Unstable graphs: A fresh outlook via TF-automorphisms

Josef Lauri, Russell Mizzi, Raffaele Scapellato


In this paper, we first establish the very close link between stability of graphs, a concept first introduced by Marušič, Scapellato and Zagaglia Salvi and studied most notably by Surowski and Wilson, and two-fold automorphisms. The concept of two-fold isomorphisms, as far as we know, first appeared in Zelinka’s work on isotopies of digraphs and later studied formally by the authors with a greater emphasis on undirected graphs. We then turn our attention to the stability of graphs which have every edge on a triangle, but with the fresh outlook provided by TF-automorphisms. Amongst such graphs are strongly regular graphs with certain parameters. The advantages of this fresh outlook are highlighted when we ultimately present a method of constructing and generating unstable graphs with large diameter having every edge lying on a triangle. This was a rather surprising outcome.


Graphs, canonical double covers, two-fold isomorphisms.

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DOI: https://doi.org/10.26493/1855-3974.534.934

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications