# News about this project

- August 27, 2020: Fundamental theorems now with 200 theorems.
- August 23, 2020: Division algebra valued energized simplicial complexes, local version [PDF]
- August 17, 2020: A blog entry about a new geometric phase phenomenon which we experiment with.
- August 6, 2020: after teaching a summer course, some blog entry.
- June 28, 2020: a bit more reflection on Positive curvature and physics and a current local version [PDF] (updated last 7/7/2020).
- June 19, 2020:
On a theorem of Grove and Searle [PDF] and Local version [PDF].
And some slides:
- May 24-May 31, 2020:
A Dehn type quantity for Riemannian manifolds (V2) and a
Local copy [PDF]. There is an entry
Chopping up Riemannian manifolds in
my quantum calculus blog. I also keep a mini blog roll on this project. And a few slides
- April 24, 2020: The
Energy article is up:
**O. knill, The energy of a simplicial complex, Linear Algebra and its Applications 600 (2020) 96-129**contains results obtained from 2016-2018. The main proof was presented here The paper is behind a paywall, but there is a share link from Elsevier, for a few days. - January 19, 2020: In Index expectation curvature for manifolds [ArXIV] local version [PDF]
- January 5, 2020: Integral geometric Hopf conjectures [ArXiv], local [PDF] Some slides (mostly about the history of the conjectures):
- December 23, 2019: Constant index expectation curvature for graphs or Riemannian manifolds [PDF], local [PDF]
- December 1, 2019: More on Poincare-Hopf and Gauss-Bonnet [ArXiv] updated local version [PDF]
- November 10, 2019: Poincaré-Hopf for vector fields on graphs [ArXiv] and local [PDF]
- October 26, 2019: updated Mathematica Demonstration .nb file for Gauss-Bonnet/Poincare-Hopf (2012). Runs in Mathematica 12 now.
- October 6, 2019: A simple sphere theorem for graphs [PDF] (The Mickey mouse theorem). local version [PDF].
- August 18, 2019:
Energized Simplicial Complexes
with local updated version.
Simplicial complexes can be equipped with divisor like functions. The unimodularity and energy
theorem generalizes to this. Some slides
- July 21, 2019: The counting matrix of a simplicial complex, (local [PDF]). About a Bosonic sibling of the connection Laplacian. The matrix is in SL(n,Z) and (very exciting) leads to a zeta function which always satisfies a functional equation.
- July 7, 2019: The energy of a simplicial complex local copy [PDF]. This is a new clean write-up containing the unimodularity theorem, energy theorem, hear Euler characteristic theorem, the Green-Star formula and the tensor representation theorem. See the page about this subproject.
- June 15, 2019:
Parametrized Poincare-Hopf theorem,
(local copy [PDF]). And some slides:
- May 30, 2019:
On Numbers and Graphs, and
local copy [PDF]
- May 12, 2019:
Dehn-Sommerville from Gauss-Bonnet,
and updated local copy [PDF].
- May 5, 2019:
The average simplex cardinality of a finite abstract simplicial complex.
Updated local copy [PDF].
- March 24, 2019: A Reeb sphere theorem in graph theory [ArXiv]. ( local version [PDF].
- December 18, 2018: A bit about discrete calculus which is calculus on graphs, a topic which appeared in Math 22a.
- November 25, 2018: Cartan's Magic Formula for Simplicial Complexes [PDF], Local [PDF]
- November 14, 2018: An interesting article of Santos, Raposo, Coutinho-Filho,Copielli,Stam and Douw about phase transitions of brain networks uses the Euler characteristic functional. The animation to the right is by those authors: .
- [August 21, 2018]Eulerian edge refinements, geodesics, billiards and sphere colorin and local [PDF].
- [August 4, 2018] A Sphere bouquet symphony (recorded in April 2018).
- [July 22, 2018]Some theorems in math (updated August 22) and local version.
- [June 18, 2018]Combinatorial manifolds are Hamiltonian. (Updated on a local copy [PDF])
- [April 22, 2018] The amazing world of simplicial complexes. (Updated on a local copy [PDF]) for an AMS meeting on April 22, 2018.
