arXiv subjects

Ivan Aidun

Publications and source records attributed to Ivan Aidun.

5 recordsLinked to original sources

arXiv2026

A statistical investigation of a divisor-sum function

The sum of proper divisors function $s(n)$ has been studied for more than 2000 years. In this paper we study statistical properties of the related function $S_s(n) \coloneqq \sum_{d \mid n} s(d)$. This function arises from a generalization of the practical numbers. Although $S_s(n)$ is neither additive nor multiplicative, we prove that $S_s(n)/n$ has a continuous asymptotic distribution function, and that its values are dense in the interval $[0,\infty)$. We evaluate its mean and establish the existence of all higher moments. Moreover, if $\mu_k$ denotes the $k^{th}$ moment of $S_s(n)/n$, we show that $$\mu_k^{1/k} \sim e^{2\gamma} (\log k)^2$$ as $k \rightarrow \infty$, where $\gamma$ is the Euler-Mascheroni constant.

math.NT↗

arXiv2020

Gonality sequences of graphs

To any graph we associate a sequence of integers called the gonality sequence of the graph, consisting of the minimum degrees of divisors of increasing rank on the graph. This is a tropical analogue of the gonality sequence of an algebraic curve. We study gonality sequences for graphs of low genus, proving that for genus up to $5$, the gonality sequence is determined by the genus and the first gonality. We then prove that any reasonable pair of first two gonalities is achieved by some graph. We also develop a modified version of Dhar's burning algorithm more suited for studying higher gonalities.

math.CO↗

arXiv2019

On the gonality of Cartesian products of graphs

In this paper we study Cartesian products of graphs and their divisorial gonality, which is a tropical version of the gonality of an algebraic curve. We present an upper bound on the gonality of the Cartesian product of any two graphs, and provide instances where this bound holds with equality, including for the $m\times n$ rook's graph with $\min\{m,n\}\leq 5$. We use our upper bound to prove that Baker's gonality conjecture holds for the Cartesian product of any two graphs with two or more vertices each, and we determine precisely which nontrivial product graphs have gonality equal to Baker's conjectural upper bound.

math.CO↗

arXiv2018

Graphs of gonality three

In 2013, Chan classified all metric hyperelliptic graphs, proving that divisorial gonality and geometric gonality are equivalent in the hyperelliptic case. We show that such a classification extends to combinatorial graphs of divisorial gonality three, under certain edge- and vertex-connectivity assumptions. We also give a construction for graphs of divisorial gonality three, and provide conditions for determining when a graph is not of divisorial gonality three.

math.CO↗

arXiv2018

Treewidth and gonality of glued grid graphs

We compute the treewidth of a family of graphs we refer to as the glued grids, consisting of the stacked prism graphs and the toroidal grids. Our main technique is constructing strict brambles of large orders. We discuss connections to divisorial graph theory coming from tropical geometry, and use our results to compute the divisorial gonality of these graphs.

math.CO↗