  Let G be a simple undirected graph on n vertices, and let I(G) \subseteq R =
k[x_1,...,x_n] denote its associated edge ideal. We show that all chordal
graphs G are sequentially Cohen-Macaulay; our proof depends upon showing that
the Alexander dual of I(G) is componentwise linear. Our result complements
Faridi's theorem that the facet ideal of a simplicial tree is sequentially
Cohen-Macaulay and implies Herzog, Hibi, and Zheng's theorem that a chordal
graph is Cohen-Macaulay if and only if its edge ideal is unmixed. We also
characterize the sequentially Cohen-Macaulay cycles and produce some examples
of nonchordal sequentially Cohen-Macaulay graphs.
