Operations on Well-Covered Graphs and the Roller-Coaster Conjecture

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Operations on Well-Covered Graphs and the Roller-Coaster Conjecture
A graph G is well-covered if every maximal independent set has the same cardinality. Let sk denote the number of independent sets of cardinality k, and define the independence polynomial of G to be S(G, z) = skzk. This paper develops a new graph theoretic operation called power magnification that preserves well-coveredness and has the effect of multiplying an independence polynomial by zc where c is a positive integer. We will apply power magnification to the recent Roller-Coaster Conjecture of Michael and Traves, proving in our main theorem that for sufficiently large independence number , it is possible to find well-covered graphs with the last (.17) terms of the independence sequence in any given linear order. Also, we will give a simple proof of a result due to Alavi, Malde, Schwenk, and Erdos on possible linear orderings of the independence sequence of not-necessarily well-covered graphs, and we will prove the Roller-Coaster Conjecture in full for independence
Philip Matchett
Added 17 Dec 2010
Updated 17 Dec 2010
Type Journal
Year 2004
Authors Philip Matchett
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