- [March 22, 2018] Some slides:
- [March 18, 2018] The Cohomology for Wu Characteristic local copy.
- [March 4, 2018] The Hydrogen identity for Laplacian (local copy).
- [February 6, 2018] Pre Euler Day Math table Polishing Euler's Gem [PDF], talk rehearsal and slides.
- [February 4, 2018] In Listening to the cohomology of graphs [ArXiv], a relation between the Hodge Laplacian and the connection Laplacian. Also the Green star formula is covered.
- [January 28, 2018] The Green star formula expresses the Green function g(x,y) in terms of the intersection of the stars of x and y.
- [January 14, 2018] The paper Elementary Dyadic Riemann zeta function is on the (ArXiv). Local PDF.
- [January 3, 2017] Start to blog about the Dyadic Riemann zeta function.
- [December 10, 2017] Perron Frobenius connection,
- [November 26, 2017] Paper One can hear the Euler characteristic of a simplicial complex [ArXiv]. Local PDF. Is there physics for connection Laplacian> and Quest for Green function formula.
- [November 12, 2017] More about the Green function values g(x,y) if one of the simplices is maximal.
- [November 4, 2017] No, one can not hear a complex: Examples of Isospectral complexes and something about Wen-Tsun Wu who passed away in 2017.
- [October 22,2017] Can one hear the sound of a complex?.
- [October 9, 2017] Getting ready for a talk about the
Energy theorem. A rehearsal:
- [August 20, 2017] A follow-up to the strong ring: the note "Atiyah-Singer and Atiyah-Bott for simplicial complexes", a first attempt in the discrete. (local copy [PDF]).
- [August 5, 2017] The paper The strong ring of simplicial complexes introduces a ring of geometric objects in which one can compute quantities like cohomologies faster. local copy.
- [June 18, 2017] The paper On the arithmetic of graphs (ArXiv) and [PDF] looks at rings of graphs or simplicial complexes. It deals with various rings of network with main focus on the Zykov Ring which is dual and isomorphic to the Sabidussi ring. The main result is that the Kuenneth formula holds for the strong Sabidussi product. (There had been some blogging about this: Jan 15, 2017 Jan 27, 2017, June 10,2017 and Jun 9, 2017.) Some Updates.
- [June 4, 2017] Some slides about the Hardy-Littlewood prime race:
- [May 29, 2017] The paper On a Dehn-Sommerville functional for simplicial complexes establishes a connection between the f-vector rsp. the generating function of a simplicial complex with the
trace of the Green function operator. It also relates the trace of the "hydrogen operator" L-L
^{-1}with the sum of the Euler characteristic of the unit spheres. There had been some blogging about this already. Local copy. - [April-May, 2017] Various pages were added on the quantum calculus blog: this includes also something about the mass gap and quantum plane which prompted me to speak some text to older unpublished slides. In order not to have them rot on my harddrive, they have been thrown to youtube now:
- [March 25, 2017} For upcoming technology demo, a experiment with sigma library.
- [March 19, 2017] The paper Helmholtz free energy for finite abstract simplicial complexes [ArXiv] uses a new Gauss-Bonnet formula to prove that the internal energy of a complex, the total sum over all Green function entries is Euler characteristic. Taking the energy analogy seriously, we add entropy S(p) to the internal energy U(p) of a probability measure p and minimize free energy F=U-T S. We observe catastrophes, discontinuous changes of the free energy in dependence of T. local copy. Apropos Entropy: clip from Movie "arrival".
- [February 12, 2017] The paper
Sphere Topology
and Invariants [ArXiv] proves that the Green function values are
a combinatorial invariant of simplicial complexes.
See
Quantum Calculus Blog.
Here are Some slides.
[Local PDF]
- [January 14, 2016] While finishing writing down the proof that Green functions form a topological invariant: Something about arithmetic in networks. The join operation plays an important role in the proof but it leads to an interesting quest: is there a fundamental theorem of graph arithmetic?
- [December 26, 2016]
ArXIV version and some slides:

- [December 24, 2016] On Fredholm determinants in topology gives the proof of the unimodularity theorem as announced in October.
