arXiv subjects

Xiaodan Chen

Publications and source records attributed to Xiaodan Chen.

8 recordsLinked to original sources

arXiv2026

Characterizing the equality case in Brouwer's inequality for Laplacian eigenvalues

Brouwer conjectured that the sum of the $k$ largest Laplacian eigenvalues of an $n$-vertex graph is less than or equal to the number of its edges plus $\binom{k+1}{2}$ for every $k\in \{1,2,\dots,n\}$, which has been confirmed by Kothari and Tudose (2026) recently. In this note, we characterize the equality case in this inequality. Our main result is that for every $n$-vertex graph $G=(V,E)$ and for every $k\in \{1,2,\dots,n-1\}$, the equality $\sum_{i=1}^k\mu_i(G)=|E(G)|+\binom{k+1}{2}$ holds if and only if $G$ is a threshold graph with clique number $k+1$, where $\mu_1(G)\geq \mu_2(G)\geq \cdots\geq \mu_{n}(G)$ are the Laplacian eigenvalues of $G$. This, together with the confirmed Brouwer's conjecture, would yield a complete solution to the full Brouwer's conjecture posed by Li and Guo (2022). Our proof relies on the projection method of Kothari and Tudose and shows directly that the equality case can occur only for threshold graphs.

math.CO↗

arXiv2026

Structure from rank: Rank-order coding as a bridge from sequence to structure

Understanding how structured sequence information can be represented and generalized in neural systems is key to modeling the transition from acoustic input to emergent structure. In this study, we propose a rank-order based neural network inspired by the STG-LIFG-PMC pathway, modeling the bottom-up transition from acoustic input to abstract rank representation and the top-down generation from that representation to motor execution. Building on previous work in rank coding, we first demonstrate that this model efficiently compresses input while retaining the capacity to reconstruct full utterances from partial cues, revealing emergent structure-sensitive generation process that reflects context-general representations of sensorimotor states, which are later shaped into context-specific motor plans during speech planning. We then show that the network exhibits global-level novelty detection similar to the P3B novelty wave, replicating the global-sequence-sensitive mechanism. As a supplement, we also compare the model's behavior under local (index-level) and global (rank-level) perturbations, revealing robustness to superficial variation and sensitivity to abstract structural violation, key features associated with hierarchical generalization. These results suggest that rank-order coding not only serves as a compact encoding scheme but also captures hierarchical structure in acoustic sequences.

cs.NE↗

arXiv2025

Confidence-Based Self-Training for EMG-to-Speech: Leveraging Synthetic EMG for Robust Modeling

Voiced Electromyography (EMG)-to-Speech (V-ETS) models reconstruct speech from muscle activity signals, facilitating applications such as neurolaryngologic diagnostics. Despite its potential, the advancement of V-ETS is hindered by a scarcity of paired EMG-speech data. To address this, we propose a novel Confidence-based Multi-Speaker Self-training (CoM2S) approach, along with a newly curated Libri-EMG dataset. This approach leverages synthetic EMG data generated by a pre-trained model, followed by a proposed filtering mechanism based on phoneme-level confidence to enhance the ETS model through the proposed self-training techniques. Experiments demonstrate our method improves phoneme accuracy, reduces phonological confusion, and lowers word error rate, confirming the effectiveness of our CoM2S approach for V-ETS. In support of future research, we will release the codes and the proposed Libri-EMG dataset-an open-access, time-aligned, multi-speaker voiced EMG and speech recordings.

cs.SD↗

arXiv2025

More on the full Brouwer Laplacian spectrum conjecture

Brouwer conjectured that the sum of the first $k$ largest Laplacian eigenvalues of an $n$-vertex graph is less than or equal to the number of its edges plus $\binom{k+1}{2}$ for each $k\in \{1,2,\cdots,n\}$, which has come to be known as Brouwer's conjecture. Recently, Li and Guo further considered the case when the equalities hold in these conjectured inequalities, and proposed the full version of Brouwer's conjecture. In this paper, we first present a concise version of the full Brouwer's conjecture. Then we show that the full Brouwer's conjecture holds for two families of spanning subgraphs of complete split graphs and for $c$-cyclic graphs with $c\in\{0,1,2\}$. We also consider the Nordhaus-Gaddum version of the full Brouwer's conjecture and present partial solutions to it.

math.CO↗

arXiv2024

Developmental Predictive Coding Model for Early Infancy Mono and Bilingual Vocal Continual Learning

