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

The chromatic polynomial   P   (   G  ,  x   )    of a graph   G   of order   n   can be expressed as    ∑   i  =  1   n(   −  1   )    n  −  ia  i    x  i   , where    a  i    is interpreted as the number of broken‐cycle‐free spanning subgraphs of   G   with exactly   i   components. The parameter    ϵ   (  G  )   =   ∑   i  =  1   n    (   n  −  i   )    a  i   ∕   ∑   i  =  1   n    a  iis the mean size of a broken‐cycle‐free spanning subgraph of   G  . In this article, we confirm and strengthen a conjecture proposed by Lundow and Markström in 2006 that    ϵ   (   T  n   )   <  ϵ   (  G  )   <  ϵ   (   K  n   )holds for any connected graph   G   of order   n   which is neither the complete graph    K  n    nor a tree    T  n    of order   n  . The most crucial step of our proof is to obtain the interpretation of all    a  i   's by the number of acyclic orientations of   G  .

Proving a conjecture on chromatic polynomials by counting the number of acyclic orientations