arXiv subjects

Liying Kang

Publications and source records attributed to Liying Kang.

At least 19 recordsLinked to original sources

arXiv2026

Tur\'an-good monotonicity thresholds

A graph $H$ is $K_{r+1}$-Tur\'an-good if, for every sufficiently large $n$, the Tur\'an graph $T_r(n)$ maximizes the number of copies of $H$ among all $n$-vertex $K_{r+1}$-free graphs. It is strictly $K_{r+1}$-Tur\'an-good if $T_r(n)$ is the unique extremal graph. Morrison, Nir, Norin, Rz\k{a}\.zewski and Wesolek [\emph{JCTB}, 2023] proved that every graph $H$ is $K_{r+1}$-Tur\'an-good whenever $r\ge 300v(H)^9$. They raised the following two questions: 1.Can the sufficient condition $r\ge 300v(H)^9$ be reduced to a condition of quadratic order in $v(H)$? 2.Is the Tur\'an-good property monotone in $r$? More precisely, if a graph $H$ is $K_r$-Tur\'an-good, must it also be $K_{r+1}$-Tur\'an-good? We affirmatively resolve the first question and derive an even stronger bound linear in the edge number: every graph $H$ with at least one edge is strictly $K_{r+1}$-Tur\'an-good and $K_{r+1}$-Tur\'an-stable whenever $r\ge 168e(H)$. This condition is quadratic in $v(H)$ for arbitrary graphs and linear in $v(H)$ for every sparse graph family with $e(H)=O(v(H))$. We answer the second question negatively. For every $r\ge3$, there exists a graph that is strictly $K_r$-Tur\'an-good but not $K_{r+1}$-Tur\'an-good. More quantitatively, for every sufficiently large $h$, there exists a graph $H$ with $v(H)\le h$ and an integer $r=h-2\sqrt h+O(1)$ such that $H$ is strictly $K_r$-Tur\'an-good but not $K_{r+1}$-Tur\'an-good. The monotonicity threshold $\lambda(H)$ is the least integer $R\ge 2$ such that, for every $r\ge R$, the graph $H$ is $K_{r+1}$-Tur\'an-good whenever it is $K_r$-Tur\'an-good. For \[ \lambda_{\max}(h)=\max\{\lambda(H)\mid v(H)\le h\}, \] our two results yield \[ h-2\sqrt h-O(1)\le \lambda_{\max}(h)\le 84h^2. \]

math.CO↗

arXiv2026

A Spectral Confirmation of the Erd\H{o}s Matching Conjecture

The Erd\H{o}s Matching Conjecture concerns the maximum number of hyperedges in an $r$-uniform hypergraph with bounded matching number. In this paper, we study a spectral counterpart of this conjecture. For sufficiently large $n$, we determine the maximum spectral radius over all $n$-vertex $r$-uniform hypergraphs whose matching number is less than $s$, and characterize the unique extremal hypergraph. To establish the main theorem, we first apply the shifting method to reduce the problem to shifted hypergraphs. We then derive several spectral upper bounds through hypergraph decomposition and related variational estimates for tensor spectral radii. With these estimates, we analyze the structural properties of shifted-saturated hypergraphs and prove the spectral extremal theorem for shifted hypergraphs with bounded matching numbers. Finally, we drop the shifted condition and extend our spectral bound to general $r$-uniform hypergraphs. Our main theorem states that for any $n$-vertex $r$-uniform hypergraph $H$ with matching number $\nu(H)<s$, the inequality $\rho(H)\leq \rho(\mathcal{F}_{s-1}(n))$ holds whenever $n$ is sufficiently large. Here $\mathcal{F}_{a}(n)$ denotes the family of all $r$-subsets of $[n]$ intersecting the vertex set $[a]$, and equality is attained if and only if $H$ is isomorphic to $\mathcal{F}_{s-1}(n)$. As an immediate corollary, we derive a spectral counterpart of the classical Erd\H{o}s-Ko-Rado theorem for intersecting hypergraph families.