Understanding how infants perceive speech sounds and language structures is still an open problem. Previous research in artificial neural networks has mainly focused on large dataset-dependent generative models, aiming to replicate language-related phenomena such as ''perceptual narrowing''. In this paper, we propose a novel approach using a small-sized generative neural network equipped with a continual learning mechanism based on predictive coding for mono-and bilingual speech sound learning (referred to as language sound acquisition during ''critical period'') and a compositional optimization mechanism for generation where no learning is involved (later infancy sound imitation). Our model prioritizes interpretability and demonstrates the advantages of online learning: Unlike deep networks requiring substantial offline training, our model continuously updates with new data, making it adaptable and responsive to changing inputs. Through experiments, we demonstrate that if second language acquisition occurs during later infancy, the challenges associated with learning a foreign language after the critical period amplify, replicating the perceptual narrowing effect.

cs.AI↗

arXiv2023

Extremal results on degree powers in some classes of graphs

Let $G$ be a simple graph of order $n$ with degree sequence $(d_1,d_2,\cdots,d_n)$. For an integer $p>1$, let $e_p(G)=\sum_{i=1}^n d^{p}_i$ and let $ex_p(n,H)$ be the maximum value of $e_p(G)$ among all graphs with $n$ vertices that do not contain $H$ as a subgraph (known as $H$-free graphs). Caro and Yuster proposed the problem of determining the exact value of $ex_2(n,C_4)$, where $C_4$ is the cycle of length $4$. In this paper, we show that if $G$ is a $C_4$-free graph having $n\geq 4$ vertices and $m\leq \lfloor 3(n-1)/2\rfloor$ edges and no isolated vertices, then $e_p(G)\leq e_p(F_n)$, with equality if and only if $G$ is the friendship graph $F_n$. This yields that for $n\geq 4$, $ex_p(n,\mathcal{C}^*)=e_p(F_n)$ and $F_n$ is the unique extremal graph, which is an improved complement of Caro and Yuster's result on $ex_p(n,\mathcal{C}^*)$, where $\mathcal{C}^*$ denotes the family of cycles of even lengths. We also determine the maximum value of $e_p(\cdot)$ among all minimally $t$-(edge)-connected graphs with small $t$ or among all $k$-degenerate graphs, and characterize the corresponding extremal graphs. A key tool in our approach is majorization.

math.CO↗

arXiv2023

Extremal spectral radius of weighted adjacency matrices of bicyclic graphs

The weighted adjacency matrix $A_{f}(G)$ of a simple graph $G=(V,E)$ is the $|V|\times|V|$ matrix whose $ij$-entry equals $f(d_{i},d_j)$, where $f(x,y)$ is a symmetric function such that $f(d_i,d_j)>0$ if $ij\in E$ and $f(d_i,d_j)=0$ if $ij\notin E$ and $d_i$ is the degree of the vertex $i$. In this paper, we determine the unique graph having the largest spectral radius of $A_{f}(G)$ among all the bicyclic graphs under the assumption that $f(x,y)$ is increasing and convex in $x$ and $f(x_1,y_1)\geq f(x_2,y_2)$ when $|x_1-y_1|>|x_2-y_2|$ and $x_1+y_1=x_2+y_2$. Moreover, we determine the unique graph having the second largest spectral radius of $A_{f}(G)$ among all the bicyclic graphs when $f(x,y)=x+y$, $(x+y)^2$ or $x^2+y^2$, which corresponds to the well-known first Zagreb index, first hyper-Zagreb index, and forgotten index, respectively. In addition, we also characterize the bicyclic graphs with the first two largest spectral radii of $A_{f}(G)$ when $f(x,y)=\frac{1}{2}(x/y+y/x)$, corresponding to the extended index.

math.CO↗

arXiv2021

On trees with extremal extended spectral radius

Let G be a simple connected graph with n vertices, and let d_i be the degree of the vertex v_i in G. The extended adjacency matrix of G is defined so that the ij-entry is 1/2(d_i/d_j+d_j/d_i) if the vertices v_i and v_j are adjacent in G, and 0 otherwise. This matrix was originally introduced for developing novel topological indices used in the QSPR/QSAR studies. In this paper, we consider extremal problems of the largest eigenvalue of the extended adjacency matrix (also known as the extended spectral radius) of trees. We show that among all trees of order n>= 5, the path Pn(resp., the star Sn) uniquely minimizes (resp., maximizes) the extended spectral radius. We also determine the first five trees with the maximal extended spectral radius.

math.CO↗