Crossing estimates from metric graph and discrete GFF

Crossing Minimization within Graph Embeddings Crossing Minimization within Graph Embeddings

We propose a novel optimization-based approach to embedding heterogeneous high-dimensional data characterized by a graph. The goal is to create a two-dimensional visualization of the graph structure such that edge-crossings are minimized while preserving proximity relations between nodes. This paper provides a fundamentally new approach for addressing the crossing minimization criteria that exp...

On the Crossing Number of the Join of Five Vertex Graph with the Discrete Graph

In this paper, we show the values of crossing numbers for join products of graph G on five vertices with the discrete graph Dn and the path Pn on n vertices. The proof is done with the help of software. The software generates all cyclic permutations for a given number n. For cyclic permutations, P1 – Pm will create a graph in which to calculate the distances between all vertices of the graph. T...

The Non-Crossing Graph

Two sets are non-crossing if they are disjoint or one contains the other. The noncrossing graph NCn is the graph whose vertex set is the set of nonempty subsets of [n] = {1, . . . , n} with an edge between any two non-crossing sets. Various facts, some new and some already known, concerning the chromatic number, fractional chromatic number, independence number, clique number and clique cover nu...

Estimates from the discrete-time Lyapunov equation

The Quasihyperbolic Metric and Associated Estimates on the Hyperbolic Metric

In this paper we investigate the geometry of the quasihyperbolic metric of domains in R". This metric arises from the conformally flat generalized Riemannian metric d(x, aD)' ldx I. Due to the fact that the density cl(x, aD) -t is not necessarily differentiable, the classical theories of Riemannian geometry do not apply to this metric. The quasihyperbolic metric has been found to have many inte...