English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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

Decreasing behavior of the depth functions of edge ideals