Acknowledgements

Let G be a finite simple graph on the vertex set VG with the edge set EG. Using the paths of G of length t, we can generate a square-free monomial ideal It(G). Let It(G) denote the square-free Alexander dual of It(G). In this project, we study properties of the sets of associated primes Ass(R/(It(G) ∨)s), as s increases. We will establish a lower bound on the index of stability N for Ass(R/(I2(G) ∨)s), when G is a star. As well, we will show how our result can be applied to answer a question raised by Francisco, Hà, an