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

We describe a new method to remove short cycles on regular graphs whilemaintaining spectral bounds (the nontrivial eigenvalues of the adjacencymatrix), as long as the graphs have certain combinatorial properties. Thesecombinatorial properties are related to the number and distance between shortcycles and are known to happen with high probability in uniformly randomregular graphs.  Using this method we can show two results involving high girth spectralexpander graphs. First, we show that given $d \geq 3$ and $n$, there exists anexplicit distribution of $d$-regular $\Theta(n)$-vertex graphs where with highprobability its samples have girth $\Omega(\log_{d - 1} n)$ and are$\epsilon$-near-Ramanujan; i.e., its eigenvalues are bounded in magnitude by$2\sqrt{d - 1} + \epsilon$ (excluding the single trivial eigenvalue of $d$).Then, for every constant $d \geq 3$ and $\epsilon > 0$, we give a deterministicpoly$(n)$-time algorithm that outputs a $d$-regular graph on$\Theta(n)$-vertices that is $\epsilon$-near-Ramanujan and has girth$\Omega(\sqrt{\log n})$, based on the work of arXiv:1909.06988 .

Spectrum preserving short cycle removal on regular graphs