Papers and updates
 [June 18, 2017] Paper: On the arithmetic of graphs ArXIV,
and local copy PDF) and Updates.
(There had been some blogging about this: Jan 15, 2017)
Jan 27, 2017,
June 10,2017 and
Jun 9, 2017.)
 [May 29, 2017] The paper "On a DehnSommerville functional for simplicial complexes"
is posted. There had been some blogging about this.
 [March 19, 2017] The paper Helmholtz free energy for finite abstract simplicial complexes [ARXIV]
is posted. [Local PDF]
 [February 12, 2017] The paper
"Sphere Topology and Invariants" is posted.
[Local PDF]
 [December 24, 2016]
ArXIV version">
On Fredholm determinants in topology
gives the proof of the unimodularity theorem as announced in October. [Local PDF]
 [October 18, 2016] Handout [PDF] for a Mathtable talk on
Bowen Lanford Zeta functions of graphs.
 [August 21, 2016]
Primes, Graphs and Cohomology
(local copy [PDF]):
counting is a Morse theoretical
process. It also provides a prototype of graphs for which all cohomology groups can be computed
and where Morse cohomology is equivalent to simplicial cohomology.
 [August 21, 2016]
Particles and Primes: primes in the two complete associative
division algebras C and H show some affinities with Leptons and Hadrons.
 [June 19, 2016] Something about primes,
Goldbach in Division algebras [ArXiv],
local copy [PDF]
A larger report on experiments:
[ArXiv]
local copy [PDF].
 [March 18, 2016]Interaction cohomology
Interaction cohomology [PDF]
A case study: like StiefelWhitney classes, interaction cohomology is able to
distinguish the cylinder from the Möbius strip.
The cohomology also admits the Lefschetz theorem.
 [March 8, 2016] Wu Characteristic Handout [PDF] for a Mathtable talk.
 January 17, 2016 GaussBonnet for multilinear valuations [ArXiv]
develops multilinear valuations on graphs. An example of a quadratic valuation was constructed by Wu 1959.
We prove that the Wu characteristic is multiplicative,
invariant under Barycentric refinements and that for dgraphs (discrete dmanifolds),
the formula w(G) = X(G) X(dG) holds, where dG is the boundary. After developing GaussBonnet
and PoincareHopf theorems for multilinear valuations, we prove the existence of multilinear
DehnSommerville invariants, settling a conjecture of Gruenbaum from 1970.
And here is a miniblog.
 [October 4, 2015 Barycentric characteristic numbers.
We give a proof that for dgraphs, the kth Barycentric characteristic number is zero if
k+d is even. This is a two page, math only, writeup, a detailed writeup will need some time.
October 6: The document has now references included.
 [September 20, 2015] A complete rewrite
of the universality result using the Barycentric operator A for which the eigenvectors of
A^{T} produce invariants for which Euler characteristic is an example
Updates are on the same miniblog.
 [August 23, 2015]
A Sard theorem for Graph Theory:
(See a miniblog with updates).
Sard works surprisingly well in a discrete setup. The topic also allows to
do a bit of quantum multivariable calculus like looking at level surfaces,
or doing Lagrange extremization. local copy.
 [August 9, 2015]
The graph spectrum of barycentric refinements [ArXiv]:
We look at the graph spectral
distribution of barycentric refinements G_{m} of a finite simple graph.
The spectrum converges to a distribution which only depends on the maximal dimension of a complete
subgraph. For graphs without triangles, the distribution is related to the smooth
equilibrium measure of the Julia set of the quadratic map z^{2} 2. In higher
dimensions d>1, the universal distributions are still unidentified but appears do not look smooth.
local copy,
Mathematica code and
Miniblog.
 [June 21, 2015]
The JordanBrouwer theorem for graphs [ArXiv]
We prove a general JordanBrouerSchoenflies separation theorem for knots of codimension one.
The proof is close to Jordan's, Brouwer's or Alexander's papers.
(local copy).
 [May 27, 2015]
Kuenneth formula in graph theory.
While procrastinating a bit on programming the geometric coloring algorithm,
we found a new product for finite graphs. Its pretty exciting as it allows to define
de Rham cohomology for general finite simple graphs graphs (only a good product allows
even to talk about de Rham cohomology for graphs), show a discrete de Rham theorem,
prove the Kuenneth formula rsp. EilenbergZilber theorem and prove that the dimension
is super additive dim(G x H) >= dim(G) + dim(H) like Hausdorff dimension in the continuum.
Products GxH have explicitly computable chromatic numbers, and if
G,H are geometric, then G x H is geometric and even Eulerian. The product allows to define
joins, homotopy, discrete manifolds or fibre bundles. (local copy).
Updates.
