arXiv subjects

Jicheng Ma

Publications and source records attributed to Jicheng Ma.

14 recordsLinked to original sources

arXiv2026

Matching complements in subcubic graphs and a proof of the 3-Decomposition Conjecture

We prove the 3-Decomposition Conjecture: every finite connected cubic loopless multigraph decomposes into a spanning tree, a 2-regular subgraph, and a matching. The proof rests on a new theorem on matching complements in subcubic graphs. Let H be a finite connected bridgeless simple graph of maximum degree three, and let S be its set of degree-two vertices, with |S| = k >= 2. We show that H has a matching of size equal to its cyclomatic number, |E(H)| - |V(H)| + 1, whose deletion leaves a single tree containing all of S, together with cycles disjoint from S. The proof is by induction on k, using an alternating-path exchange that stops at the first entry into the growing tree component.

math.CO↗

arXiv2026

Evolving edge weights via local entropy flow and cohesion flow on graphs

In this paper, we first propose two different quantities on graphs, namely local entropy and cohesion, then design two corresponding flows for edge weights: the local entropy flow and the cohesion flow. We establish the global existence and uniqueness of solutions for both flows and investigate their asymptotic behaviors, including the case that the limit goes to positive infinity. Moreover, they can be applied to fundamental network analysis tasks, including community detection and node classification. Empirical evaluations demonstrate that our method achieves performance competitive with Ollivier Ricci flow and Lin-Lu-Yau Ricci flow on benchmark network analysis tasks. In experimental scenarios, we first apply the cohesion flow to evolve the edge weights of the graph, and then apply the local entropy flow to further update the resulting weighted graph. Both flows are computationally efficient, leading to a significant reduction in overall computational cost and improved scalability.

math.CA↗

arXiv2026

Large point-degrees in intersecting families of finite vector spaces

Let $V$ be an $n$-dimensional vector space over the finite field $\Fq$, and let $\Gr{V}{k}$ denote the family of all $k$-dimensional subspaces of $V$. A family $\cF\subseteq\Gr{V}{k}$ is called intersecting if $\dim(F\cap F')\ge1$ for all $F,F'\in\cF$. For a point $P\le V$, let $d_P(\cF)$ denote the number of members of $\cF$ that contain $P$, and order the point-degrees as $d_1(\cF)\ge d_2(\cF)\ge\cdots\ge d_{\points{n}}(\cF)$, where $\points{m}=(q^m-1)/(q-1)$ is the number of points in an $m$-dimensional subspace. Recent work of Frankl and Wang~\cite{FW2025} and of Huang and Rao~\cite{HR2026} established that for $k$-uniform intersecting families $\cF\subseteq\binom{[n]}{k}$ with $n\ge2k+1$, the bound $\binom{n-2}{k-2}$ governs the order statistic $d_{2k+1}(\cF)$. We prove that every intersecting family $\cF\subseteq\Gr{V}{k}$ with $n\ge2k+1$ satisfies $d_{\points{k}^{2}}(\cF)\le\qbinom{n-2}{k-2}$. The naive $q$-analog of the Huang--Rao $(k+2)$-th degree theorem fails, as a vector-space Hilton--Milner construction has $\points{k+1}$ points of degree larger than $\qbinom{n-2}{k-2}$; we prove the corrected bound $d_{1+\points{k+1}}(\cF)\le\qbinom{n-2}{k-2}$ for fixed $q$, sufficiently large $k$, and $n>3k$, using a structural theorem of Ihringer and Kupavskii~\cite{IK2026}. For larger degree indices $i$, a saturated Frankl--Hilton--Milner family of Ihringer and Kupavskii identifies the conjectural sharp bound on $d_{\points{i+1}}(\cF)$, and we prove two necessary conditions that any strict counterexample to this conjecture must satisfy.

math.CO↗

arXiv2026

On Permutation Groups of Cyclic Codes over Finite Fields

The permutation groups of cyclic codes are widely applicable in determining the weight distribution of codes, decoding theory and various other areas. In this paper, by employing two distinct matrix representations, we can relate cyclic codes with very long lengths and special generator polynomials to those with prime lengths. Consequently, we mainly determine the permutation groups of certain cyclic codes over $\mathbb{F}_{r^\alpha}$ with lengths $hp$, $r^mp^n$ and $pq$ and special generator polynomials where $h$ is a positive integer and $p$, $q$ and $r$ are distinct prime numbers. For length $pq$, we manage to provide the permutation groups of cyclic codes with generator polynomials $Q_{pq}(x)$(the $pq$-th cyclotomic polynomial) or others, which seems to be the first work about permutation groups of cyclic codes with generator polynomials that are factors of $x^{pq}-1$ but not factors of $x^p-1(\text{or }x^q-1)$.

