arXiv subjects

Zhishuo Zhang

Publications and source records attributed to Zhishuo Zhang.

11 recordsLinked to original sources

arXiv2026

Asymptotic stability of Cayley graphs on abelian groups

For a finite group $G$, we say that a Cayley graph $Γ$ on $G$ is a most rigid representation (MRR) of $G$ if its full automorphism group has the smallest possible order among all Cayley graphs on $G$, and say that $Γ$ is stable if every automorphism of $Γ\times K_2$ comes from $\mathrm{Aut}(Γ)\times\Aut(K_2)$. Although the study of stability has attracted significant attention, particularly regarding Cayley graphs on abelian groups, a complete classification is currently out of reach even for Cayley graphs on cyclic groups. In this paper, we prove that almost all Cayley graphs on finite abelian groups are stable MRRs. This strengthens the main result of Dobson, Spiga and Verret [Combinatorica, 36 (2016), no.~4, 371--393], which states that almost all Cayley graphs on finite abelian groups are MRRs.

math.CO↗

arXiv2026

DPDSyn: Improving Differentially Private Dataset Synthesis for Model Training by Downstream Task Guidance

How to synthesize a dataset while achieving differential privacy for AI model training is a meaningful but challenging problem. To address this problem, state-of-the-art methods first select original private dataset's multiple low-dimensional distributions that have the potential to approximate the distribution of original private dataset with high precision, and then synthesize a dataset obeying all selected low-dimensional distributions as the synthetic dataset. However, it is difficult to select suitable low-dimensional distributions, which in turn degrades the data utility of resulting synthetic dataset. To improve differentially private dataset synthesis, we propose to train a differentially private AI model for downstream tasks on the original private dataset and utilize the trained model to synthesize datasets. In particular, on the one hand, the AI model satisfies differential privacy so no matter how to use the model does not disclose private information of original private dataset. On the other hand, the AI model is trained to complete the downstream task so the AI model preserves critical information for completing downstream tasks. We utilize the AI model to synthesize datasets to achieve the goal of improving data utility while preserving privacy. Empirical evaluations on four benchmark datasets demonstrate that our proposed DPDSyn consistently outperforms eight state-of-the-art baselines with a maximum improvement of 2.40x in accuracy and 333.73x in synthesis efficiency. Further experiments also validate that DPDSyn has strong scalability across varying data scales.

cs.CR↗

arXiv2025

Which maximal subgroups are perfect codes?

A perfect code in a graph $Γ=(V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A subgroup $H$ of a group $G$ is called a subgroup perfect code of $G$ if it is a perfect code in some Cayley graph of $G$. In this paper, we undertake a systematic study of which maximal subgroups of a group can be perfect codes. Our approach highlights a characterization of subgroup perfect codes in terms of their ``local'' complements.

math.CO↗

arXiv2025

Determining the vertex stabilizers of 4-valent half-arc-transitive graphs

We say that a group is a $4$-HAT-stabilizer if it is the vertex stabilizer of some connected $4$-valent half-arc-transitive graph. In 2001, Marušič and Nedela proved that every $4$-HAT-stabilizer must be a concentric group. However, over the past two decades, only a very small proportion of concentric groups have been shown to be $4$-HAT-stabilizers. This paper develops a theory that provides a general framework for determining whether a concentric group is a $4$-HAT-stabilizer. With this approach, we significantly extend the known list of $4$-HAT-stabilizers. As a corollary, we confirm that $\mathcal{H}_7\times C_2^{m-7}$ are $4$-HAT-stabilizers for $m\geq 7$, achieving the goal of a conjecture posed by Spiga and Xia.

math.CO↗

arXiv2025

Isomorphisms of bi-Cayley graphs on generalized quaternion groups

Let $G$ be a finite group and $S$ be a subset of $G$. The bi-Cayley graph $\mathrm{BCay}(G,S)$ is the graph with vertex set $G\times \{0,1\}$ and edge set $\{\{(x,0),(sx,1)\}\mid x\in G,s\in S\}$. A bi-Cayley graph $\mathrm{BCay}(G,S)$ is called a BCI-graph if for every $T\subseteq G$, the isomorphism $\mathrm{BCay}(G,S)\cong \mathrm{BCay}(G,T)$ implies that $T=gS^α$ for some $g\in G$ and $α\in \mathrm{Aut}(G)$. We say a group $G$ an $m$-BCI-group if every bi-Cayley graphs of $G$ with valency at most $m$ is a BCI-graph. In this paper, we show that for $m\in\{2,3\}$, the generalized quaternion group of order $4n$ with $n\geq 2$ is an $m$-BCI-group if and only if it is an $m$-DCI-group if and only if it is an $m$-CI-group if and only if $n$ is odd or $n=2$.

math.CO↗

arXiv2025

On subgroup perfect codes in vertex-transitive graphs

A subset $C$ of the vertex set $V$ of a graph $Γ$ is called a perfect code in $Γ$ if every vertex in $V\setminus C$ is adjacent to exactly one vertex in $C$. Given a group $G$ and a subgroup $H$ of $G$, a subgroup $A$ of $G$ containing $H$ is called a perfect code of the pair $(G,H)$ if there exists a coset graph $\mathrm{Cos}(G,H,U)$ such that the set of left cosets of $H$ in $A$ is a perfect code in $\mathrm{Cos}(G,H,U)$. In particular, $A$ is called a perfect code of $G$ if $A$ is a perfect code of the pair $(G,1)$. In this paper, we give a characterization of $A$ to be a perfect code of the pair $(G,H)$ under the assumption that $H$ is a perfect code of $G$. As a corollary, we derive an additional sufficient and necessary condition for $A$ to be a perfect code of $G$. Moreover, we establish conditions under which $A$ is not a perfect code of $(G,H)$, which is applied to construct infinitely many counterexamples to a question posed by Wang and Zhang [\emph{J.~Combin.~Theory~Ser.~A}, 196 (2023) 105737]. Furthermore, we initiate the study of determining which maximal subgroups of $S_n$ are perfect codes.