 [Jan 11,2015] "Graphs with Eulerian Unit spheres"
addresses questions like "what are lines and spheres" in graph theory. We define
dspheres inductively as homotopy spheres for which every unit sphere is a (d1) sphere.
To define lines in a graph, we need a unique geodesic flow. Because such a flow defines
a fixed point free involution on each unit sphere, we must restrict to a subclass of Eulerian
graphs. Such graphs with Eulerian unit spheres are the topic of this. Eulerian spheres are
exciting since if we could extend a general 0sphere to an Eulerian 3 sphere, it
would prove the four color theorem. The paper also gives a short independent classification
of all Platonic solids in ddimensions these are dspheres for which all unit spheres are
Platonic solids. (local copy)
 [Dec 21,2014] Coloring graphs using topology.
A geometric approach to graph coloring. We hope it could be a path towards
``seeing why the four color theorem is true". The idea is to embed the graph in a higher
dimensional graph and made 4 colorable by cutting it up. It works in examples but not yet
systematically. (local copy kept updated) and
updates and
some illustration.
 Oct 12, 2014: We look at various variational problems and especially
Characteristic Length and Clustering [ArXiv].
(local copy).
We see more indications that Euler characteristic is the most interesting functional and see
correlations between dimension and lengthcluster coefficient. A tiny mathematical lemma proven expresses clustering coefficient with
a relative characteristic length allowing to look at clustering and lengthcluster coefficient in general metric spaces.
 Oct 5, 2014: the short note Curvature from Graph Colorings (local copy)
extends the index expectation theorem but with graph colorings
a much smaller probability space.
 Jun 17, 2014: The Binet paper
is finally updated, actually substantially enhanced. We use now more natural
definitions and include Pseudo Pfaffians and also mention ChebotarevShamis,
since it is so nice. [See the mini update blog]
Local copy of the revision.
 Mar 23, 2014: "If Archimedes would have known functions ..."
ArXiv"ArXiv and
local copy [PDF] with
updates.
 Feb 8, 2014: "Classical mathematical structures within topological graph theory",
ArXiv, and
Local copy kept up to date [PDF], notes for
a AMS session in January.
and slides.
updates.
 Jan 12, 2014:
A notion of graph homeomorphism [ARXIV] and
local [PDF].
[Updates]
 Dec 15, 2013:The zeta function of circular graphs [ARXIV] and local [PDF]. [Updates]
 Dec 1, 2013: On quadratic orbital networks [ARXIV],
and local [PDF].
 Nov 25, 2013: Natural orbital networks [ARXIV], local file [PDF].
 Nov 17, 2013: Dynamically generated networks [ArXiv] local file [PDF].
See the project page.
 Counting rooted forests in a network.
The number of rooted spanning forests in a finite simple graph is det(1+L) where L is the combinatorial Laplacian.
ArXiv. [update blog]
 The Euler characteristic of an evendimensional graph. We argue that Euler
characteristic is an interesting functional because Euler curvature as an average of two dimensional
curvatures of random two dimensional geometric subgraphs.
ArXiv.
 Isospectral deformations of the Dirac operator (ArXiv). More details about the
integrable dynamical system in geometry.
 The Dirac operator of a graph [PDF], consists of notes to the talk
It is also on the [ArXiv]. The Slides [PDF],
[updates]
 CauchyBinet for PseudoDeterminants [PDF], [updates], ArXiv, Jun 1, 2013
 An integrable evolution equation in geometryArXiv, Jun 1, 2013
 The McKeanSinger Formula in Graph Theory [PDF], ArXiv, Jan 8, 2013
 The LusternikSchnirelmann theorem for graphs [PDF], ArXiv, Nov 4 (updated Nov 13), 2012 and updates.
 A Brouwer fixed point theorem for graph endomorphisms [PDF], ArXiv, June 4, 2012 and updates.
Fixed Point Theory and Applications.2013, 2013:85. DOI: 10.1186/16871812201385.
 An index formula for simple graphs [PDF], ArXiv May 2012 and updates.
 On index expectation and curvature for networks [PDF], ArXiv Feb 2012 and updates.
 The theorems of GreenStokes,GaussBonnet and PoincareHopf in Graph Theory[PDF]. ArXiv Jan 2012, and updates.
 A graph theoretical PoincareHopf Theorem [PDF]. ArXiv Jan 2012, updates
 On the Dimension and Euler characteristic of random graphs [PDF]. ArXiv Dec 2011 and Updates
 ChernGaussBonnet theorem for graphs [PDF], On ArXiv Nov 2011 and Updates
 A discrete GaussBonnet type theorem [PDF].On ArXiv [Sep 2010].Elemente der Mathematik, 67,1, pp144, 2012

Code
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