- [December 1,2016] About Euler and Fredholm illustrating the unimodularity theorem using prime and GCD graphs introduced here.
- [November 24, 2016] A discussion written over thanksgiving about the unimodularity theorem status on Quantum calculus. The result is now more general and the proof more transparent based on a generalized multiplicative Poincaré-Hopf lemma assuring that the Fredholm characteristic is the product of Euler-Poincare-indices φ(G) = ∏
_{x}ω(x), similarly as Euler characteristic is the sum χ(G) = ∑_{x}ω(x). The Fredholm determinant is a multiplicative analogue of Euler characteristic. - [October 18, 2016] Handout [PDF] for a Mathtable talk on Bowen Lanford Zeta functions of graphs. The proof comes soon.
- [August 28, 2016] Some updates of the Morse theoretical genesis of numbers.
- [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. Some updates (miniblog).
- [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] Got a bit distracted by primes, for which there is also some graph theory. About Goldbach in division algebras: ArXiv, local copy [PDF] And a larger report with experiments in number theory, local copy [PDF].
- [May 26, 2016] Keiji Miura shared a movie showing an application of Poincaré-Hopf for touch screen devices. Pretty cool.
- [March 18, 2016]
**Interaction cohomology [PDF]**is a case study: like Stiefel-Whitney classes, interaction cohomology is able to distinguish the cylinder from the Möbius strip. The cohomology also admits the Lefschetz fixed point theorem. More on the miniblog. - [March 8, 2016]
**Wu Characteristic**Handout for Mathtable. - [January 23, 2016],
Some Slides about Wu characteristic.
- [January 17, 2016] Gauss-Bonnet for multi-linear valuations [ArXiv] develops multi-linear 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 d-graphs (discrete d-manifolds), the formula w(G) = X(G) -X(dG) holds, where dG is the boundary. After developing Gauss-Bonnet and Poincare-Hopf theorems for multilinear valuations, we prove the existence of multi-linear Dehn-Sommerville invariants, settling a conjecture of Gruenbaum from 1970. Local version [PDF]. And here is a miniblog.
- [October 13, 2015] A rehearsal for a seminar.
- [October 4, 2015] Wave evolution animation (we happen to teach a lecture on PDEs).
- [October 4, 2015] Barycentric characteristic numbers. We outline a proof that for d-graphs, the k-th Barycentric characteristic number is zero if k+d is even. This is a two page, math only writeup. 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 as described on the same miniblog. local copy. - [September 7, 2015] Some Chladni figures (nodal surfaces) in the case of Barycentric refinements of the triangle G in the case m=1,2 and m=3, a rounded discrete square, or larger and even larger square. A cylinder. More updates in the miniblog.
- [August 31, 2015] For a lecture of the course MathE320 in the extension school, a worksheet on Barycentric refinement.
- [August 26, 2015] Some slides about level surfaces of graphs.
- [August 23, 2015] A Sard theorem for Graph Theory: , with a miniblog for updates. How to we define level surfaces, discrete algebraic sets in graph theory? How large is the set of critical values? What is the second derivative test or how does one do Lagrange extremization in a network? Things go pretty much in the same way as in the continuum, but there are some twists. Local copy.
- [August 9, 2015] The graph spectrum of barycentric refinements:
See a miniblog with code and updates.
The graph spectra of Barycentric refinements G
_{m}of a finite simple graph show a remarkable universality: the graph spectra converge 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 dimension, the universal distributions are unidentified, but appears to be non-smooth with discrete or singular continuous components. Local copy. - [July 25, 2015] Some pictures of eigenfunctions: A gallery of pictures.
- [June 21, 2015] The Jordan-Brouwer theorem for graphs. The theme is well suited to test definitions and geometric notions. We prove a general Jordan-Brouer-Schoenflies separation theorem for knots of codimension one. The inductive definition of spheres (as we found out during this research put forward already by Alexander Evako) works very well. The proof would not have been possible without the tool of the graph product found earlier. (Local copy).
