  Let $G$ be a simple graph on $n$ vertices. We introduce the notion of
bipartite connectivity of $G$, denoted by $\operatorname{bc}(G)$ and prove that
  $$\lim_{s \to \infty} \operatorname{depth} (S/I(G)^{(s)}) \le
\operatorname{bc}(G),$$
  where $I(G)$ denotes the edge ideal of $G$ and $S = \mathrm{k}[x_1, \ldots,
x_n]$ is a standard graded polynomial ring over a field $\mathrm{k}$. We
further compute the depth of symbolic powers of edge ideals of several classes
of graphs, including odd cycles and whisker graphs of complete graphs to
illustrate the cases where the above inequality becomes equality.
