Volume
21
Issue
1-2
Abstract
This paper formalizes the stepping stone problem introduced Ladoucer and Rebenstock [G] to the setting of simple graphs. This paper considers the set of functions from the vertices of our graph to N, which assign a fixed number of 1’s to some vertices and assign higher numbers to other vertices by adding up their neighbors’ assignments. The stepping stone solution is defined as an element obtained from the argmax of the maxima of these functions, and the maxima as its growth. This work is organized into work on the bounded and unbounded degree graph cases. In the bounded case, sufficient conditions are obtained for superlinear and sublinear stepping stone solution growth. Furthermore this paper demonstrates the existence of a basis of graphs which characterizes superlinear growth. In the unbounded case, properly sublinear and superlinear stepping stone solution growth are obtained.
First Page
42
Last Page
53
Recommended Citation
O'Dwyer, Rory
(2024)
"Stepping Stone Problem on Graphs,"
The Mathematics Enthusiast: Vol. 21
:
No.
1
, Article 4.
DOI: https://doi.org/10.54870/1551-3440.1616
Available at:
https://scholarworks.umt.edu/tme/vol21/iss1/4
Digital Object Identifier (DOI)
10.54870/1551-3440.1616
Publisher
University of Montana, Maureen and Mike Mansfield Library
