Maximum ratio of (graph) irregularities

TL;DR

This paper determines the maximum ratio between two graph irregularity measures for graphs and trees, answering existing conjectures and providing bounds related to graph variance.

Contribution

It establishes the maximum ratio bounds for graph and tree irregularities, resolving a question and a conjecture by Filipovski et al.

Findings

Maximum ratio for graphs is bounded by Θ(n^{5/2})

Maximum ratio for trees is bounded by n-2

Irregularity measure bounds the graph variance for trees

Abstract

We estimate the maximum ratio between the $\sigma_t$- and $\sigma$-irregularity for graphs and trees of order $n$, which are respectively bounded by $\Theta(n^{5/2})$ and $n-2$. This answers a question and a conjecture by Filipovski et al. in an elegant way. For trees, we obtain that the (Albertson) irregularity measure $\irr$ is an upper bound for the graph variance (normalised with the order).