cs.IT↗

arXiv2026

An Efficient Entropy Flow on Weighted Graphs: Theory and Applications

We propose a novel entropy flow on weighted graphs, which provides a principled framework that characterizes the evolution of probability distributions over graph structures while sharing geometric intuition with discrete Ricci flow. We provide its rigorous formulation, establish its fundamental theoretical properties, and prove the long-time existence and convergence of its solutions. To demonstrate its applicability, we employ entropy flow for community detection in real-world networks. Empirically, it achieves detection accuracy fully comparable to that of discrete Ricci flow. Crucially, by avoiding computations of optimal transport distances and shortest paths, our approach overcomes the fundamental computational bottleneck of Ollivier and Lin-Lu-Yau Ricci flows. As a result, entropy flow requires only $1.61\%$-$3.20\%$ of the computation time of Ricci flow. These results indicate that entropy flow provides a theoretically rigorous and computationally efficient framework for large-scale graph analysis.

math.CA↗

arXiv2026

Geometric Evolution Graph Convolutional Networks: Enhancing Graph Representation Learning via Ricci Flow

We introduce the Geometric Evolution Graph Convolutional Network (GEGCN), a novel framework that enhances graph representation learning through explicit modeling of geometric evolution on graph structures. Specifically, GEGCN leverages a Long Short-Term Memory (LSTM) network to capture the dynamic structural sequence generated by discrete Ricci flow, and infuses the learned dynamic representations into a graph convolutional network. Extensive experiments demonstrate that GEGCN achieves excellent performance on classification tasks across various benchmark datasets, including homophilic/heterophilic graphs, filtered graphs, and large-scale graphs.

cs.LG↗

arXiv2026

EternalMath: A Living Benchmark of Frontier Mathematics that Evolves with Human Discovery

Current evaluations of mathematical reasoning in large language models (LLMs) are dominated by static benchmarks, either derived from competition-style problems or curated through costly expert effort, resulting in limited coverage of research-level mathematics and rapid performance saturation. We propose a fully automated, theorem-grounded pipeline for evaluating frontier mathematical reasoning, which directly transforms recent peer-reviewed mathematical literature into executable and verifiable reasoning tasks. The pipeline identifies constructive or quantitative results, instantiates them into parameterized problem templates, and generates deterministic solutions through execution-based verification, enabling scalable, reproducible, and continuously updatable evaluation without reliance on large-scale expert authoring. By design, this approach supports temporal extensibility, intrinsic correctness checking, and domain-specific customization across mathematical subfields. Applying this pipeline yields \textbf{EternalMath}, an evolving evaluation suite derived from contemporary research papers. Experiments with state-of-the-art LLMs reveal substantial performance gaps, indicating that mathematical reasoning at the research frontier remains far from saturated and underscoring the need for evaluation methodologies that evolve in step with human mathematical discovery.

cs.CL↗

arXiv2025

Finding core subgraphs of directed graphs via discrete Ricci curvature flow

Ricci curvature and its associated flow offer powerful geometric methods for analyzing complex networks. While existing research heavily focuses on applications for undirected graphs such as community detection and core extraction, there have been relatively less attention on directed graphs. In this paper, we introduce a definition of Ricci curvature and an accompanying curvature flow for directed graphs. Crucially, for strongly connected directed graphs, this flow admits a unique global solution. We then apply this flow to detect strongly connected subgraphs from weakly connected directed graphs. (A weakly connected graph is connected overall but not necessarily strongly connected). Unlike prior work requiring graphs to be strongly connected, our method loosens this requirement. We transform a weakly connected graph into a strongly connected one by adding edges with very large artificial weights. This modification does not compromise our core subgraph detection. Due to their extreme weight, these added edges are automatically discarded during the final iteration of the Ricci curvature flow. For core evaluation, our approach consistently surpasses traditional methods, achieving better results on at least two out of three key metrics. The implementation code is publicly available at https://github.com/12tangze12/Finding-core-subgraphs-on-directed-graphs.

cs.SI↗

arXiv2025

Core detection via Ricci curvature flows on weighted graphs

Graph Ricci curvature is crucial as it geometrically quantifies network structure. It pinpoints bottlenecks via negative curvature, identifies cohesive communities with positive curvature, and highlights robust hubs. This guides network analysis, resilience assessment, flow optimization, and effective algorithm design. In this paper, we derived upper and lower bounds for the weights along several kinds of discrete Ricci curvature flows. As an application, we utilized discrete Ricci curvature flows to detect the core subgraph of a finite undirected graph. The novelty of this work has two aspects. Firstly, along the Ricci curvature flow, the bounds for weights determine the minimum number of iterations required to ensure weights remain between two prescribed positive constants. In particular, for any fixed graph, we conclude weights can not overflow and can not be treated as zero, as long as the iteration does not exceed a certain number of times; Secondly, it demonstrates that our Ricci curvature flow method for identifying core subgraphs outperforms prior approaches, such as page rank, degree centrality, betweenness centrality and closeness centrality. The codes for our algorithms are available at https://github.com/12tangze12/core-detection-via-Ricci-flow.

