Searcharxiv⌕ Search

arXiv subjects

Zilin Jiang

Publications and source records attributed to Zilin Jiang.

At least 19 recordsLinked to original sources
arXiv2026

Bounds on median eigenvalues of graphs of bounded degree

We prove that for every integer $d \ge 3$, the median eigenvalues of any graph of maximum degree $d$ are bounded above by $\sqrt{d-1}$. We also prove that, in three separate cases, the median eigenvalues of a graph of maximum degree $d$ are bounded below by $-\sqrt{d-1}$: when the graph is triangle-free, when $d-1$ is a perfect square, or when $d \ge 75$. These results resolve, for all but finitely many values of $d$, an open problem of Mohar on median eigenvalues of graphs of maximum degree $d$. As a byproduct, we establish an upper bound on the average energy of graphs of maximum degree at most $d$, generalizing a previous result of van Dam, Haemers, and Koolen for $d$-regular graphs.

math.CO↗
arXiv2026

Classification of equiangular lines with fixed angle $\arccos(1/(1+2\sqrt2))$

We determine the maximum number $N_α(d)$ of equiangular lines with fixed angle $\arccosα$ for $α= 1/(1+2\sqrt2)$ in $d$-dimensional Euclidean space: $2,3,4,6,8,10,14,15,16,17,18,20,22$ for $d \in \{2,\dots,14\}$, and $\max(24, \lfloor 3(d-1)/2 \rfloor)$ for $d \ge 15$. This appears to be the first complete determination of $N_α(d)$ in all dimensions $d$ for a fixed nontrivial $α$, since the work of Lemmens and Seidel for $α= 1/3$ in 1973.

math.CO↗
arXiv2026

Subcubic graphs without eigenvalues in $(-1, 1)$

Guo and Royle recently classified the connected cubic graphs without eigenvalues of their adjacency matrix in the open interval $(-1, 1)$, and raised the question of extending their classification to graphs of maximum degree at most $3$. Together with their cubic classification, our result fully answers this question by characterizing all connected subcubic graphs that are not cubic and have no eigenvalues in $(-1,1)$. We show that exactly two infinite families and seven sporadic examples occur, and that every sporadic graph has at most $18$ vertices. To obtain this complete classification, we build a bridge between spectral graph theory and structural graph theory for graphs whose adjacency matrices, after selected diagonal entries are changed to $-1$, have smallest eigenvalue at least $-2$. This generalizes the classical theorem of Cameron, Goethals, Seidel and Shult for graphs with smallest eigenvalue at least $-2$. As a consequence, we prove that $(-1,1)$ is a maximal spectral gap set for the class of connected subcubic graphs. Guo and Royle, answering a question of Koll\'ar and Sarnak, established this maximality for connected cubic graphs. Our result generalizes their conclusion to the subcubic setting.

math.CO↗
arXiv2025

Beyond the classification theorem of Cameron, Goethals, Seidel, and Shult

In 1976, Cameron, Goethals, Seidel, and Shult classified all the graphs whose smallest eigenvalue is at least $-2$ by relating such graphs to root systems that appear in the classification of semisimple Lie algebras. In this paper, extending their beautiful theorem, we give a complete classification of all connected graphs whose smallest eigenvalue lies in $(-λ^*, -2)$, where $λ^* = ρ^{1/2} + ρ^{-1/2} \approx 2.01980$, and $ρ$ is the unique real root of $x^3 = x + 1$. Our result is the first classification of infinitely many connected graphs with their smallest eigenvalue in $(-λ, -2)$ for any constant $λ> 2$.

math.CO↗
arXiv2025

Forbidden induced subgraphs for graphs and signed graphs with eigenvalues bounded from below

The smallest eigenvalue of a graph is the smallest eigenvalue of its adjacency matrix. We show that the family of graphs with smallest eigenvalue at least $-λ$ can be defined by a finite set of forbidden induced subgraphs if and only if $λ< λ^*$, where $λ^* = ρ^{1/2} + ρ^{-1/2} \approx 2.01980$, and $ρ$ is the unique real root of $x^3 = x + 1$. This resolves a question raised by Bussemaker and Neumaier. As a byproduct, we find all the limit points of smallest eigenvalues of graphs, supplementing Hoffman's work on those limit points in $[-2, \infty)$. We also prove that the same conclusion about forbidden subgraph characterization holds for signed graphs. Our impetus for the study of signed graphs is to determine the maximum cardinality of a spherical two-distance set with two fixed angles (one acute and one obtuse) in high dimensions. Denote by $N_{α, β}(n)$ the maximum number of unit vectors in $\mathbb{R}^d$ where all pairwise inner products lie in $\{α, β\}$ with $-1 \le β< 0 \le α< 1$. Very recently Jiang, Tidor, Yao, Zhang and Zhao determined the limit of $N_{α, β}(d)/d$ as $d\to\infty$ when $α+ 2β< 0$ or $(1-α)/(α-β) \in \{1,\sqrt2,\sqrt3\}$, and they proposed a conjecture on the limit in terms of eigenvalue multiplicities of signed graphs. We establish their conjecture whenever $(1-α)/(α- β) < λ^*$.

