Abstract
In this paper we give a detailed survey of stability properties of various combinations of graphs.We review previous work on unions, joins and (cartesian) products of graphs, and supply further evidence of the unpredictability of the stability index function under cartesian products in that we show that for r>2, the r-cube has stability index 1, which for most values of m and n the product Pm × Pn of two paths has stability index mn-7.Finally, we discuss the stability properties of compositions (lexicographic products) and coronas of graphs, in particular finding infinite families of such graphs which are stable.
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