math.AP↗

arXiv2025

Piecewise-linear Ricci curvature flows on weighted graphs

Community detection is an important problem in graph neural networks. Recently, algorithms based on Ricci curvature flows have gained significant attention. It was suggested by Ollivier (2009), and applied to community detection by Ni et al (2019) and Lai et al (2022). Its mathematical theory was due to Bai et al (2024) and Li-M\"unch (2025). In particular, solutions to some of these flows have existence, uniqueness and convergence. However, a unified theoretical framework has not yet been established in this field. In the current study, we propose several unified piecewise-linear Ricci curvature flows with respect to arbitrarily selected Ricci curvatures. First, we prove that the flows have global existence and uniqueness. Second, we show that if the Ricci curvature being used is homogeneous, then after undergoing multiple surgeries, the evolving graph has a constant Ricci curvature on each connected component. Note that five commonly used Ricci curvatures, which were respectively defined by Ollivier, Lin-Lu-Yau, Forman, Menger and Haantjes, are all homogeneous, and that the proof of all these results is independent of the choice of the specific Ricci curvature. Third, as an application, we apply the discrete piecewise-linear Ricci curvature flow with surgeries to the problem of community detection. On three real-world datasets, the flow consistently outperforms baseline models and existing methods. Complementary experiments on synthetic graphs further confirm its scalability and robustness. Compared with existing algorithms, our algorithm has two advantages: it does not require curvature calculations at each iteration, and the iterative process converges.

math.AP↗

arXiv2025

Community detection of hypergraphs by Ricci flow

Community detection in hypergraphs is both instrumental for functional module identification and intricate due to higher-order interactions among nodes. We define a hypergraph Ricci flow that directly operates on higher-order interactions of hypergraphs and prove long-time existence of the flow. Building on this theoretical foundation, we develop HyperRCD-a Ricci-flow-based community detection approach that deforms hyperedge weights through curvature-driven evolution, which provides an effective mathematical representation of higher-order interactions mediated by weighted hyperedges between nodes. Extensive experiments on both synthetic and real-world hypergraphs demonstrate that HyperRCD exhibits remarkable enhanced robustness to topological variations and competitive performance across diverse datasets.

cs.SI↗

arXiv2024

Evolution of weights on a connected finite graph

On a connected finite graph, we propose an evolution of weights including Ollivier's Ricci flow as a special case. During the evolution process, on each edge, the speed of change of weight is exactly the difference between the Wasserstein distance related to two probability measures and certain graph distance. Here the probability measure may be chosen as an $\alpha$-lazy one-step random walk, an $\alpha$-lazy two-step random walk, or a general probability measure. Based on the ODE theory, we show that the initial value problem has a unique global solution. A discrete version of the above evolution is applied to the problem of community detection. Our algorithm is based on such a discrete evolution, where probability measures are chosen as $\alpha$-lazy one-step random walk and $\alpha$-lazy two-step random walk respectively. Note that the later measure has not been used in previous works [2, 16, 21, 24]. Here, as in [21], only one surgery needs to be performed after the last iteration. Moreover, our algorithm is much easier than those of [2, 16, 21], which were all based on Lin-Lu-Yau's Ricci curvature. The code is available at https://github.com/mjc191812/Evolution-of-weights-on-a-connected-finite-graph.

math.CA↗

arXiv2024

A modified Ricci flow on arbitrary weighted graph

In this paper, we propose a modified Ricci flow, as well as a quasi-normalized Ricci flow, on arbitrary weighted graph. Each of these two flows has a unique global solution. In particular, these global existence and uniqueness results do not require an exit condition proposed by Bai et al in a recent work [2]. As applications, these two Ricci flows are applied to community detection for complex networks, including Karate Club, American football games, Facebook, as well as artificial networks. In our algorithms, unlike in [5,15], there is no need to perform surgery at every iteration, only one surgery needs to be performed after the last iteration. From three commonly used criteria for evaluating community detection algorithms, ARI, NMI and Q, we conclude that our algorithms outperform existing algorithms, including Ollivier's Ricci flow [5], normalized Ollivier's Ricci flow and normalized Lin-Lu-Yau's Ricci flow [15]. The codes for our algorithms are available at https://github.com/mjc191812/Modified-Ricci-Flow.

math.AP↗

arXiv2024

On automorphism groups of polar codes

Over the past years, Polar codes have arisen as a highly effective class of linear codes, equipped with a decoding algorithm of low computational complexity. This family of codes share a common algebraic formalism with the well-known Reed-Muller codes, which involves monomial evaluations. As useful algebraic codes, more specifically known as decreasing monomial codes, a lot of decoding work has been done on Reed-Muller codes based on their rich code automorphisms. In 2021, a new permutation group decoder, referred to as the automorphism ensemble (AE) decoder, was introduced. This decoder can be applied to Polar codes and has been shown to produce similar decoding effects. However, identifying the right set of code automorphisms that enhance decoding performance for Polar codes remains a challenging task. This paper aims to characterize the full automorphism group of Polar codes. We will prove a reduction theorem that effectively reduces the problem of determining the full automorphism group of arbitrary random Polar codes to that of a specified class of Polar codes. Besides, we give exact classification of the full automorphism groups of families of Polar codes that are constructed using the Reed-Muller codes.

math.CO↗