math.CO↗

arXiv2026

Three characterizations of the weighted center of imputations value

The weighted center of imputations (CIS) value allocates the surplus of the grand coalition equally after granting each player a fixed proportion of his individual worth. This paper provides three axiomatic characterizations of this value by generalizing the individual rationality and subgame order preservation axioms. The first characterization employs individual rationality with respect to the weights together with the equal surplus increment property. The second relies on efficiency, additivity, symmetry adjusted by the weights, and a dummifying player property adapted to the weights. The third builds on efficiency and a weak subgame order preservation axiom that incorporates the weights. These results unify and extend recent findings, covering both the equal division and the standard CIS values as special cases.

econ.TH↗

arXiv2026

The spectral inducibility of graphs

We introduce a spectral version of the classical inducibility problem. Given an $\ell$-vertex graph $F$ and an $n$-vertex graph $G$, let $H_F(G)$ be the $\ell$-uniform hypergraph whose edges are the $\ell$-sets inducing a copy of $F$ in $G$. We study the maximum possible $\alpha$-spectral radius of $H_F(G)$ over all $n$-vertex graphs $G$. For fixed $G$, this spectral parameter tends to $\ell!$ times the number of induced copies of $F$ in $G$ as $\alpha\to\infty$, and therefore refines the usual induced-copy count. Our main result is a spectral analogue of the Brown--Sidorenko reduction: for every complete multipartite graph $F$, every $n$, and every $\alpha\ge1$, a spectral extremal graph can be chosen to be complete multipartite. We also show that the leading asymptotic constant is the ordinary inducibility $i(F)$, and obtain exact multipartite reductions for stars $K_{1,t}$ and balanced complete $r$-partite graphs $K_{a,\ldots,a}$ with $r\le 2^a-1$.

math.CO↗

arXiv2026

Preference Tuning as Spectral Update Reorganization

Preference-based post-training is usually understood through endpoint behavior, yet the learned update that produces this behavior remains largely opaque. We study RLHF and related preference optimization through the spectral structure of their induced parameter updates. By decomposing effective LoRA updates and reloading their spectral components as plug-in modules, we turn preference-induced updates into objects that can be isolated, recomposed, and directly intervened on. Across model families, optimization algorithms, and supervision regimes, these updates consistently develop a spectral head--tail organization. A compact head emerges early and carries the dominant endpoint shift, while a heterogeneous residual tail remains. The split is functional rather than merely descriptive. Plug-in intervention shows that the head accounts for the visible behavioral departure from the base model, while the tail is weak in isolation. Cross-run recomposition further shows that mixed adapters follow the source of the head, indicating that the head carries run-level solver bias. This endpoint dominance does not imply learning sufficiency. Head-only learning is non-vacuous but fails to recover the full solution, especially on out-of-distribution behavior. Tail-only learning yields little visible gain, yet the full solution is not recovered without the tail. These findings recast preference post-training as structured update reorganization rather than a monolithic behavioral correction, and suggest that alignment gain and coverage loss are tied to how the learned update itself is organized.

