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 show that for a certain class of convex functions $f$, including theexponential functions $x\mapsto e^{\lambda x}$ with $\lambda>0$ a real number,and all the powers $x\mapsto x^\beta$, $x\geq 0$ and $\beta\geq 2$ a realnumber, with a unique small exception, if $(d_1,\ldots,d_n)$ ranges over thedegree sequences of graphs with $n$ vertices and $m$ edges and $m\leq n-1$,then the maximum of $\sum_i f(d_i)$ is uniquely attained by the degree sequenceof a quasi-star graph, namely, a graph consisting of a star plus possiblyadditional isolated vertices. This result significantly extends a similarresult in [D.~Ismailescu, D.~Stefanica, Minimizer graphs for a class ofextremal problems, J.~Graph Theory,~39~(4)~(2002)]. Dually, we show that for acertain class of concave functions $g$, including the negative exponentialfunctions $x\mapsto 1-e^{-\lambda x}$ with $\lambda>\ln(2)$ a real number, allthe powers $x\mapsto x^\alpha$, $x\geq 0$ and $0<\alpha\leq \frac{1}{2}$ a realnumber, and the function $x\mapsto \frac{x}{x+1}$ for $x\geq 0$, if$(d_1,\ldots,d_n)$ ranges over the degree sequences of graphs with $n$ verticesand $m$ edges, then the minimum of $\sum_i g(d_i)$ is uniquely attained by thedegree sequence of a quasi-complete graph, i.e., a graph consisting of acomplete graph plus possibly an additional vertex connected to some but not allvertices of the complete graph, plus possibly isolated vertices. This resultextends a similar result in the same paper.

Functions that are uniquely maximized by sparse quasi-star graphs, and uniquely minimized by quasi-complete graphs