arXiv subjects

Jinlong Shu

Publications and source records attributed to Jinlong Shu.

14 recordsLinked to original sources

arXiv2026

Clique supersaturation under a chromatic constraint below the Tur\'{a}n threshold

A central theme in extremal graph theory is the supersaturation problem, which investigates the minimum number of copies of a target subgraph forced by prescribed edge conditions. This line of research goes back to Rademacher and Erd\H{o}s for triangles, and was later extended to cliques by Lov\'asz and Simonovits in the regime above the Tur\'an threshold. Mubayi further extended this theory to color-critical graphs. Below the Tur\'an threshold, a closely related existence-threshold phenomenon arises in the non-$p$-partite setting: a classical result of Brouwer shows that, for $n\ge 2p+1$, every $n$-vertex non-$p$-partite $K_{p+1}$-free graph has at most $e(T_{n,p})-\lfloor n/p\rfloor+1$ edges. Motivated by this threshold, we investigate a sharp clique-counting problem below the Tur\'an threshold under the non-$p$-partite assumption. Let $p\ge 2$ and $s\ge 1$ be fixed integers. Let $Y_{n,p,s}$ be the graph obtained from $T_{n,p}$ by adding an edge inside a largest part and deleting all but $s$ of the edges from one endpoint of this new edge to a smallest part. Then $e(Y_{n,p,s})=e(T_{n,p})-\lfloor n/p\rfloor+s+1$. We prove that, for all sufficiently large $n$, every $n$-vertex non-$p$-partite graph $G$ with $e(G)\ge e(Y_{n,p,s})$ contains at least as many copies of $K_{p+1}$ as $Y_{n,p,s}$ does. The bound is sharp, as it is attained by the construction $Y_{n,p,s}$. Thus our result provides the exact clique-counting analogue of Brouwer's threshold for non-$p$-partite $K_{p+1}$-free graphs.

math.CO↗

arXiv2026

On a spectral booksize problem fo non bipartite graphs

The $\text{bk}(G)$ of a graph $G$ is the maximum number of triangles sharing a common edge. Motivated by a classical conjecture of Erd\H{o}s, spectral lower bounds for the booksize have received considerable attention. For a positive divisor $s$ of $m-1$ with $\frac{m-1}{s}\ge2$, let $S_{m,s}^{+}$ be obtained from $K_{s,\frac{m-1}{s}}$ by adding one edge inside the part of order $\frac{m-1}{s}$. Zhai et al. proved that, apart from this explicit family, every $m$-edge non-bipartite graph satisfying $\rho(G)^2\ge m-1+\frac{2}{\rho(G)-1}$ has booksize greater than $\frac{1}{240}\sqrt{m}$, and they asked for the best possible constant. We answer this question asymptotically. For every $0<\varepsilon<\frac{1}{4}$ and all sufficiently large $m$, every $m$-edge non-bipartite graph $G$ without isolated vertices satisfying the same spectral condition either is isomorphic to $S_{m,s}^{+}$ for some such integer $s$, or satisfies $\text{bk}(G)>\left(\frac{1}{4}-\varepsilon\right)\sqrt{m}$. We also give infinitely many graphs outside the exceptional family showing that no constant larger than $\frac{1}{4}$ is possible. Thus $\frac{1}{4}$ is the optimal asymptotic constant in the problem of Zhai et al.

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arXiv2025

Toughness in regular graphs from eigenvalues

The {\it toughness} $τ(G)=\mathrm{min}\{\frac{|S|}{c(G-S)}: S~\mbox{is a vertex cut in}~G\}$ for $G\ncong K_n,$ which was initially proposed by Chvátal in 1973. A graph $G$ is called {\it $t$-tough} if $τ(G)\geq t.$ Let $λ_i(G)$ be the $i$-th largest eigenvalue of the adjacency matrix of a graph $G$. In 1996, Brouwer conjectured that $τ(G)\geq\frac{d}λ-1$ for a connected $d$-regular graph $G,$ where $λ=\mathrm{max}\{|λ_2|, |λ_n|\}.$ Gu [SIAM J. Discrete Math. 35 (2021) 948-952] completely confirmed this conjecture. From Brouwer and Gu's result $τ(G)\geq\frac{d}λ-1,$ we know that if $G$ is a connected $d$-regular graph and $λ\leq\frac{bd}{b+1}$, then $τ(G)\geq\frac{1}{b}$ for an integer $b\geq1.$ Inspired by the above result and utilizing typical spectral techniques and graph construction methods from Cioabă et al. [J. Combin. Theory Ser. B 99 (2009) 287-297], we prove that if $G$ is a connected $d$-regular graph and $λ_2(G)<ϕ(d,b)$, then $τ(G)\geq\frac{1}{b}$. Meanwhile, we construct graphs implying that the upper bound on $λ_2(G)$ is best possible. Our theorem strengthens the result of Chen et al. [Discrete Math. 348 (2025) 114404]. Finally, we also prove an upper bound of $λ_{b+1}(G)$ to guarantee a connected $d$-regular graph to be $\frac{1}{b}$-tough.

