Florian Lesny | Searcharxiv

arXiv2026

On topological indices of random caterpillar graphs

We consider the asymptotic behaviour of topological indices of random caterpillar graphs, such as the Zagreb, $α$-Randić, Wiener, Gini and level indices, as the size of the graph grows. Our simple, general framework, which is based on a multivariate central limit theorem, the continuous mapping theorem and Slutsky's theorem, can also be used to cover other topological indices. We derive limit laws for these indices and provide exact and asymptotic results on their means and variances. The $α$-Randić index exhibits a discontinuity in its limit distribution at $α=\frac{1}{2}$.

math.PR↗

arXiv2026

Probabilistic analysis of optimal multi-pivot QuickSort

We consider a multi-pivot QuickSort algorithm using $K\in\mathbb{N}$ pivot elements to partition a nonsorted list into $K+1$ sublists in order to proceed recursively on these sublists. For the partitioning stage, various strategies are in use. We focus on the strategy that minimizes the expected number of key comparisons in the standard random model, where the list is given as a uniformly permuted list of distinct elements. We derive asymptotic expansions for the expectation and variance of the number of key comparisons as well as a limit law for all $K\in\mathbb{N}$, where the convergence holds for all (exponential) moments. For $K\le 4$ we also bound the rate of convergence within the Wasserstein and Kolmogorov--Smirnov distance. Our analysis of the expectation is based on classical results for random $m$-ary search trees. For the remaining results, combinatorial considerations are used to make the contraction method applicable.

math.PR↗