30.07.2024 16:00 Kurt Klement Gottwald: Šoltés’s problem for the Kirchhoff index of a graph
A good vertex of a graph is a vertex whose removal doesn't change the Wiener Index of the graph. Šoltés posed the problem of finding all simple graphs with only good vertices. He found that the cycle on 11 vertices does the trick and to this day it is still the only known graph with this property. Due to the challenge of finding more examples of such graphs, the relaxed version of the problem was tackled, namely the problem of finding graphs with a large proportion of good vertices. We consider a similar problem, but instead of the shortest path distance as in the Wiener Index, we use the resistance distance and the Kirchhoff Index. Similarly to the original problem, we find only the cycle on 5 vertices to solve the full problem. We construct several families of graphs with large proportions of good vertices.
Quelle
30.07.2024 16:30 Thomas Jahn: Minkowski chirality of triangles
The Minkowski asymmetry of a convex body K in R^d, i.e., the smallest dilation factor λ for which K is contained in a translate of λ(-K), is known to be at most d. The maximizers of the Minkowski symmetry are precisely the simplices, in particular those which are mirror symmetric with respect to some hyperplane. In order to quantify how mirror symmetric a given convex body K in R^d is, we may study the smallest dilation factor λ for which K is contained in a translate of some λ A_L(K) where A_L is the reflection about some j-dimensional linear subspace L of R^d. The Minkowski asymmetry is the j=0 case, and in this talk we focus on the case where d=2, j=1, and K is a triangle. We present some numerical evidence and discuss our analytical findings.
Quelle