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 Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

In this article, we extend Moon's classic formula for counting spanning trees in complete graphs containing a fixed spanning forest to complete bipartite graphs. Let(   X  ,  Y   )$(X,Y)$    be the bipartition of the complete bipartite graphK   m  ,  n${K}_{m,n}$    with∣  X  ∣  =  m$| X| =m$    and∣  Y  ∣  =  n$| Y| =n$   . We prove that for any given spanning forestF$F$    ofK   m  ,  n${K}_{m,n}$    with componentsT  1   ,   T  2   ,  …  ,   T  k${T}_{1},{T}_{2},\ldots ,{T}_{k}$   , the number of spanning trees inK   m  ,  n${K}_{m,n}$    which contain all edges inF$F$    is equal to1   m  n∏   i  =  1   k    (    m  i   n  +   n  i   m   )1  −   ∑   i  =  1   km  i    n  im  i   n  +   n  i   m,$\frac{1}{mn}\left(\prod _{i=1}^{k}({m}_{i}n+{n}_{i}m)\right)\left(1-\sum _{i=1}^{k}\frac{{m}_{i}{n}_{i}}{{m}_{i}n+{n}_{i}m}\right),$wherem  i   =  ∣  V   (   T  i   )   ∩  X  ∣${m}_{i}=| V({T}_{i})\cap X| $    andn  i   =  ∣  V   (   T  i   )   ∩  Y  ∣${n}_{i}=| V({T}_{i})\cap Y| $    fori  =  1  ,  2  ,  …  ,  k$i=1,2,\ldots ,k$   .

Counting spanning trees in a complete bipartite graph which contain a given spanning forest