Unexpected symmetries in unstable graphs

Authors:
Steve Wilson
Affiliations:
Department of Mathematics and Statistics, Northern Arizona University, Box 5717, Flagstaff, AZ 86011, USA
Venue:
Journal of Combinatorial Theory Series B
Year:
2008

Citing 7
Cited 0

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Abstract

This paper considers instability of graphs in all of its possible forms. First, four theorems (one with two interesting special cases) are presented, each of which shows that graphs satisfying certain conditions are unstable. Several infinite families of graphs are investigated. For each family, the four general theorems are used to find and prove enough theorems particular for the family to explain the instability of all graphs in the family up to a certain point. A very dense family of edge-transitive unstable graphs is constructed and, at the other extreme, an unstable graph is constructed whose only symmetry is trivial. Finally, the four theorems are shown to be able to explain all instability.