Maximum ratio of (graph) irregularities

Abstract

We estimate the maximum ratio between the $σ_t$- and $σ$-irregularity for graphs and trees of order $n$, which are respectively bounded by $Θ(n^{5/2})$ 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).