45 citations · 674 across the 102 of their papers we have counts for
10 papers · 1 filter
Cohomology of open sets
Oliver Knill
If G is a finite abstract simplicial complex and K is a subcomplex of G and U=G-K is the open complement of K in G, the Betti vectors of K and U and G satisfy the inequality b(G) l…
Remarks about the Arithmetic of Graphs
Oliver Knill
The arithmetic of N, Z, Q, R can be extended to a graph arithmetic where N is the semiring of finite simple graphs and where Z and Q are integral domains, culminating in a Banach a…
Eulerian edge refinements, geodesics, billiards and sphere coloring
Oliver Knill
A finite simple graph is called a 2-graph if all of its unit spheres S(x) are cyclic graphs of length 4 or larger. A 2-graph G is Eulerian if all vertex degrees of G are even. An e…
Combinatorial manifolds are Hamiltonian
Oliver Knill
Extending a theorem of Whitney of 1931 we prove that all connected d-graphs are Hamiltonian for positive d. A d-graph is a type of combinatorial manifold which is inductively defin…
Listening to the cohomology of graphs
Oliver Knill
We prove that the spectrum of the Kirchhoff Laplacian H0 of a finite simple Barycentric refined graph and the spectrum of the connection Laplacian L of G determine each other: we p…
Gauss-Bonnet for multi-linear valuations
Oliver Knill
We prove Gauss-Bonnet and Poincare-Hopf formulas for multi-linear valuations on finite simple graphs G=(V,E) and answer affirmatively a conjecture of Gruenbaum from 1970 by constru…