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Kyle Huang

Publications and source records attributed to Kyle Huang.

5 recordsLinked to original sources
arXiv2026

Exact Flatness Constant for One-Point Convex Bodies and the Discrete Isominwidth Problem: The Planar Case

A variant of the flatness problem from integer programming is studied, in which one considers convex bodies in $\mathbb{R}^d$ with at most $k$ interior lattice points. The maximum lattice width of such a body is denoted by Flt(d,k) and it is related to the classical flatness constant as well as a conjectural dual version of Minkowski's convex body theorem due to Makai. Moreover, it is shown that Flt(2, 1) = 3, i.e., any planar convex body with at most one interior point has lattice width at most three. This leads to an isominwidth inequality for the lattice point enumerator of planar convex bodies.

math.MG↗
arXiv2021

The Prime Graphs of Some Classes of Finite Groups

In this paper we study prime graphs of finite groups. The prime graph of a finite group $G$, also known as the Gruenberg-Kegel graph, is the graph with vertex set {primes dividing $|G|$} and an edge $p$-$q$ if and only if there exists an element of order $pq$ in $G$. In finite group theory, studying the prime graph of a group has been an important topic for the past almost half century. Only recently prime graphs of solvable groups have been characterized in graph theoretical terms only. In this paper, we continue this line of research and give complete characterizations of several classes of groups, including groups of square-free order, metanilpotent groups, groups of cube-free order, and, for any $n\in \mathbb{N}$, solvable groups of $n^\text{th}$-power-free order. We also explore the prime graphs of groups whose composition factors are cyclic or $A_5$ and draw connections to a conjecture of Maslova. We then propose an algorithm that recovers the prime graph from a dual prime graph.

math.GR↗
arXiv2020

Minimal Prime Graphs of Solvable Groups

We explore graph theoretical properties of minimal prime graphs of finite solvable groups. In finite group theory studying the prime graph of a group has been an important topic for the past almost half century. Recently prime graphs of solvable groups have been characterized in graph theoretical terms only. This now allows the study of these graphs with methods from graph theory only. Minimal prime graphs turn out to be of particular interest, and in this paper we pursue this further by exploring, among other things, diameters, Hamiltonian cycles and the property of being self-complementary for minimal prime graphs. We also study a new, but closely related notion of minimality for prime graphs and look into counting minimal prime graphs.

math.CO↗
arXiv2020

The Adjacency Spectra of Some Families of Minimally Connected Prime Graphs

In finite group theory, studying the prime graph of a group has been an important topic for almost the past half-century. Recently, prime graphs of solvable groups have been characterized in graph theoretical terms only. This now allows the study of these graphs without any knowledge of the group theoretical background. In this paper we study prime graphs from a linear algebra angle and focus on the class of minimally connected prime graphs introduced in earlier work on the subject. As our main results, we determine the determinants of the adjacency matrices and the spectra of some important families of these graphs.

math.CO↗