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 $x\mapsto x^\beta$, $x\geq 0$ and $\beta\geq 2$ a real number, with a unique small exception, if $(d_1,\ldots,d_n)$ ranges over the degree 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 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 [D.~Ismailescu, D.~Stefanica, Minimizer graphs for a class of extremal problems, J.~Graph Theory,~39~(4)~(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 $\lambda>\ln(2)$ a real number, all the powers $x\mapsto x^\alpha$, $x\geq 0$ and $0<\alpha\leq \frac{1}{2}$ a real number, 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 $n$ vertices and $m$ edges, then the minimum of $\sum_i g(d_i)$ 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.