Functions that are uniquely maximized by sparse quasi-star graphs, and uniquely minimized by quasi-complete graphs
Nicola Apollonio
TL;DR
The paper studies extremal problems for degree-based graph functionals Σ_f(d) over graphs with fixed n and m edges. It proves that threshold graphs maximize Σ_f for any f in a broad class 𝔽 satisfying an integer strict convexity condition, and that, in the very sparse regime m ≤ n−1, the unique maximizer is the sparse quasi-star QS(n,m) (with a precise exception at m=3 under a specific equality). Dually, it shows that for a wide class 𝔊 of concave-type functions, the quasi-complete graph QK(n,m) uniquely minimizes Σ_g, leveraging a complementary mapping. The results extend prior work on Σ_2 and its maximizers/minimizers by encompassing exponential and power functions, their nonnegative combinations, and dual constructions, and they rely on Chebyshev’s inequality and majorization/threshold graph structure. Together, these findings delineate when highly structured, threshold-like graphs uniquely optimize degree-sum functions, with implications for graph-design problems under convex/concave criteria.
Abstract
We show that for a certain class of convex functions $f$, including the exponential functions $x\mapsto e^{λx}$ with $λ>0$ a real number, and all the powers $x\mapsto x^β$, $x\geq 0$ and $β\geq 2$ 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 $x\mapsto 1-e^{-λx}$ with $λ>\ln(2)$ a real number, all the powers $x\mapsto x^α$, $x\geq 0$ and $0<α\leq \frac{1}{2}$ 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.