Two comments on balls in vertex transitive graphs
Itai Benjamini, Gady Kozma
TL;DR
This paper investigates the geometric properties of balls in vertex transitive graphs, highlighting limitations in finite graphs and conditions for their appearance in infinite graphs.
Contribution
It provides new insights into the realization of local neighborhoods in finite versus infinite vertex transitive graphs.
Findings
A ball of radius 1 in the grandfather graph cannot be realized in any finite vertex transitive graph.
Conditions are identified under which a ball in a finite vertex transitive graph appears in an infinite one.
Abstract
We observe that a ball of radius in the grandfather graph can not be realized as a ball of radius in a finite vertex transitive graph. We remark on when a ball in a finite vertex transitive graph appears as a ball in an infinite vertex transitive graph.
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph Labeling and Dimension Problems · Limits and Structures in Graph Theory