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Chenhui Lv

Publications and source records attributed to Chenhui Lv.

11 recordsLinked to original sources
arXiv2026

The Aldous property for normal Cayley graphs on symmetric groups

Aldous' spectral gap conjecture states that the random walk and the interchange process on any connected graph have the same spectral gap, or, equivalently, the second largest eigenvalue of any connected Cayley graph on the symmetric group $S_n$ with respect to a set of transpositions is achieved by the standard representation of $S_n$. This celebrated conjecture, proved in its general form in 2010, has inspired much interest in searching for other Cayley graphs on $S_n$ possessing this property, now known as the Aldous property. In this paper, we first prove that for $n \ge 5$ at most one of a normal Cayley graph on $S_n$ and its complement can possess the Aldous property except when these two graphs are $2K_{n!/2}$ and $K_{n!/2,n!/2}$ respectively. We then determine, for sufficiently large $n$, all normal Cayley graphs $\mathrm{Cay}(S_n, S)$ that have the Aldous property, except for the case when $S$ contains a permutation with support size in $\{2, 3, \dots, n-2\}$ and a permutation with support size in $\{n-1, n\}$, but not all permutations with support size $n$ are contained in $S$. In particular, we show that a non-complete normal Cayley graph $\mathrm{Cay}(S_n, S)$ does not have the Aldous property if all permutations in $S$ have support size $n-1$ or $n$, or all permutations with support size $n$ are contained in $S$, thereby solving an open problem posed by Li, Xia and Zhou in 2023. Along the way we determine all normal Cayley graphs on $S_n$ that are line graphs, and classify all normal Cayley graphs on $S_n$ with the strictly second largest eigenvalue at most $1$.

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arXiv2026

Amply regular graphs with $\mu$ close to half the valency and group divisible designs

In this paper, we classify connected amply regular graphs with diameter $d \geq 4$ and parameters $(v, k, \lambda, \mu)$ satisfying $\mu = \frac{k-1}{2}$, where $k\geq 5$ is odd. We prove that such a graph must be exactly one of the following: the $5$-cube, the graph $\K_2 \square \Lambda$, where $\Lambda$ is the unique bipartite $(0,2)$-graph on $14$ vertices, or the point--block incidence graph of a group divisible design with the dual property, namely a $GDDDP\left(2, k+1;\, k;\, 0, \frac{k-1}{2}\right)$. For the last family, we give equivalent characterizations in terms of bipartite $Q$-regular graphs and relation graphs of symmetric association schemes with five classes. Furthermore, we present constructions of such amply regular graphs, yielding infinite families of examples derived from Paley graphs, Peisert graphs, and Paley digraphs.

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arXiv2026

Kleitman's theorem over vector spaces: parity phenomena in canonical and global stability

In 1966, Kleitman determined the maximum size of a family of subsets of $[n]$ with bounded symmetric difference. Liao, Liu and Yan recently established a vector-space analogue in the cases $n=d+1$ and $n>2d$, and asked for the sharp bound in the remaining range. We resolve this problem completely by proving the exact vector-space analogue of Kleitman's theorem for every $n\ge d+1$, and we also determine all extremal configurations. We further develop a stability theory for the vector-space diameter problem. Unlike the Boolean cube, the lattice of subspaces has no translation symmetry, and this makes the stability theory substantially different from its classical counterpart. The geometry of subspace balls leads to two natural notions: canonical stability, which forbids containment only in the canonical extremal configurations, and global stability, which forbids containment in arbitrary balls or adjacent double balls of the corresponding radius. We determine sharp canonical stability in even diameter, sharp canonical and global stability in odd diameter, and prove a nontrivial general upper bound for global stability in even diameter. In particular, these two notions exhibit a sharp parity split: in odd diameter they collapse to the same problem, whereas in even diameter they lead to genuinely different extremal behavior.

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arXiv2026

On the characterization of geometric distance-regular graphs

In 2010, Koolen and Bang proposed the following conjecture: For a fixed integer $m \geq 2$, any geometric distance-regular graph with smallest eigenvalue $-m$, diameter $D \geq 3$ and $c_2 \geq 2$ is either a Johnson graph, a Grassmann graph, a Hamming graph, a bilinear forms graph, or the number of vertices is bounded above by a function of $m$. In this paper, we obtain some partial results towards this conjecture.