- [May 27, 2015] Kuenneth formula in graph theory. Slides [PDF], 2014. Having stumbled over new product for finite graphs, we introduce de Rham cohomology for general finite simple graphs graphs, show a discrete de Rham theorem, prove the Kuenneth formula and Eilenberg-Zilber 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, new notions of homotopy, discrete manifolds or fibre bundles. (local copy). Updates.
- [Feb 1, 2015] Some diary notes on the miniblog.
Theoretical obstacles seem gone. The problem is to implement the procedure and see it work.
- [Jan 11,2015] "Graphs with Eulerian Unit spheres" is written in the context of coloring problems but addresses the fundamental question "what are lines and spheres" in graph theory. We define d-spheres inductively as homotopy spheres for which each unit sphere is a (d-1) sphere. In order to define lines in a graph, we need a unique geodesic flow. Because such a flow requires a fixed point free involution on each unit sphere, we restrict to the subclass of Eulerian graphs. Such graphs with Eulerian unit spheres are the topic of this paper. Eulerian spheres are very exciting since if we could extend a general 2-sphere to an Eulerian 3-sphere, it would prove the 4-color theorem. The paper also gives a short independent classification of all Platonic solids in d-dimensions, which only uses Gauss-Bonnet-Chern: these are d-spheres for which all unit spheres are (d-1)-dimensional 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, containing updates), mini blog, some illustration.
- [Oct 12,2014] We looked at various variational problems and especially Characteristic Length and Clustering [ArXiv]. (local copy) and [update log] We see more indications that Euler characteristic is the most interesting functional and see correlations between dimension and length-cluster coefficient. A tiny mathematical lemma proven expresses clustering coefficient with a relative characteristic length allowing to look at clustering and length-cluster coefficient in general metric spaces.
- [Oct 5, 2014] Curvature from Graph Colorings and (Local copy). This is an extension of the Index expectation theorem but with a much smaller probability space: the set of colorings. It uses the remark that the discrete Poincaré-Hopf theorem holds also for locally injective functions aka colorings. Averaging over all colorings gives curvature. The topic mixes chromatic graph theory, integral geometry and is motivated by results known in differential geometry (like the Fary-Milnor theorem of 1950 which writes total curvature of a knot as an index expectation) and is elementary.
- Link to Binet article
- [July, 2014:] A summer HCRP project with Jenny Nitishinskaya on graph coloring problems
seen from a differential geometric and topological point of view.
A summary report [PDF].
It has been turbulent as we were in uncharted territory.
We conjecture that 4 colors suffice, as for any orientable surface like sphere or torus.
(note that we look for graphs where every unit sphere
is a cyclic graph, disqualifying the K
_{7}example, which is 6 dimensional for us). We also believe that chromatic number 5 is maximal for surfaces (attained only for nonorientable surfaces like the projective plane (an example found by Jenny)). [Dec. 2014/Jan. 2015 updates there are examples due to Fisk showing that the chromatic number 5 can occur for tori. It really seems to matter that the complement of a torus in a 3 sphere is not simply connected. There is evidence that the chromatic number of any surface is 3,4 or 5: any 2D surface S can be placed into a closed 4D unit ball B, so that the complement of S intersected with int(B) is simply connected. For orientable surfaces we can place S even into the 3-dimensional boundary of B. By coloring int(B)-S (the problem being to make the interior 5 colorable by subdivision or collaps), we could color S.] - [June 17, 2014:] Update on Binet paper Local copy. Substantially enhanced, including PseudoPaffians and Chebotarev-Shamis. [See the mini update blog]
- [Apr 3, 2014:] Orbital graphs with Sigma graphics library.
- [Mar 23, 2014:] "If Archimedes would have known functions ..." contains a Pecha-Kucha talk, a short summary of calculus on finite simple graph, a collection of calculus problems and some historical remarks. ArXiv and local copy [PDF] with updates.
- [February 8, 2014:] "Classical mathematical structures within topological graph theory", ArXiv, and Local copy [PDF] consists of some preparation notes for a talk at the a AMS session in January. See also the slides.