cs.CL↗

arXiv2026

Spectral Turán Problems for Expanded hypergraphs

Given a graph $F$, the expansion $F^{(r)}$ of $F$ is defined as the $r$-uniform hypergraph obtained from $F$ by adding a set of $(r-2)$ distinct new vertices to each edge of $F$. In this paper, we investigate spectral stability results for hypergraphs and their applications.We first establish a spectral stability property: for any $r$-uniform hypergraph containing no copy of the expansion $F^{(r)}$ of a $(k+1)$-chromatic graph $F$, if its $p$-spectral is close to the extremal value, then the hypergraph is structurally close to $T_r(n, k)$, the complete $k$-partite $r$-uniform hypergraph on $n$ vertices where sizes of any two parts differ by at most one.Using this spectral stability result, we determine the unique extremal hypergraph that maximizes the $p$-spectral radius among all $n$-vertex $r$-uniform hypergraphs without $t$ vertex-disjoint copies of the expansion $K_{k+1}^{(r)}$ of $K_{k+1}$. We prove that this extremal hypergraph is isomorphic to $K_{t-1}^{r} \,\vee\, T_r(n-t+1, k)$, the join of the complete $r$-uniform hypergraph $K_{t-1}^{r}$ and $T_r(n-t+1, k)$.As a corollary, we show that $K_{t-1}^{r} \,\vee\, T_r(n-t+1, k)$ is the unique extremal hypergraph for $tK_{k+1}^{(r)}$, which extends a result of Pikhurko [J. Combin. Theory Ser. B, 103 (2013) 220--225] for expanded complete graphs.

math.CO↗

arXiv2026

The signless Laplacian spectral Turán problems for hypergraphs

Let $\mathcal{H}=(V, E)$ be an $r$-uniform hypergraph on $n$ vertices. The signless Laplacian spectral radius of $\mathcal{H}$ is defined as the maximum modulus of the eigenvalues of the tensor $\mathcal{Q}(\mathcal{H})=\mathcal{D}(\mathcal{H})+\mathcal{A}(\mathcal{H})$, where $\mathcal{D}(\mathcal{H})$ and $\mathcal{A}(\mathcal{H})$ are the degree diagonal tensor and the adjacency tensor of $\mathcal{H}$, respectively. In this paper, we establish a general theorem that extends the spectral Turán result of Keevash, Lenz and Mubayi [SIAM J. Discrete Math., 28 (4) (2014)] to the setting of signless Laplacian spectral Turán problems. We prove that if a family $\mathcal{F}$ of $r$-uniform hypergraphs is degree-stable with respect to a family $\mathcal{H}_n$ of $r$-uniform hypergraphs and its extremal constructions satisfy certain natural assumptions, then the signless Laplacian spectral Turán problem for $\mathcal{F}$ can be effectively reduced to the corresponding problem restricted to the family $\mathcal{H}_n$. As a concrete application, we completely determine the extremal hypergraph that maximizes the signless Laplacian spectral radius among all Fano plane-free $3$-uniform hypergraphs, showing that the unique extremal hypergraph is the balanced complete bipartite $3$-uniform hypergraph.

math.CO↗

arXiv2025

The $α$-spectral Turán type problems for graphs

For $0 \leq α< 1$, the $α$-spectral radius of a graph $G$ is defined as the largest eigenvalue of $A_α(G)=αD(G)+(1-α)A(G)$, where $D(G)$ and $A(G)$ are the diagonal matrix of degrees and adjacency matrix of $G$, respectively. A graph is called color-critical if it contains an edge whose deletion reduces its chromatic number. The celebrated Erdős-Stone-Simonovits theorem asserts that $ \mathrm{ex}(n,\mathcal{F})=\left(1-\frac{1}{χ(\mathcal{F})-1}+o(1)\right)\frac{n^2}{2},$ where $χ(\mathcal{F})$ is the chromatic number of $\mathcal{F}$. Nikiforov and Zheng et al. established the adjacency spectral and signless Laplacian spectral versions of this theorem, respectively. In this paper, we present the $α$-spectral version of this theorem, which unifies the aforementioned results. Furthermore, we characterize the $α$-spectral extremal graphs for color-critical graphs, thereby extending the existing results on adjacency spectral and signless Laplacian spectral extremal graphs for such graphs.

math.CO↗

arXiv2025

Spectral extremal problems for degenerate graphs

A family of graphs is called degenerate if it contains at least one bipartite graph. In this paper, we investigate the spectral extremal problems for a degenerate family of graphs $\mathcal{F}$. By employing covering and independent covering of graphs, we establish a spectral stability result for $\mathcal{F}$. Using this stability result, we prove two general theorems that characterize spectral extremal graphs for a broad class of graph families $\mathcal{F}$ and imply several new and known results. Meanwhile, we establish the correlation between extremal graphs and spectral extremal graphs for $\mathcal{F}$.

math.CO↗

arXiv2025

GuardVal: Dynamic Large Language Model Jailbreak Evaluation for Comprehensive Safety Testing

Jailbreak attacks reveal critical vulnerabilities in Large Language Models (LLMs) by causing them to generate harmful or unethical content. Evaluating these threats is particularly challenging due to the evolving nature of LLMs and the sophistication required in effectively probing their vulnerabilities. Current benchmarks and evaluation methods struggle to fully address these challenges, leaving gaps in the assessment of LLM vulnerabilities. In this paper, we review existing jailbreak evaluation practices and identify three assumed desiderata for an effective jailbreak evaluation protocol. To address these challenges, we introduce GuardVal, a new evaluation protocol that dynamically generates and refines jailbreak prompts based on the defender LLM's state, providing a more accurate assessment of defender LLMs' capacity to handle safety-critical situations. Moreover, we propose a new optimization method that prevents stagnation during prompt refinement, ensuring the generation of increasingly effective jailbreak prompts that expose deeper weaknesses in the defender LLMs. We apply this protocol to a diverse set of models, from Mistral-7b to GPT-4, across 10 safety domains. Our findings highlight distinct behavioral patterns among the models, offering a comprehensive view of their robustness. Furthermore, our evaluation process deepens the understanding of LLM behavior, leading to insights that can inform future research and drive the development of more secure models.

cs.LG↗

arXiv2025

REVOLVE: Optimizing AI Systems by Tracking Response Evolution in Textual Optimization

Recent advancements in large language models (LLMs) have significantly enhanced the ability of LLM-based systems to perform complex tasks through natural language processing and tool interaction. However, optimizing these LLM-based systems for specific tasks remains challenging, often requiring manual interventions like prompt engineering and hyperparameter tuning. Existing automatic optimization methods, such as textual feedback-based techniques (e.g., TextGrad), tend to focus on immediate feedback, analogous to using immediate derivatives in traditional numerical gradient descent. However, relying solely on such feedback can be limited when the adjustments made in response to this feedback are either too small or fluctuate irregularly, potentially slowing down or even stalling the optimization process. To overcome these challenges, more adaptive methods are needed, especially in situations where the system's response is evolving slowly or unpredictably. In this paper, we introduce REVOLVE, an optimization method that tracks how "R"esponses "EVOLVE" across iterations in LLM systems. By focusing on the evolution of responses over time, REVOLVE enables more stable and effective optimization by making thoughtful, progressive adjustments at each step. Experimental results demonstrate that REVOLVE outperforms competitive baselines, achieving a 7.8% improvement in prompt optimization, a 20.72% gain in solution refinement, and a 29.17% increase in code optimization. Additionally, REVOLVE converges in fewer iterations, resulting in significant computational savings. Beyond its practical contributions, REVOLVE highlights a promising direction, where the rich knowledge from established optimization principles can be leveraged to enhance LLM systems, which paves the way for further advancements in this hybrid domain.

cs.CL↗

arXiv2025

On generalized Tur{á}n problems with bounded matching number and circumference

Let $ \mathcal{F} $ be a family of graphs. The generalized Turán number $ \operatorname{ex}(n, K_r, \mathcal{F}) $ is the maximum number of $K_r$ in an $ n $-vertex graph that does not contain any member of $ \mathcal{F} $ as a subgraph. Recently, Alon and Frankl initiated the study of Turán problems with bounded matching number. In this paper, we determine the generalized Turán number of $ C_{\geq k} $ with bounded matching number.

math.CO↗

arXiv2025

Spectral bipartite Turan problems on linear hypergraphs

Let $F$ be a graph, and let $\mathcal{B}_r(F)$ be the class of $r$-uniform Berge-$F$ hypergraphs. In this paper, we establish a relationship between the spectral radius of the adjacency tensor of a uniform hypergraph and its local structure through walks. Based on the relationship, we give a spectral asymptotic bound for $\mathcal{B}_{r}(C_3)$-free linear $r$-uniform hypergraphs and upper bounds for the spectral radii of $\mathcal{B}_{r}(K_{2,t})$-free or $\{\mathcal{B}_{r}(K_{s,t}),\mathcal{B}_{r}(C_{3})\}$-free linear $r$-uniform hypergraphs, where $C_{3}$ and $K_{s,t}$ are respectively the triangle and the complete bipartite graph with one part having $s$ vertices and the other part having $t$ vertices. Our work implies an upper bound for the number of edges of $\{\mathcal{B}_{r}(K_{s,t}),\mathcal{B}_{r}(C_{3})\}$-free linear $r$-uniform hypergraphs and extends some of the existing research on (spectral) extremal problems of hypergraphs.

math.CO↗

arXiv2024

On generalized Turán problems with bounded matching number

The generalized Turán number $\mathrm{ex}(n, H, \mathcal{F})$ is defined as the maximum number of copies of a graph $H$ in an $n$-vertex graph that does not contain any graph $F \in \mathcal{F}$. Alon and Frankl initiated the study of Turán problems with a bounded matching number.In this paper, we establish stability results for generalized Turán problems with bounded matching number.Using the stability results, we provide exact values of $\ex(n,K_r,\{F,M_{s+1}\})$ for $F$ being any non-bipartite graph or a path on $k$ vertices.

math.CO↗

arXiv2024

Spectral Extremal Graphs of Planar Graphs with Fixed Size

Tait and Tobin [J. Combin. Theory Ser. B 126 (2017) 137--161] determined the unique spectral extremal graph over all outerplanar graphs and the unique spectral extremal graph over all planar graphs when the number of vertices is sufficiently large. In this paper we consider the spectral extremal problems of outerplanar graphs and planar graphs with fixed number of edges. We prove that the outerplanar graph on $m \geq 64$ edges with the maximum spectral radius is $S_m$, where $S_m$ is a star with $m$ edges. For planar graphs with $m$ edges, our main result shows that the spectral extremal graph is $K_2 \vee \frac{m-1}{2} K_1$ when $m$ is odd and sufficiently large, and $K_1 \vee (S_{\frac{m-2}{2}} \cup K_1)$ when $m$ is even and sufficiently large. Additionally, we obtain spectral extremal graphs for path, cycle and matching in outerplanar graphs and spectral extremal graphs for path, cycle and complete graph on $4$ vertices in planar graphs.

math.CO↗

arXiv2024

Hypergraph Extensions of Spectral Turán Theorem

The spectral Turán theorem states that the $k$-partite Turán graph is the unique graph attaining the maximum adjacency spectral radius among all graphs of order $n$ containing no the complete graph $K_{k+1}$ as a subgraph. This result is known to be stronger than the classical Turán theorem. In this paper, we consider hypergraph extensions of spectral Turán theorem. For $k\geq r\geq 2$, let $H_{k+1}^{(r)}$ be the $r$-uniform hypergraph obtained from $K_{k+1}$ by enlarging each edge with a new set of $(r-2)$ vertices. Let $F_{k+1}^{(r)}$ be the $r$-uniform hypergraph with edges: $\{1,2,\ldots,r\} =: [r]$ and $E_{ij} \cup\{i,j\}$ over all pairs $\{i,j\}\in \binom{[k+1]}{2}\setminus\binom{[r]}{2}$, where $E_{ij}$ are pairwise disjoint $(r-2)$-sets disjoint from $[k+1]$. Generalizing the Turán theorem to hypergraphs, Pikhurko [J. Combin. Theory Ser. B, 103 (2013) 220--225] and Mubayi and Pikhurko [J. Combin. Theory Ser. B, 97 (2007) 669--678] respectively determined the exact Turán number of $H_{k+1}^{(r)}$ and $F_{k+1}^{(r)}$, and characterized the corresponding extremal hypergraphs. Our main results show that $T_r(n,k)$, the complete $k$-partite $r$-uniform hypergraph on $n$ vertices where no two parts differ by more than one in size, is the unique hypergraph having the maximum $p$-spectral radius among all $n$-vertex $H_{k+1}^{(r)}$-free (resp. $F_{k+1}^{(r)}$-free) $r$-uniform hypergraphs for sufficiently large $n$. These findings are obtained by establishing $p$-spectral version of the stability theorems. Our results offer $p$-spectral analogues of the results by Mubayi and Pikhurko, and connect both hypergraph Turán theorem and hypergraph spectral Turán theorem in a unified form via the $p$-spectral radius.