math.CO↗

arXiv2025

On the Tur\'{a}n number of odd-ballooning of $3$-chromatic graphs

Given a graph $F$, the Tur\'{a}n number ${\rm ex}(n,F)$ is the maximum number of edges in any $n$-vertex $F$-free graph. The odd-ballooning of $F$, denoted by $F^{o}$, is a graph obtained by replacing each edge of $F$ with an odd cycle, where all new vertices of the odd cycles are distinct. The Tur\'{a}n number of the odd-ballooning of $F$ has been established for several important cases. For a star, it was determined by Erd\H{o}s, F\"{u}redi, Gould, and Gunderson (1995), Hou, Qiu, and Liu (2018), and Yuan (2018); for trees under certain conditions, by Zhu and Chen (2023); and for complete bipartite graphs $K_{s,t}$ ($t\geq s \geq 2$) where each substituted odd cycle has length at least five, by Peng and Xia (2024). In this paper, we apply Simonovits' celebrated method of progressive induction to determine the Tur\'{a}n number for the odd-ballooning of a class of $3$-chromatic graphs. Specifically, let $F$ be a graph formed by connecting a single vertex to all vertices of another graph whose components are either non-trivial trees or even cycles. We determine ${\rm ex}(n,F^{o})$ when each substituted odd cycle in $F^{o}$ has length at least five. As corollaries, we obtain the Tur\'{a}n number for the odd-ballooning of several well-known graph classes, including odd wheels, fan graphs, book graphs, and friendship graphs, where each substituted odd cycle in the ballooning has length at least five.

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arXiv2025

Let's Be Self-generated via Step by Step: A Curriculum Learning Approach to Automated Reasoning with Large Language Models

While Chain of Thought (CoT) prompting approaches have significantly consolidated the reasoning capabilities of large language models (LLMs), they still face limitations that require extensive human effort or have performance needs to be improved. Existing endeavors have focused on bridging these gaps; however, these approaches either hinge on external data and cannot completely eliminate manual effort, or they fall short in effectively directing LLMs to generate high-quality exemplary prompts. To address the said pitfalls, we propose a novel prompt approach for automatic reasoning named \textbf{LBS3}, inspired by curriculum learning which better reflects human learning habits. Specifically, LBS3 initially steers LLMs to recall easy-to-hard proxy queries that are pertinent to the target query. Following this, it invokes a progressive strategy that utilizes exemplary prompts stemmed from easy-proxy queries to direct LLMs in solving hard-proxy queries, enabling the high-quality of the proxy solutions. Finally, our extensive experiments in various reasoning-intensive tasks with varying open- and closed-source LLMs show that LBS3 achieves strongly competitive performance compared to the SOTA baselines.

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arXiv2024

Privacy-Preserving Federated Learning with Consistency via Knowledge Distillation Using Conditional Generator

Federated Learning (FL) is gaining popularity as a distributed learning framework that only shares model parameters or gradient updates and keeps private data locally. However, FL is at risk of privacy leakage caused by privacy inference attacks. And most existing privacy-preserving mechanisms in FL conflict with achieving high performance and efficiency. Therefore, we propose FedMD-CG, a novel FL method with highly competitive performance and high-level privacy preservation, which decouples each client's local model into a feature extractor and a classifier, and utilizes a conditional generator instead of the feature extractor to perform server-side model aggregation. To ensure the consistency of local generators and classifiers, FedMD-CG leverages knowledge distillation to train local models and generators at both the latent feature level and the logit level. Also, we construct additional classification losses and design new diversity losses to enhance client-side training. FedMD-CG is robust to data heterogeneity and does not require training extra discriminators (like cGAN). We conduct extensive experiments on various image classification tasks to validate the superiority of FedMD-CG.