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arXiv2025

Comparing classes of highly symmetric graphs: From $2$-arc-transitive to $2$-distance-transitive

A $2$-distance-transitive graph is a vertex-transitive graph whose vertex stabilizer is transitive on both the first- and second-step neighborhoods. This concept simultaneously generalizes both distance-transitive graphs and $2$-arc-transitive graphs. In this paper, we first determine the vertex-quasiprimitive types of $2$-distance-transitive graphs of odd order, partially answering a question posed by A. Devillers, M. Giudici, C. H. Li and C. E. Praeger in 2012. We then prove that a $2$-distance-transitive graph of valency $p+1$, where $p$ is a prime, is $2$-arc-transitive if and only if it has girth at least $4$. We also show that every locally-primitive $2$-distance-transitive graph of valency at most $8$ is $2$-arc-transitive, with the icosahedron as the unique exception. Finally, we prove that if $\Gamma$ is a $G$-locally-primitive, $(G,2)$-distance-transitive graph of valency at least $3$ and $G$ is soluble, then either $\Gamma\cong \K_{p,p}$ for some prime $p$, or the order of $\Gamma$ is not square-free.

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arXiv2025

An improved bound for strongly regular graphs with smallest eigenvalue $-m$

In 1979, Neumaier gave a bound on $\lambda$ in terms of $m$ and $\mu$, where $-m$ is the smallest eigenvalue of a primitive strongly regular graph, unless the graph in question belongs to one of the two infinite families of strongly regular graphs. We improve this result. We also indicate how our methods can be used to give an alternate derivation of Bruck's Completion Theorem for orthogonal arrays.

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arXiv2025

A Bose-Laskar-Hoffman theory for $\mu$-bounded graphs with fixed smallest eigenvalue

In 2018, by Ramsey and Hoffman theory, Koolen, Yang, and Yang presented a structural result on graphs with smallest eigenvalue at least $-3$ and large minimum degree. In this study, we depart from the conventional use of Ramsey theory and instead employ a novel approach that combines the Bose-Laskar type argument with Hoffman theory to derive structural insights into $\mu$-bounded graphs with fixed smallest eigenvalue. Our method establishes a reasonable bound on the minimum degree. Note that local graphs of distance-regular graphs are $\mu$-bounded. We apply these results to characterize the structure for any local graph of a distance-regular graph with classical parameters $(D,b,\alpha,\beta)$. Consequently, we show that the parameter $\alpha$ is bounded by a cubic polynomial in $b$ if $D \geq 9$ and $b \geq 2$. We also show that $\alpha \leq 2$ if $b =2$ and $D \geq 12$.

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arXiv2024

Bounding the parameter $\beta$ of a distance-regular graph with classical parameters

Let $\Gamma$ be a distance-regular graph with classical parameters $(D, b, \alpha, \beta)$ satisfying $b\geq 2$ and $D\geq 3$. Let $r=1+b+b^2+\cdots+b^{D-1}$. In 1999, K. Metsch showed that there exists a positive constant $C(\alpha,b)$ only depending on $\alpha$ and $b$, such that if $\beta \geq C(\alpha, b)r^2$, then either $\Gamma$ is a Grassmann graph or a bilinear forms graph. In this work, we show that for $b\geq 2$ and $D\geq 3$, then there exists a constant $C_1(\alpha, b)$ only depending on $\alpha$ and $b$, such that if $\beta \geq C_1(\alpha, b)r$, then either $\Gamma$ is a Grassmann graph, or a bilinear forms graph.

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arXiv2020

Literature Triage on Genomic Variation Publications by Knowledge-enhanced Multi-channel CNN

Background: To investigate the correlation between genomic variation and certain diseases or phenotypes, the fundamental task is to screen out the concerning publications from massive literature, which is called literature triage. Some knowledge bases, including UniProtKB/Swiss-Prot and NHGRI-EBI GWAS Catalog are created for collecting concerning publications. These publications are manually curated by experts, which is time-consuming. Moreover, the manual curation of information from literature is not scalable due to the rapidly increasing amount of publications. In order to cut down the cost of literature triage, machine-learning models were adopted to automatically identify biomedical publications. Methods: Comparing to previous studies utilizing machine-learning models for literature triage, we adopt a multi-channel convolutional network to utilize rich textual information and meanwhile bridge the semantic gaps from different corpora. In addition, knowledge embeddings learned from UMLS is also used to provide extra medical knowledge beyond textual features in the process of triage. Results: We demonstrate that our model outperforms the state-of-the-art models over 5 datasets with the help of knowledge embedding and multiple channels. Our model improves the accuracy of biomedical literature triage results. Conclusions: Multiple channels and knowledge embeddings enhance the performance of the CNN model in the task of biomedical literature triage. Keywords: Literature Triage; Knowledge Embedding; Multi-channel Convolutional Network

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arXiv2018

The spectral radius of graphs without trees of diameter at most four

Nikiforov (LAA, 2010) conjectured that for given integer $k$, any graph $G$ of sufficiently large order $n$ with spectral radius $μ(G)\geq μ(S_{n,k})$ contains all trees of order $2k+2$, unless $G=S_{n,k}$, where $S_{n,k}=K_k\vee \overline{K_{n-k}}$, the join of a complete graph of order $k$ and an empty graph of order $n-k$. In this paper, we show that the conjecture is true for trees of diameter at most four.

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