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

Consider an undirected multigraph G = (V,E) with n vertices and m edges, and let Ni denote the number of connected spanning subgraphs with i(m ≥ i ≥ n) edges in G. Recently, we showed in [3] the validity of (m-i + 1)Ni-1 (i-n + ⌊3+√9+8(i-n)/2⌋)Ni for a simple graph and each i(m ≥ i ≥ n). Note that, from this inequality, (m-n)Nn/2Nn+1 + Nn/(m-n+1)Nn-1 ≥ 2 is easily derived. In this paper, for a multigraph G and all i(m ≥ i ≥ n), we prove (m-i + 1)Ni-1 ≥ (i-n + 2)Ni, and give a necessary and sufficient condition by which (m-i + 1)Ni-1 = (i-n + 2)Ni. In particular, this means that (m-i + 1)Ni-1 (i-n + ⌊3+√9+8(i-n)/2⌋)Ni is not valid for all multigraphs, in general. Furthermore, we prove (m-n)Nn/2Nn+1 + Nn/(m-n+1)Nn-1 ≥ 2, which is not straightforwardly derived from (m-i + 1)Ni-1 ≥ (i-n+2)Ni, and also introduce a necessary and sufficent condition by which (m-n)Nn/2Nn+1 + Nn(m-n + 1)Nn-1 = 2. Moreover, we show a sufficient condition for a multigraph to have Nn2 Nn-1Nn+1. As special cases of the sufficient condition, we show that if G contains at least ⌈2/3(m-n)⌉+1 multiple edges between some pair of vertices, or if its underlying simple graph has no cycle with length more than 4, then Nn2 Nn-1Nn+1.

Inequalities on the Number of Connected Spanning Subgraphs in a Multigraph