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

Abstract

We show that for a certain class of convex functions $f$, including the exponential functions $x\mapsto e^{\lambda x}$ with $\lambda >0$ a real number, and all the powers $x\mapsto x^{\beta }$, $x\ge 0$ and $\beta \ge 2$ a real number, with a unique small exception, if $(d_1,\dots ,d_n)$ ranges over the degree sequences of graphs with $n$ vertices and $m$ edges and $m\le n-1$, then the maximum of ${\sum }_{i}f\left(d_i\right)$ is uniquely attained by the degree sequence of a quasi-star graph, namely, a graph consisting of a star plus possibly additional isolated vertices. This result significantly extends a similar result in Ismailescu and Stefanica (2002). Dually, we show that for a certain class of concave functions $g$, including the negative exponential functions $x\mapsto 1-e^{-\lambda x}$ with $\lambda >ln\left(2\right)$ a real number, all the powers $x\mapsto x^{\alpha }$, $x\ge 0$ and $0<\alpha \le \frac{1}{2}$ a real number, and the function $x\mapsto \frac{x}{x+1}$ for $x\ge 0$, if $(d_1,\dots ,d_n)$ ranges over the degree sequences of graphs with $n$ vertices and $m$ edges, then the minimum of ${\sum }_{i}g\left(d_i\right)$ is uniquely attained by the degree sequence of a quasi-complete graph, i.e., a graph consisting of a complete graph plus possibly an additional vertex connected to some but not all vertices of the complete graph, plus possibly isolated vertices. This result extends a similar result in the same paper.