Max-plus objects to study the complexity of graphs

TL;DR

This paper introduces the mp-chart, a max-plus algebraic object for graphs, to analyze graph complexity through statistical measures and a central limit theorem, applicable to both the graph and its complement.

Contribution

It defines the mp-chart in max-plus algebra, relates it to graph complexity, and derives formulas for its mean, variance, and a central limit theorem.

Findings

Derived formulas for mean and variance of the mp-chart.

Established a central limit theorem for the mp-chart.

Showed tractability of the mp-chart for complement graphs.

Abstract

Given an undirected graph $G$, we define a new object $H_G$, called the mp-chart of $G$, in the max-plus algebra. We use it, together with the max-plus permanent, to describe the complexity of graphs. We show how to compute the mean and the variance of $H_G$ in terms of the adjacency matrix of $G$ and we give a central limit theorem for $H_G$. Finally, we show that the mp-chart is easily tractable also for the complement graph.