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 Tools annual

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 Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

The $k$-independence number of a graph is the maximum size of a set ofvertices at pairwise distance greater than $k$. A graph is called $k$-partiallywalk-regular if the number of closed walks of a given length $l\le k$, rootedat a vertex $v$, only depends on $l$. In particular, a distance-regular graphis also $k$-partially walk-regular for any $k$. In this note, we introduce anew family of polynomials obtained from the spectrum of a graph. Thesepolynomials, together with the interlacing technique, allow us to give tightspectral bounds on the $k$-independence number of a $k$-partially walk-regulargraph. Together with some examples where the bounds are tight, we also showthat the odd graph $O_{\ell}$ with $\ell$ odd has no $1$-perfect code.

A new class of polynomials from to the spectrum of a graph, and its application to bound the $k$-independence number