The Mickey mouse theorem assures that a connected positive curvature graph of positive dimension is a sphere.
The simplest construct in mathematics is probably a finite set of sets. Unlike a simple set alone, it has natural algebraic, geometric, analytic and order structures built in. In the finite case, there is lot of overlap but still, there is a rich variety of structure. Algebraic structure An example of a an algebraic structure is to have the symmetric … ….
If a set of set is equipped with an energy function, one can define integer matrices for which the determinant, the eigenvalue signs are known. For constant energy the matrix is conjugated to its inverse and defines two isospectral multi-graphs.
The counting matrix of a simplicial complex has determinant 1 and is isospectral to its inverse. The sum of the matrix entries of the inverse is the number of elements in the complex.
The f-function of a graph minus 1 is the sum of the antiderivatives of the f-function anti-derivatives evaluated on the unit spheres.
Dehn-Sommerville relations are a symmetry for a class of geometries which are of Euclidean nature.
Some update about recent activities: a new calculus course, the Cartan magic formula and some programming about the coloring algorithm.
We prove that any discrete surface has an Eulerian edge refinement. For a 2-disk, an Eulerian edge refinement is possible if and only if the boundary length is divisible by 3
We prove that connected combinatorial manifolds of positive dimension define finite simple graphs which are Hamiltonian.
About the origin of the definitino of shellability.