- [January 12, 2014:] A notion of graph homeomorphism., (local [PDF]) We find a notion of homeomorphism between finite simple graphs which preserves basic properties like connectivity, dimension, cohomology and homotopy type and which for triangle free graphs includes the standard notion of homeomorphism of graphs. The notion is inspired by pointless topology and Cech constructions. The fact that homeomorphisms with non-zero Lefschetz numbers have fixed open invariant sets, can be seen as a Kakutani fixed point theorem for finite simple graphs.
- [December 15, 2013:]
The zeta function of circular graphs [ARXIV]
(local [PDF].
The Riemann zeta function is the Dirac zeta function of the circle. We study the roots of the zeta function
of the circular graphs C
_{n}, which are entire functions. We prove that the roots converge to the axes Re(s)=1. This is equivalent that the roots of the Laplace zeta function of the circular graphs converge to the axes Re(s)=1/2. We also derive discrete Basel problem values like zeta(2)=(n^{2}-1)/12 or zeta(4) = (n^{2}-1)(n^{2}+11)/45 which lead in the limit to the classical Basel values zeta(2) = pi^{2}/6 or zeta(4)=pi^{4}/90 for the circle. [Updates: Dec 18: The Kubert connection with Milnor's results. Dec 24: the local maximum of the imaginary part in Figure 3 is at the height of the first root of the Riemann zeta function. - [December 1, 2013:] On quadratic orbital networks [ARXIV],
and local [PDF].
Some remarks in the case of quadratic orbital networks. Was written after finding a disconnected quadratic
network (Z
_{p},z^{2}+a,z^{2}+b,z^{2}+c) with prime p. The computer is since still looking for more. [Update January 22, 2014: Some slides] - [November 26, 2013:] Natural orbital networks [ARXIV], local file [PDF]. This is part of the project on dynamical graphs. It contains questions about the connectivity of orbital networks generated by polynomial maps.
- [November 17, 2013] Dynamically generated networks. Project page. Arxiv. This is a project started with Montasser Ghachem in September 2013. This paper shows some pictures and states some results related to elementary number theory. The project page shows some pictures, movies.
- [July 13, 2013] Counting rooted forests in a network.
We prove that the number of rooted spanning forests in a finite simple graph is det(1+L) where L is the combinatorial
Laplacian of the graph. Compare that with the tree theorem of Kirchhoff which tells that
the pseudo determinant Det(L) is the number of rooted spanning trees in a finite simple graph.
The result can also be interpreted as a voting count: assume that in a social network everybody can vote one of the
friends as "president". If the network forbids any cyclic nominations to prevent groups from tempering with the vote,
then the number of possible voting patterns is det(L+1).
See ArXiv.

July 18: We found that Cheboratev and Shamis have proven the forest theorem already. We are of course disappointed but also reassured. The paper is now upgraded to count colored trees. The linear algebra results are much stronger and give this too. The update will appear also on the ArXiv. update blog. - [July 13, 2013] The Euler characteristic of an even-dimensional graph. We argue that Euler characteristic is an interesting functional on four dimensional geometric graphs because Euler curvature as an average of two dimensional curvatures of random two dimensional geometric subgraphs. Since Euler curvature is conceptionally close to scalar curvature, which integrates to the Hilbert action, the Euler characteristic should be an interesting analogue. ArXiv.
- [June 24, 2013] A Isospectral deformation of the Dirac operator: More details about the
integrable system which deforms D=d+d
^{*}on a graph or manifold. This is the first writeup on this system. Its rough on the edges, chatty and repetitive and maybe even has a forbidding style, but details to most computations should be there. ArXiv. Source code to experiment with the system will be posted later. - [June 9, 2013] Some expanded notes [PDF] from a talk given on June 5 at an ILAS meeting. ArXiv and Slides [PDF]. The talk covered on some linear algebra related to the Dirac operator D of a graph and to demonstrate how natural this object is. The language of graphs is also a natural frame work in which one can see essential ideas of multi-variable calculus in arbitrary dimensions. Stokes theorem on graphs was covered in this talk in even less than 6 minutes 40 seconds.
- [May 31, 2013] A Cauchy-Binet theorem for Pseudo-Determinants [PDF],
ArXiv, Jun 1, 2013.