math.CO↗
arXiv2025

On the smallest eigenvalues of $3$-colorable graphs

We prove that the set of the smallest eigenvalues attained by $3$-colorable graphs is dense in $(-\infty, -λ^*)$, where $λ^* = ρ^{1/2} + ρ^{-1/2} \approx 2.01980$ and $ρ$ is the positive real root of $x^3 = x + 1$. As a consequence, in the context of spherical two-distance sets, our result precludes any further refinement of the forbidden-subgraph method through the chromatic number of signed graphs.

math.CO↗
arXiv2025

Median eigenvalues of subcubic graphs

We show that the median eigenvalues of every connected graph of maximum degree at most three, except for the Heawood graph, are at most $1$ in absolute value, resolving open problems posed by Fowler and Pisanski, and by Mohar.

math.CO↗
arXiv2025

On symmetric hollow integer matrices with eigenvalues bounded from below

A hollow matrix is a square matrix whose diagonal entries are all equal to zero. Define $λ^* = ρ^{1/2} + ρ^{-1/2} \approx 2.01980$, where $ρ$ is the unique real root of $x^3 = x + 1$. We show that for every $λ< λ^*$, there exists $n \in \mathbb{N}$ such that if a symmetric hollow integer matrix has an eigenvalue less than $-λ$, then one of its principal submatrices of order at most $n$ does as well. However, the same conclusion does not hold for any $λ\ge λ^*$.

math.CO↗
arXiv2024

A survey on fairness of large language models in e-commerce: progress, application, and challenge

This survey explores the fairness of large language models (LLMs) in e-commerce, examining their progress, applications, and the challenges they face. LLMs have become pivotal in the e-commerce domain, offering innovative solutions and enhancing customer experiences. This work presents a comprehensive survey on the applications and challenges of LLMs in e-commerce. The paper begins by introducing the key principles underlying the use of LLMs in e-commerce, detailing the processes of pretraining, fine-tuning, and prompting that tailor these models to specific needs. It then explores the varied applications of LLMs in e-commerce, including product reviews, where they synthesize and analyze customer feedback; product recommendations, where they leverage consumer data to suggest relevant items; product information translation, enhancing global accessibility; and product question and answer sections, where they automate customer support. The paper critically addresses the fairness challenges in e-commerce, highlighting how biases in training data and algorithms can lead to unfair outcomes, such as reinforcing stereotypes or discriminating against certain groups. These issues not only undermine consumer trust, but also raise ethical and legal concerns. Finally, the work outlines future research directions, emphasizing the need for more equitable and transparent LLMs in e-commerce. It advocates for ongoing efforts to mitigate biases and improve the fairness of these systems, ensuring they serve diverse global markets effectively and ethically. Through this comprehensive analysis, the survey provides a holistic view of the current landscape of LLMs in e-commerce, offering insights into their potential and limitations, and guiding future endeavors in creating fairer and more inclusive e-commerce environments.

cs.CL↗
arXiv2023

Negligible obstructions and Turán exponents

We show that for every rational number $r \in (1,2)$ of the form $2 - a/b$, where $a, b \in \mathbb{N}^+$ satisfy $\lfloor b/a \rfloor^3 \le a \le b / (\lfloor b/a \rfloor +1) + 1$, there exists a graph $F_r$ such that the Turán number $\operatorname{ex}(n, F_r) = Θ(n^r)$. Our result in particular generates infinitely many new Turán exponents. As a byproduct, we formulate a framework that is taking shape in recent work on the Bukh--Conlon conjecture.

math.CO↗
arXiv2022

Spherical two-distance sets and eigenvalues of signed graphs

We study the problem of determining the maximum size of a spherical two-distance set with two fixed angles (one acute and one obtuse) in high dimensions. Let $N_{α,β}(d)$ denote the maximum number of unit vectors in $\mathbb R^d$ where all pairwise inner products lie in $\{α,β\}$. For fixed $-1\leqβ<0\leqα<1$, we propose a conjecture for the limit of $N_{α,β}(d)/d$ as $d \to \infty$ in terms of eigenvalue multiplicities of signed graphs. We determine this limit when $α+2β<0$ or $(1-α)/(α-β) \in \{1, \sqrt{2}, \sqrt{3}\}$. Our work builds on our recent resolution of the problem in the case of $α= -β$ (corresponding to equiangular lines). It is the first determination of $\lim_{d \to \infty} N_{α,β}(d)/d$ for any nontrivial fixed values of $α$ and $β$ outside of the equiangular lines setting.

