Decreasing behavior of the depth functions of edge ideals

Let $I$ be the edge ideal of a connected non-bipartite graph and $R$ the base polynomial ring. Then $\mathrm{depth}R\mathrm{/}I\ge 1$ and $\mathrm{depth}R\mathrm{/}{I}^t=0$ for $t\gg 1$. We give combinatorial conditions for $\mathrm{depth}R\mathrm{/}{I}^t=1$ for some $t$ in between and show that the depth function is non-increasing thereafter. Especially, the depth function quickly decreases to 0 after reaching 1. We show that if $\mathrm{depth}R\mathrm{/}I=1$ then $\mathrm{depth}R\mathrm{/}{I}^2=0$ and if $\mathrm{depth}R\mathrm{/}{I}^2=1$ then $\mathrm{depth}R\mathrm{/}{I}^5=0$. Other similar results suggest that if $\mathrm{depth}R\mathrm{/}{I}^t=1$ then $\mathrm{depth}R\mathrm{/}{I}^{t+3}=0$. This a surprising phenomenon because the depth of a power can determine a smaller depth of another power. Furthermore, we are able to give a simple combinatorial criterion for $\mathrm{depth}R\mathrm{/}{I}^{(t)}=1$ for $t\gg 1$ and show that the condition $\mathrm{depth}R\mathrm{/}{I}^{(t)}=1$ is persistent, where $I^{(t)}$ denotes the $t$-th symbolic powers of $I$.