Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups

We study the quasi-isometric rigidity of a large family of finitely generated groups that split as graphs of groups with virtually free vertex groups and two-ended edge groups. Let G be a group that is one-ended, hyperbolic relative to virtually abelian subgroups, and has JSJ decomposition over two-ended subgroups containing only virtually free vertex groups that are not quadratically hanging. Our main result is that any group quasi-isometric to G is abstractly commensurable to G. In particular, our result applies to certain “generic” HNN extensions of a free group over cyclic subgroups.

Free-Vertex Tetrahedral Finite-Element Representation and Its Use for Estimating Density Distribution of Irregularly-Shaped Asteroids

In this article, we present a free-vertex tetrahedral finite-element representation of irregularly shaped small bodies, which provides an alternative solution for estimating asteroid density distribution. We derived the transformations between gravitational potentials expressed by the free-vertex tetrahedral finite elements and the spherical harmonic functions. Inversely, the density of each free-vertex tetrahedral finite element can be estimated via the least-squares method, assuming a spherical harmonic gravitational function is present. The proposed solution is illustrated by modeling gravitational potential and estimating the density distribution of the simulated asteroid 216 Kleopatra.

A Deviation-Based Dynamic Vertex Reordering Technique for 2D Mesh Quality Improvement

We propose a novel deviation-based vertex reordering method for 2D mesh quality improvement. We reorder free vertices based on how likely this is to improve the quality of adjacent elements, based on the gradient of the element quality with respect to the vertex location. Specifically, we prioritize the free vertex with large differences between the best and the worst-quality element around the free vertex. Our method performs better than existing vertex reordering methods since it is based on the theory of non-smooth optimization. The downhill simplex method is employed to solve the mesh optimization problem for improving the worst element quality. Numerical results show that the proposed vertex reordering techniques improve both the worst and average element, compared to those with existing vertex reordering techniques.

Maximum value of conflict-free vertex-connection number of graphs

A path in a vertex-colored graph is called conflict-free if there is a color used on exactly one of its vertices. A vertex-colored graph is said to be conflict-free vertex-connected if any two vertices of the graph are connected by a conflict-free path. The conflict-free vertex-connection number, denoted by [Formula: see text], is defined as the smallest number of colors required to make [Formula: see text] conflict-free vertex-connected. Li et al. [Conflict-free vertex-connections of graphs, preprint (2017), arXiv:1705.07270v1[math.CO]] conjectured that for a connected graph [Formula: see text] of order [Formula: see text], [Formula: see text]. We confirm that the conjecture is true and poses two relevant conjectures.

The 1-Good-Neighbor Conditional Diagnosability of Some Regular Graphs

The g-good-neighbor conditional diagnosability is the maximum number of faulty vertices a network can guarantee to identify, under the condition that every fault-free vertex has at least g fault-free neighbors. In this paper, we study the 1-good-neighbor conditional diagnosabilities of some general k-regular k-connected graphs G under the PMC model and the MM* model. The main result [Formula: see text] under some conditions is obtained, where l is the maximum number of common neighbors between any two adjacent vertices in G. Moreover, the following results are derived: [Formula: see text] for the hierarchical star networks, [Formula: see text] for the BC networks, [Formula: see text] for the alternating group graphs [Formula: see text].