Geodesic
Unstable graphs: A fresh outlook via TF-automorphisms
Ars Mathematica Contemporanea, Tome 8 (2015) no. 1, pp. 115-131.

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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.
DOI : 10.26493/1855-3974.534.934
Keywords: Graphs, canonical double covers, two-fold isomorphisms. 
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Josef Lauri; Russell Mizzi; Raffaele Scapellato. Unstable graphs: A fresh outlook via TF-automorphisms. Ars Mathematica Contemporanea, Tome 8 (2015) no. 1, pp. 115-131. doi : 10.26493/1855-3974.534.934. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.534.934/

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