This paper generalizes the classical Cauchy-Binet theorem
for pseudo determinants and more: it gives an expression for the coefficients of the characteristic
polynomial of the matrix F
^{T}G in terms of products of minors of F and G, where F,G are arbitrary matrices of the same size. The proof is done using the exterior algebra. An update of June 10, 2013 includes Mathematica code. July 6: added that the main result implies an identity for usual determinants: for any two matrices F,G of the same shape det(1+F^{T}G) = sum_P det(F_P) det(G_P), where P runs over all possible minors, with 1 for the empty minor. See also the [ update log with Mathematica code to copy paste. ] August 6: article. - [May 31, 2013] An integrable evolution equation in geometry ,
[ArXiv, Jun 1, 2013].
A bit more back to the roots when working on integrable systems in grad school. It introduces a Noether symmetry by doing an isospectral deformation of the
Dirac operator D=d+d
^{*}on any compact Riemannian manifold or finite simple graph. It also deforms the exterior derivative d but the Laplacian L=D^{2}stays the same as does cohomology. Classical wave or heat evolution on the geometry are not affected neither. Besides the deformed D(t) = d(t) + d(t)^{*}+ b(t) the new exterior derivative defines a new Dirac operator C(t) = d(t) + d(t)^{*}which in the spirit of noncommutative geometry defines a new geometry on the manifold or graph. We prove that the geometry always expands, with a fast inflationary start - as in cosmology. The McKean-Singer supersymmetry relation still holds: the nonlinear unitary evolution U(t) - which naturally replaces the Dirac wave evolution - has the property that str(U(t))= chi(G) at all times. However, supersymmetry is not visible. At t=0, a fermion f and its partner Df are orthogonal at t=0. Already after a short time, the super partner D(t) f is so close to the fermionic subspace that it must be taken as a fermion. Supersymmetry is not broken, but invisible. This holds we take symmetries of quantum mechanics serious. An other feature of the system is that if we do not constrain the evolution to the real, a complex structure evolves. It is absent at t=0 and asymptotically for large t, but it is important in the early part of the evolution. We illustrate in the simplest case like the circle or the two point graph but have computer code which evolves any graph. - [January 6, 2013] The The McKean-Singer Formula in Graph Theory [PDF]
[ArXiv].
This paper deals with the Dirac operator D on general finite simple graphs G. It is a matrix
associated with G and contains geometric information. The square L=D
^{2}is a block matrix, where each block is the Laplacian on p-forms. The McKean-Singer formula telling that str(exp(-t L) is the Euler characteristic for all t reflects a symmetry. It has combinatorial consequences for counting paths in the simplex space. It also helped to construct graphs which are Dirac isospectral. The matrix is also valuable for doing computations in geometry. Already Poincare has used it in 1900. Today, one can with a dozen lines of computer algebra system code produce the cohomology groups for any graph. The Dirac operator also allows to to see the graph theoretical Gauss-Bonnet-Chern theorem as an example of a discrete index theorem. - [November 4, 2012] The Lusternik-Schnirelmann theorem for graphs
[ArXiv].
With Frank Josellis,
we prove cup(G) ≤ tcat(G) ≤ crit(G) for a general finite simple graph G where cup(G) is the cup length,
tcat(G) is the minimal number of in G contractible subgraphs covering G and crit(G) is the minimal number of
critical points an injective function can have on G.
This implies cup(G) ≤ cat(G) ≤ cri(G) for a general finite simple graph G,
where cat(G) is the minimum over all tcat(H) with H homotopic to G and cri(G) is the minimal
crit(H) for an graph H homotopic to G.
The Lusternik-Schnirelmann theorem links an
**algebraic invariant**(cup) with a**topological invariant**(cup) and an**analytic invariant**(cri). [Update Nov 13, 2012: The original cat was renamed topological category tcat(G) since it is - similarly than the geometric category gcat(G) - not yet a homotopy invariant. (While the Fox graph is an example with gcat(G)=3, tcat(G)=cat(G)=2, the dunce hat G is homotopic to a point and satisfies cat(G)=1 but tcat(G)=2 because it is not contractible). - [June 4, 2012:] A fixed point theorem for graphs [ArXiv, June 4, 2012] proves a general Lefschetz formula for graph endomorphisms, leading to fixed point results like a discrete Brouwer theorem which generalizes the edge theorem of Nowakowski-Rival. Unlike in the continuum, we have to look at simplices as the basic "points". With the right notion of "degree" of a simplex with respect to T, the proof is pretty close to Hopfs proof in the classical case, which essentially boils down to "circular graphs have Euler characteristic 0" and "fixed points have Euler characteristic 1" and "every attractor of an endomorphism is either a circular graph or fixed point". The paper also gives a formula for the zeta function of T which involves the signature and dimension of prime simplex orbits.
- [May 1, 2012:] A continuation shows that curvature K(x) is zero for odd dimensional geometric graphs. This is proven by showing that the symmetric index j(f,x) = [i(f,x) + i(-f,x)]/2 is constant zero for odd dimensional geometric graphs, a result which holds for odd dimensional Riemannian manifolds. In the discrete, we need to define level surfaces B(f,x) = { f=c } in unit spheres S(x). We show that each B(f,x) is a polytop which can be completed to become geometric. For general simple graphs, the symmetric index j(f,x) satisfies j(f,x) = [2-chi(S(x))-chi(B(x))]/2 (a formula which also holds in the manifold case). For odd dimensional graphs in particular, j(f,x) = -chi(B(f,x))/2 which is zero by Poincaré-Hopf and induction. Curvature K(x) as the expectation E[j(f,x)] over a probability space of scalar functions f is therefore zero too.
- [Feb 20, 2012:] Index expectation (ArXiv brings in some probability theory. It will be used to show that curvature for odd dimensional geometric graphs is constant zero: if we integrate over all Morse functions on a graph and average the indices, we get curvature. Since each individual index function adds up to Euler characteristic, simply taking expectation over all fields gives Gauss-Bonnet. While this does not simplify the proof of Gauss-Bonnet in the discrete, it most likely will simplify Gauss-Bonnet-Chern for Riemannian manifolds.
- [Jan 31, 2012:] Wolfram Demo of Dimension and Euler characteristic and Wolfram Demo of GaussBonnetPoincareHopf.
- [Jan 29, 2012:] An expository paper [PDF] which might be extended more in the future. It deals with Gauss-Bonnet, Poincare-Hopf and Green Stokes in a graph theoretical setting.
- [Jan 4, 2012:] A discrete analogue of Poincaré Hopf (ArXiv ) for a general simple graphs G. Computer experimentation were essential to try different approaches, starting with small dimensions and guided by the continuum to find the index which works for random graphs. Discretisation would have been difficult because the index is classically defined as the degree of a sphere map (needing algebraic topology to be understood properly) and the analogue of spheres in graph theory can be pretty arbitrary graphs. Even with a computer, it needed months of experimentation. Morse theory is relief also in the continuum.
- [Dec 19, 2011:] A paper (ArXiv) on the dimension and Euler characteristic of random graphs provides explicit formulas for the expectation of inductive dimension dim(G) or Euler characteristic X(G), which are considered random variables over Erdoes-Renyi probability spaces. Most results were found first experimentally using brute force computations before proving them. The paper belongs to the mathematics of complex networks. Dimension and Euler characteristic mixes in some geometry. [Update Jan 2018: this paper of 2013 mentions the expectation of Euler characteristic on page 16. Kahle obviously was unaware of my 2011 paper. I still think (2018) that the formula appeared first here.]
- [Nov 21 2011:] A paper on higher dimensional Gauss-Bonnet which fits the occasion of Chern's birthday of October 26, 1911, The result was obtained in the summer of 2009 but illustrating it with examples took time. One can define curvature K(x) which depends only on the unit sphere of a vertex x in a graph G=(V,E) such that the sum of K(x) over V is Euler characteristic X(G). To see this implemented in Mathematica visit the code page.
- [Jul 6, 2010] This project started in spring 2009. The subject is simple topology or discrete differential geometry initiated in this paper. The goal is to understand graphs on a geometric level and investigate discrete analogues of structures which are known in differential geometry. This website is up since July 2010.