The Mickey mouse theorem assures that a connected positive curvature graph of positive dimension is a sphere.
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 parametrized poincare-hopf theorem allows to see the f-vector of a graph in terms of the f-vector s of parts of the unit spheres of the graph.
Dehn-Sommerville spaces are generalized spheres as they share many properties of spheres: Euler characteristic and more generally Dehn-Sommerville symmetries.
We have calculated with graphs from the very beginning: Humans computed with pebbles like in this scene of the `Clan of the Cave Baer` (1986)) or with line graphs when writing with tally sticks (see this lecture). In all of these cases, the addition of graphs is the disjoint union which serves a nice monoid like the natural numbers. It … ….
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.
Branko Grünbaum (1929-2018) passed away last September. Here is the obituary from the university of Washington. One of his master pieces is the book “Tilings and patterns”, written with G.C. Shephard. The well illustrated book is considered the bible on Tilings. Here is a page from that book: Links: Personal website at Washington. croatia.org featuring an other picture. Wikipedia entry … ….
Some update about recent activities: a new calculus course, the Cartan magic formula and some programming about the coloring algorithm.