Let $S = K[x_1,..., x_n]$ be a polynomial ring over a field $K$. Let $I(G)
\subseteq S$ denote the edge ideal of a graph $G$. We show that the $\ell$th
symbolic power $I(G)^{(\ell)}$ is a Cohen-Macaulay ideal (i.e.,
$S/I(G)^{(\ell)}$ is Cohen-Macaulay) for some integer $\ell \ge 3$ if and only
if $G$ is a disjoint union of finitely many complete graphs. When this is the
case, all the symbolic powers $I(G)^{(\ell)}$ are Cohen-Macaulay ideals.
Similarly, we characterize graphs $G$ for which $S/I(G)^{(\ell)}$ has (FLC).

We introduce sequentially $S_r$ modules over a commutative graded ring and
sequentially $S_r$ simplicial complexes. This generalizes two properties for
modules and simplicial complexes: being sequentially Cohen-Macaulay, and
satisfying Serre's condition $S_r$. In analogy with the sequentially
Cohen-Macaulay property, we show that a simplicial complex is sequentially
$S_r$ if and only if its pure $i$-skeleton is $S_r$ for all $i$. For $r=2$, we
provide a more relaxed characterization.

We study $h$-vectors of simplicial complexes which satisfy Serre's condition
($S_r$). We say that a simplicial complex $\Delta$ satisfies Serre's condition
($S_r$) if $\tilde H_i(\lk_\Delta(F);K)=0$ for all faces $F \in \Delta$ and for
all $i < \min \{r-1,\dim \lk_\Delta(F)\}$, where $\lk_\Delta(F)$ is the link of
$\Delta$ with respect to $F$ and where $\tilde H_i(\Delta;K)$ is the reduced
homology groups of $\Delta$ over a field $K$.

We consider a class of graphs $G$ such that the height of the edge ideal
$I(G)$ is half of the number $\sharp V(G)$ of the vertices. We give
Cohen-Macaulay criteria for such graphs.