10 papers
Remarks about the Moebius-Kantor graph
Oliver Knill
The Moebius-Kantor graph MK=G(8,3) is a Cayley graph of three non-abelian groups, the Pauli group P(1), the semi-dihedral group SD(16), as well as the dihedral group D(16) of order…
God numbers for Graphs, Games and Groups
Z. Adams, M. Z. Cassim, C. Hou +3
We describe and axiomatize finite solitaire puzzles and zero sum sequential games graph theoretically. Zermelo's theorem telling that there is a win for one of the players or a dra…
Remarks on the Brouwer Conjecture
Oliver Knill
The Brouwer conjecture (BC) in spectral graph theory claims that the sum of the largest k Kirchhoff eigenvalues of a graph are bounded above by the number m of edges plus k(k+1)/2.…
Soft Barycentric Refinement
Oliver Knill
The soft Barycentric refinement preserves manifolds with or without boundary. In every dimension larger than one, there is a universal spectral central limiting measure that has af…
On Symmetries of Finite Geometries
Oliver Knill
The isospectral set of the Dirac matrix D=d+d* consists of orthogonal Q for which Q* D Q is an equivalent Dirac matrix. It can serve as the symmetry of a finite geometry G. The sym…
Morse and Lusternik-Schnirelmann for graphs
Oliver Knill
Both Morse theory and Lusternik-Schnirelmann theory link algebra, topology and analysis in a geometric setting. The two theories can be formulated in finite geometries like graph t…
Manifolds from Partitions
Oliver Knill
If f maps a discrete d-manifold G onto a (k+1)-partite complex P then H(G,f,P),the set of simplices x in G such that f(x) contains at least one facet in P defines a (d-k)-manifold.
Discrete Algebraic sets in Discrete Manifolds
Oliver Knill
A discrete d-manifold is a finite simple graph G=(V,E) where all unit spheres are (d-1)-spheres. A d-sphere is a d-manifold for which one can remove a vertex to make it contractibl…
Arboricity and Acyclic Chromatic Number
Oliver Knill
A theorem of Hakimi, Mitchem and Schmeichel from 1996 states that the edge arboricity arb(G) of a graph is bounded above by the acyclic chromatic number acy(G). We can improve this…
On the arboricity of manifolds
Oliver Knill
The arboricity of a discrete 2-sphere is always 3. The arboricity of any other discrete 2-dimensional surface is always 4. For d-manifolds of dimension larger than 2, the arboricit…