Unstable Graphs: A Fresh Outlook via TF-Automorphisms
Abstract
In this paper, we first establish the very close link between stability of graphs, a concept first introduced in Scapsalvi1 and studied most notably by Surowski Surowski1, Surowski2 and Wilson Wilson01 and two-fold automorphisms. The concept of two-fold isomorphisms, as far as we know, first appeared in literature in the form of isotopies of digraphs zelinka4, zelinka1, zelinka2, zelinka3 and later studied formally in lms1, lms2 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.