, the hydrogen identity H=L-L^(-1) holds, w...
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02/05/2018
Listening to the cohomology of graphs
We prove that the spectrum of the Kirchhoff Laplacian H0 of a finite sim...
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01/15/2018
An Elementary Dyadic Riemann Hypothesis
The connection zeta function of a finite abstract simplicial complex G i...
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11/27/2017
One can hear the Euler characteristic of a simplicial complex
We prove that that the number p of positive eigenvalues of the connectio...
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08/21/2017
On Atiyah-Singer and Atiyah-Bott for finite abstract simplicial complexes
A linear or multi-linear valuation on a finite abstract simplicial compl...
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08/05/2017
The strong ring of simplicial complexes
We define a ring R of geometric objects G generated by finite abstract s...
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05/30/2017
On a Dehn-Sommerville functional for simplicial complexes
Assume G is a finite abstract simplicial complex with f-vector (v0,v1, ....
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08/18/2007
A structure from motion inequality
We state an elementary inequality for the structure from motion problem ...
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08/17/2007
Space and camera path reconstruction for omni-directional vision
In this paper, we address the inverse problem of reconstructing a scene ...
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08/17/2007