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arXiv2024

DFDG: Data-Free Dual-Generator Adversarial Distillation for One-Shot Federated Learning

Federated Learning (FL) is a distributed machine learning scheme in which clients jointly participate in the collaborative training of a global model by sharing model information rather than their private datasets. In light of concerns associated with communication and privacy, one-shot FL with a single communication round has emerged as a de facto promising solution. However, existing one-shot FL methods either require public datasets, focus on model homogeneous settings, or distill limited knowledge from local models, making it difficult or even impractical to train a robust global model. To address these limitations, we propose a new data-free dual-generator adversarial distillation method (namely DFDG) for one-shot FL, which can explore a broader local models' training space via training dual generators. DFDG is executed in an adversarial manner and comprises two parts: dual-generator training and dual-model distillation. In dual-generator training, we delve into each generator concerning fidelity, transferability and diversity to ensure its utility, and additionally tailor the cross-divergence loss to lessen the overlap of dual generators' output spaces. In dual-model distillation, the trained dual generators work together to provide the training data for updates of the global model. At last, our extensive experiments on various image classification tasks show that DFDG achieves significant performance gains in accuracy compared to SOTA baselines.

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arXiv2024

An LLM-Enhanced Adversarial Editing System for Lexical Simplification

Lexical Simplification (LS) aims to simplify text at the lexical level. Existing methods rely heavily on annotated data, making it challenging to apply in low-resource scenarios. In this paper, we propose a novel LS method without parallel corpora. This method employs an Adversarial Editing System with guidance from a confusion loss and an invariance loss to predict lexical edits in the original sentences. Meanwhile, we introduce an innovative LLM-enhanced loss to enable the distillation of knowledge from Large Language Models (LLMs) into a small-size LS system. From that, complex words within sentences are masked and a Difficulty-aware Filling module is crafted to replace masked positions with simpler words. At last, extensive experimental results and analyses on three benchmark LS datasets demonstrate the effectiveness of our proposed method.

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arXiv2024

Spectral extremal results on trees

Let ${\rm spex}(n,F)$ be the maximum spectral radius over all $F$-free graphs of order $n$, and ${\rm SPEX}(n,F)$ be the family of $F$-free graphs of order $n$ with spectral radius equal to ${\rm spex}(n,F)$. Given integers $n,k,p$ with $n>k>0$ and $0\leq p\leq \lfloor(n-k)/2\rfloor$, let $S_{n,k}^{p}$ be the graph obtained from $K_k\nabla(n-k)K_1$ by embedding $p$ independent edges within its independent set, where `$\nabla$' means the join product. For $n\geq\ell\geq 4$, let $G_{n,\ell}=S_{n,(\ell-2)/2}^{0}$ if $\ell$ is even, and $G_{n,\ell}=S_{n,(\ell-3)/2}^{1}$ if $\ell$ is odd. Cioabă, Desai and Tait [SIAM J. Discrete Math. 37 (3) (2023) 2228--2239] showed that for $\ell\geq 6$ and sufficiently large $n$, if $ρ(G)\geq ρ(G_{n,\ell})$, then $G$ contains all trees of order $\ell$ unless $G=G_{n,\ell}$. They further posed a problem to study ${\rm spex}(n,F)$ for various specific trees $F$. Fix a tree $F$ of order $\ell\geq 6$, let $A$ and $B$ be two partite sets of $F$ with $|A|\leq |B|$, and set $q=|A|-1$. We first show that any graph in ${\rm SPEX}(n,F)$ contains a spanning subgraph $K_{q,n-q}$ for $q\geq 1$ and sufficiently large $n$. Consequently, $ρ(K_{q,n-q})\leq {\rm spex}(n,F)\leq ρ(G_{n,\ell})$, we further respectively characterize all trees $F$ with these two equalities holding. Secondly, we characterize the spectral extremal graphs for some specific trees and provide asymptotic spectral extremal values of the remaining trees. In particular, we characterize the spectral extremal graphs for all spiders, surprisingly, the extremal graphs are not always the spanning subgraph of $G_{n,\ell}$.

math.CO↗

arXiv2023

Toughness and distance spectral radius in graphs involving minimum degree

The toughness $τ(G)=\mathrm{min}\{\frac{|S|}{c(G-S)}: S~\mbox{is a cut set of vertices in}~G\}$ for $G\ncong K_n.$ The concept of toughness initially proposed by Chv$\mathrm{\acute{a}}$tal in 1973, which serves as a simple way to measure how tightly various pieces of a graph hold together. A graph $G$ is called $t$-tough if $τ(G)\geq t.$ It is very interesting to investigate the relations between toughness and eigenvalues of graphs. Fan, Lin and Lu [European J. Combin. 110 (2023) 103701] provided sufficient conditions in terms of the spectral radius for a graph to be 1-tough with minimum degree $δ$ and $t$-tough with $t\geq 1$ being an integer, respectively. By using some typical distance spectral techniques and structural analysis, we in this paper present sufficient conditions based on the distance spectral radius to guarantee a graph to be 1-tough with minimum degree $δ.$ Moreover, we also prove sufficient conditions with respect to the distance spectral radius for a graph to be $t$-tough, where $t$ or $\frac{1}{t}$ is a positive integer.