math.CO↗

arXiv2024

GPT4Rec: Graph Prompt Tuning for Streaming Recommendation

In the realm of personalized recommender systems, the challenge of adapting to evolving user preferences and the continuous influx of new users and items is paramount. Conventional models, typically reliant on a static training-test approach, struggle to keep pace with these dynamic demands. Streaming recommendation, particularly through continual graph learning, has emerged as a novel solution. However, existing methods in this area either rely on historical data replay, which is increasingly impractical due to stringent data privacy regulations; or are inability to effectively address the over-stability issue; or depend on model-isolation and expansion strategies. To tackle these difficulties, we present GPT4Rec, a Graph Prompt Tuning method for streaming Recommendation. Given the evolving user-item interaction graph, GPT4Rec first disentangles the graph patterns into multiple views. After isolating specific interaction patterns and relationships in different views, GPT4Rec utilizes lightweight graph prompts to efficiently guide the model across varying interaction patterns within the user-item graph. Firstly, node-level prompts are employed to instruct the model to adapt to changes in the attributes or properties of individual nodes within the graph. Secondly, structure-level prompts guide the model in adapting to broader patterns of connectivity and relationships within the graph. Finally, view-level prompts are innovatively designed to facilitate the aggregation of information from multiple disentangled views. These prompt designs allow GPT4Rec to synthesize a comprehensive understanding of the graph, ensuring that all vital aspects of the user-item interactions are considered and effectively integrated. Experiments on four diverse real-world datasets demonstrate the effectiveness and efficiency of our proposal.

cs.IR↗

arXiv2024

High-Frequency-aware Hierarchical Contrastive Selective Coding for Representation Learning on Text-attributed Graphs

We investigate node representation learning on text-attributed graphs (TAGs), where nodes are associated with text information. Although recent studies on graph neural networks (GNNs) and pretrained language models (PLMs) have exhibited their power in encoding network and text signals, respectively, less attention has been paid to delicately coupling these two types of models on TAGs. Specifically, existing GNNs rarely model text in each node in a contextualized way; existing PLMs can hardly be applied to characterize graph structures due to their sequence architecture. To address these challenges, we propose HASH-CODE, a High-frequency Aware Spectral Hierarchical Contrastive Selective Coding method that integrates GNNs and PLMs into a unified model. Different from previous "cascaded architectures" that directly add GNN layers upon a PLM, our HASH-CODE relies on five self-supervised optimization objectives to facilitate thorough mutual enhancement between network and text signals in diverse granularities. Moreover, we show that existing contrastive objective learns the low-frequency component of the augmentation graph and propose a high-frequency component (HFC)-aware contrastive learning objective that makes the learned embeddings more distinctive. Extensive experiments on six real-world benchmarks substantiate the efficacy of our proposed approach. In addition, theoretical analysis and item embedding visualization provide insights into our model interoperability.

cs.IR↗