Convergent and divergent numbers games for certain collections of edge-weighted graphs

TL;DR

This paper investigates the convergence and divergence of a numbers game played on edge-weighted graphs with negative integer amplitudes, using combinatorial methods to analyze the game's behavior.

Contribution

It introduces combinatorial techniques to study the convergence properties of numbers games on specific edge-weighted graphs with negative amplitudes.

Findings

Provides criteria for convergence and divergence on certain graphs

Supports previous results through combinatorial analysis

Enhances understanding of numbers games on edge-weighted graphs

Abstract

The numbers game is a one-player game played on a finite simple graph with certain "amplitudes" assigned to its edges and with an initial assignment of real numbers to its nodes. The moves of the game successively transform the numbers at the nodes using the amplitudes in a certain way. Here, the edge amplitudes will be negative integers. Combinatorial methods are used to investigate the convergence and divergence of numbers games played on certain such graphs. The results obtained here provide support for results in a companion paper.