math.CO↗

arXiv2024

A complete classification of shuffle groups

For positive integers $k$ and $n$, the shuffle group $G_{k,kn}$ is generated by the $k!$ permutations of a deck of $kn$ cards performed by cutting the deck into $k$ piles with $n$ cards in each pile, and then perfectly interleaving these cards following a certain permutation of the $k$ piles. For $k=2$, the shuffle group $G_{2,2n}$ was determined by Diaconis, Graham and Kantor in 1983. The Shuffle Group Conjecture states that, for general $k$, the shuffle group $G_{k,kn}$ contains $\mathrm{A}_{kn}$ whenever $k\notin\{2,4\}$ and $n$ is not a power of $k$. In particular, the conjecture in the case $k=3$ was posed by Medvedoff and Morrison in 1987. The only values of $k$ for which the Shuffle Group Conjecture has been confirmed so far are powers of $2$, due to recent work of Amarra, Morgan and Praeger based on Classification of Finite Simple Groups. In this paper, we confirm the Shuffle Group Conjecture for all cases using results on $2$-transitive groups and elements of large fixed point ratio in primitive groups.

math.GR↗

arXiv2023

Type-enhanced Ensemble Triple Representation via Triple-aware Attention for Cross-lingual Entity Alignment

Entity alignment(EA) is a crucial task for integrating cross-lingual and cross-domain knowledge graphs(KGs), which aims to discover entities referring to the same real-world object from different KGs. Most existing methods generate aligning entity representation by mining the relevance of triple elements via embedding-based methods, paying little attention to triple indivisibility and entity role diversity. In this paper, a novel framework named TTEA -- Type-enhanced Ensemble Triple Representation via Triple-aware Attention for Cross-lingual Entity Alignment is proposed to overcome the above issues considering ensemble triple specificity and entity role features. Specifically, the ensemble triple representation is derived by regarding relation as information carrier between semantic space and type space, and hence the noise influence during spatial transformation and information propagation can be smoothly controlled via specificity-aware triple attention. Moreover, our framework uses triple-ware entity enhancement to model the role diversity of triple elements. Extensive experiments on three real-world cross-lingual datasets demonstrate that our framework outperforms state-of-the-art methods.

cs.CL↗

arXiv2023

OTIEA:Ontology-enhanced Triple Intrinsic-Correlation for Cross-lingual Entity Alignment

Cross-lingual and cross-domain knowledge alignment without sufficient external resources is a fundamental and crucial task for fusing irregular data. As the element-wise fusion process aiming to discover equivalent objects from different knowledge graphs (KGs), entity alignment (EA) has been attracting great interest from industry and academic research recent years. Most of existing EA methods usually explore the correlation between entities and relations through neighbor nodes, structural information and external resources. However, the complex intrinsic interactions among triple elements and role information are rarely modeled in these methods, which may lead to the inadequate illustration for triple. In addition, external resources are usually unavailable in some scenarios especially cross-lingual and cross-domain applications, which reflects the little scalability of these methods. To tackle the above insufficiency, a novel universal EA framework (OTIEA) based on ontology pair and role enhancement mechanism via triple-aware attention is proposed in this paper without introducing external resources. Specifically, an ontology-enhanced triple encoder is designed via mining intrinsic correlations and ontology pair information instead of independent elements. In addition, the EA-oriented representations can be obtained in triple-aware entity decoder by fusing role diversity. Finally, a bidirectional iterative alignment strategy is deployed to expand seed entity pairs. The experimental results on three real-world datasets show that our framework achieves a competitive performance compared with baselines.

cs.CL↗

arXiv2021

Multistationarity of Reaction Networks with One-Dimensional Stoichiometric Subspaces

We study the multistationarity for the reaction networks with one-dimensional stoichiometric subspaces, and we focus on the networks admitting finitely many positive steady states. We prove that if a network admits multistationarity, then network has an embedded one-species network with arrow diagram (->,<-) and another with arrow diagram (<-,->). The inverse is also true if there exist two reactions in the network such that the subnetwork consisting of the two reactions admits at least one and finitely many positive steady states. We also prove that if a network admits at least three positive steady states, then it contains at least three bi-arrow diagrams. More than that, we completely characterize the bi-reaction networks that admit at least three positive steady states.

math.DS↗

arXiv2021

Multistability of Reaction Networks with One-Dimensional Stoichiometric Subspaces

For the reaction networks with one-dimensional stoichiometric subspaces, we show the following results. (1) If the maximum number of positive steady states is an even number N, then the maximum number of stable positive steady states is N/2. (2) If the maximum number of positive steady states is an odd number N, then we provide a condition on the network such that the maximum number of stable positive steady states is (N-1)/2 if this condition is satisfied, and this maximum number is (N+1)/2 otherwise.

math.DS↗