5 research outputs found
Discrete curvature on graphs from the effective resistance
This article introduces a new approach to discrete curvature based on the
concept of effective resistances. We propose a curvature on the nodes and links
of a graph and present the evidence for their interpretation as a curvature.
Notably, we find a relation to a number of well-established discrete curvatures
(Ollivier, Forman, combinatorial curvature) and show evidence for convergence
to continuous curvature in the case of Euclidean random graphs. Being both
efficient to calculate and highly amenable to theoretical analysis, these
resistance curvatures have the potential to shed new light on the theory of
discrete curvature and its many applications in mathematics, network science,
data science and physics.Comment: 37 pages, 7 figures. Updates in this version: Section 3.2 added,
Appendix B added, Figure 3 extended, Proof of Proposition 2 correcte
Tight bounds on angle sums of nonobtuse simplices
It is widely known that the sum of the angles of a triangle equals two right angles. Far less known are the answers to similar questions for tetrahedra and higher dimensional simplices. In this paper we review some of these less known results, and look at them from a different point of view. Then we continue to derive tight bounds on the dihedral angle sums for the subclass of nonobtuse simplices. All the dihedral angles of such simplices are less than or equal to right. They have several important applications (Brandts et al., 2009). The main conclusion is that when the spatial dimension n is even, the range of dihedral angle sums of nonobtuse simplices is n times smaller than the corresponding range for arbitrary simplices. When n is odd, it is n-1 times smaller