The Manickam-Mikl\'os-Singhi Parameter of Graphs and Degree Sequences

TL;DR

This paper introduces a new graph parameter based on vertex weightings with nonnegative total sum, providing polynomial algorithms for degree sequence realizations and characterizing extremal values for regular graphs.

Contribution

It defines the Manickam-Miklós-Singhi parameter for graphs and offers polynomial-time computation methods for degree sequences and extremal values for regular graphs.

Findings

Polynomial algorithm for degree sequence realizations.

Exact minimum and maximum values for regular graphs.

NP-hardness of the general parameter computation.

Abstract

Let $G$ be a simple graph. Consider all weightings of the vertices of $G$ with real numbers whose total sum is nonnegative. How many edges of $G$ have endpoints with a nonnegative sum? We consider the minimum number of such edges over all such weightings as a graph parameter. Computing this parameter has been shown to be NP-hard but we give a polynomial algorithm to compute the minimum of this parameter over realizations of a given degree sequence. We also completely determine the minimum and maximum value of this parameter for regular graphs.