math.CO↗

arXiv2022

The maximum spectral radius of irregular bipartite graphs

A bipartite graph is subcubic if it is an irregular bipartite graph with maximum degree three. In this paper, we prove that the asymptotic value of maximum spectral radius over subcubic bipartite graphs of order $n$ is $3-\varTheta(\frac{π^{2}}{n^{2}})$. Our key approach is taking full advantage of the eigenvalues of certain tridiagonal matrices, due to Willms [SIAM J. Matrix Anal. Appl. 30 (2008) 639--656]. Moreover, for large maximum degree, i.e., the maximum degree is at least $\lfloor n/2 \rfloor$, we characterize irregular bipartite graphs with maximum spectral radius. For general maximum degree, we present an upper bound on the spectral radius of irregular bipartite graphs in terms of the order and maximum degree.

math.CO↗

arXiv2021

Spectral extrema of graphs with fixed size: cycles and complete bipartite graphs

Nikiforov [Some inequalities for the largest eigenvalue of a graph, Combin. Probab. Comput. 179--189] showed that if $G$ is $K_{r+1}$-free then the spectral radius $ρ(G)\leq\sqrt{2m(1-1/r)}$, which implies that $G$ contains $C_3$ if $ρ(G)>\sqrt{m}$. In this paper, we follow this direction on determining which subgraphs will be contained in $G$ if $ρ(G)> f(m)$, where $f(m)\sim\sqrt{m}$ as $m\rightarrow \infty$. We first show that if $ρ(G)\geq \sqrt{m}$, then $G$ contains $K_{2,r+1}$ unless $G$ is a star; and $G$ contains either $C_3^+$ or $C_4^+$ unless $G$ is a complete bipartite graph, where $C_t^+$ denotes the graph obtained from $C_t$ and $C_3$ by identifying an edge. Secondly, we prove that if $ρ(G)\geq{\frac12+\sqrt{m-\frac34}}$, then $G$ contains pentagon and hexagon unless $G$ is a book; and if $ρ(G)>{\frac12(k-\frac12)+\sqrt{m+\frac14(k-\frac12)^2}}$, then $G$ contains $C_t$ for every $t\leq 2k+2$. In the end, some related conjectures are provided for further research.

math.CO↗

arXiv2020

On the $A_α$-spectra of graphs

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For any real $α\in [0,1]$, Nikiforov \cite{VN1} defined the matrix $A_α(G)$ as $$A_α(G)=αD(G)+(1-α)A(G).$$ In this paper, we give some results on the eigenvalues of $A_α(G)$ with $α>1/2$. In particular, we show that for each $e\notin E(G)$, $λ_i(A_α(G+e))\geqλ_i(A_α(G))$. By utilizing the result, we prove have $λ_k(A_α(G))\leqαn-1$ for $2\leq k\leq n$. Moreover, we characterize the extremal graphs with equality holding. Finally, we show that $λ_n(A_α({G}))\geq 2α-1$ if $G$ contains no isolated vertices.

math.CO↗

arXiv2017

More results on the distance (signless) Laplacian eigenvalues of graphs

Let $G$ be a connected graph with vertex set $V(G)$ and edge set $E(G)$. Let $Tr(G)$ be the diagonal matrix of vertex transmissions of $G$ and $D(G)$ be the distance matrix of $G$. The distance Laplacian matrix of $G$ is defined as $\mathcal{L}(G)=Tr(G)-D(G)$. The distance signless Laplacian matrix of $G$ is defined as $\mathcal{Q}(G)=Tr(G)+D(G)$. In this paper, we give a lower bound on the distance Laplacian spectral radius in terms of $D_1$, as a consequence, we show that $\partial_1^L(G)\geq n+\lceil\frac{n}ω\rceil$ where $ω$ is the clique number of $G$. Furthermore, we give some graft transformations, by using them, we characterize the extremal graph attains the maximum distance spectral radius in terms of $n$ and $ω$. Moreover, we also give bounds on the distance signless Laplacian eigenvalues of $G$, and give a confirmation on a conjecture due to Aouchiche and Hansen.

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