math.CO↗
arXiv2022

Equiangular lines with a fixed angle

Solving a longstanding problem on equiangular lines, we determine, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given angle. Fix $0 < α< 1$. Let $N_α(d)$ denote the maximum number of lines through the origin in $\mathbb{R}^d$ with pairwise common angle $\arccos α$. Let $k$ denote the minimum number (if it exists) of vertices in a graph whose adjacency matrix has spectral radius exactly $(1-α)/(2α)$. If $k < \infty$, then $N_α(d) = \lfloor k(d-1)/(k-1) \rfloor$ for all sufficiently large $d$, and otherwise $N_α(d) = d + o(d)$. In particular, $N_{1/(2k-1)}(d) = \lfloor k(d-1)/(k-1) \rfloor$ for every integer $k\ge 2$ and all sufficiently large $d$. A key ingredient is a new result in spectral graph theory: the adjacency matrix of a connected bounded degree graph has sublinear second eigenvalue multiplicity.

math.CO↗
arXiv2021

On the binary adder channel with complete feedback, with an application to quantitative group testing

We determine the exact value of the optimal symmetric rate point $(r, r)$ in the Dueck zero-error capacity region of the binary adder channel with complete feedback. We proved that the average zero-error capacity $r = h(1/2-δ) \approx 0.78974$, where $h(\cdot)$ is the binary entropy function and $δ= 1/(2\log_2(2+\sqrt3))$. Our motivation is a problem in quantitative group testing. Given a set of $n$ elements two of which are defective, the quantitative group testing problem asks for the identification of these two defectives through a series of tests. Each test gives the number of defectives contained in the tested subset, and the outcomes of previous tests are assumed known at the time of designing the current test. We establish that the minimum number of tests is asymptotic to $(\log_2 n) / r$ as $n \to \infty$.

cs.IT↗
arXiv2021

Rainbow odd cycles

We prove that every family of (not necessarily distinct) odd cycles $O_1, \dots, O_{2\lceil n/2 \rceil-1}$ in the complete graph $K_n$ on $n$ vertices has a rainbow odd cycle (that is, a set of edges from distinct $O_i$'s, forming an odd cycle). As part of the proof, we characterize those families of $n$ odd cycles in $K_{n+1}$ that do not have any rainbow odd cycle. We also characterize those families of $n$ cycles in $K_{n+1}$, as well as those of $n$ edge-disjoint nonempty subgraphs of $K_{n+1}$, without any rainbow cycle.

math.CO↗
arXiv2020

Cooperative colorings of trees and of bipartite graphs

Given a system $(G_1, \ldots ,G_m)$ of graphs on the same vertex set $V$, a cooperative coloring is a choice of vertex sets $I_1, \ldots ,I_m$, such that $I_j$ is independent in $G_j$ and $\bigcup_{j=1}^{m}I_j = V$. For a class $\mathcal{G}$ of graphs, let $m_{\mathcal{G}}(d)$ be the minimal $m$ such that every $m$ graphs from $\mathcal{G}$ with maximum degree $d$ have a cooperative coloring. We prove that $Ω(\log\log d) \le m_\mathcal{T}(d) \le O(\log d)$ and $Ω(\log d)\le m_\mathcal{B}(d) \le O(d/\log d)$, where $\mathcal{T}$ is the class of trees and $\mathcal{B}$ is the class of bipartite graphs.

math.CO↗
arXiv2019

Rainbow fractional matchings

We prove that any family $E_1, \ldots , E_{\lceil rn \rceil}$ of (not necessarily distinct) sets of edges in an $r$-uniform hypergraph, each having a fractional matching of size $n$, has a rainbow fractional matching of size $n$ (that is, a set of edges from distinct $E_i$'s which supports such a fractional matching). When the hypergraph is $r$-partite and $n$ is an integer, the number of sets needed goes down from $rn$ to $rn-r+1$. The problem solved here is a fractional version of the corresponding problem about rainbow matchings, which was solved by Drisko and by Aharoni and Berger in the case of bipartite graphs, but is open for general graphs as well as for $r$-partite hypergraphs with $r>2$. Our topological proof is based on a result of Kalai and Meshulam about a simplicial complex and a matroid on the same vertex set.

math.CO↗