Abstract
Given finite simple graph one can associate the edge ideal. In this paper we discuss the non-vanishingness of the graded Betti numbers of edge ideals in terms of the original graph. In particular, we give a necessary and sufficient condition for a chordal graph on which the graded Betti number does not vanish and characterize the graded Betti number for a forest. Moreover we characterize the projective dimension for a chordal graph.
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