Kernels by Monochromatic Directed Paths in 3-Colored Tournaments and Quasi-Tournaments
We call the tournament T an m-coloured tournament if the arcs of T are coloured with m colours. If v is a vertex of an m-coloured tournament T, we denote by ζ(v) the set of colours assigned to the arcs with v as endpoint, ζ − (v) the set of colours assigned to the arcs with v as final vertex, and ζ + (v) the set of colours assigned to the arcs with v as an initial vertex. In this paper, we prove that if T is a 3-coloured tournament which does not contain C3 such that: 1. for each v ∈ V(T) |ζ − (v) |≤ 2,
Tournament Directed Graphs
Paired comparison is the process of comparing objects two at a time. A tournament in Graph Theory is a representation of such paired comparison data. Formally, an n-tournament is an oriented complete graph on n vertices; that is, it is the representation of a paired comparison, where the winner of the comparison between objects x and y (x and y are called vertices) is depicted with an arrow or arc from the winner to the other. In this thesis, we shall prove several results on tournaments. In Chapter 2, we will prove that the maximum number of vertices that can beat exactly m other vertices in an n-tournament is min{2m + 1, 2n 2m 1}. The remainder of this thesis will deal with tournaments whose arcs have been colored. In Chapter 3 we will define what it means for a k-coloring of a tournament to be k-primitive. We will prove that the maximum k such that some strong n-tournament can be k-colored to be k-primitive lies in the interval ⇥ n 1
Evidence of spurious results along with spatially autocorrelated errors in the context of geographically weighted regression for two independent SAR(1) processes
AbstractThe aim of this paper is to examine the performance of the geographically weighted regression (GWR) technique in terms of recognizing linear relationships at the local level. Using a Monte Carlo analysis with two independent spatially autocorrelated SAR(1) processes, this study finds that spurious behavior exists at the local level along with spatially autocorrelated errors. Therefore, GWR estimation results must be interpreted with caution.
