graph-theory

This article discusses the relationship between Graph Engineers and Loop Engineers, emphasizing that a Loop is the smallest Graph, and references research from 'Nature Machine Intelligence' to analyze the applicable scenarios of multi-agent systems.

This paper establishes a precise mathematical correspondence between graph surgery and the do-operator in acyclic structural causal models, proving their equivalence in terms of dependency graphs.

This paper introduces RamseyGadgets, a dataset of 70 graph construction problems designed to evaluate LLMs' reasoning abilities in mathematics, specifically focusing on underexplored Ramsey-good graphs to test capabilities beyond memorization.

This article explains zero-knowledge proofs by focusing on a protocol for graph 3-coloring from the Goldreich-Micali-Widgerson paper and shares a simple implementation.

A paper reports an AI-agent-driven attack on Conway's 99-graph problem, providing partial-credit bounds, a forced-structure reduction, and a verifiable framework, without solving the open question.

An undergraduate researcher reports that GPT-5.6 Sol Max solved two open graph theory problems: proving the Imbalance Conjecture and disproving Teschner's bondage-number conjecture. The preprints have been posted but not yet peer-reviewed.

An AI system called Theo Conjecture, leveraging a large language model, solved a 35-year-old graph theory problem originally posed by mathematician Paul Erdős, discovering an unexpected term. The system works by proposing, testing, and revising mathematical ideas in a loop.

An autonomous AI agent (math-god) proved the weighted theta extension theorem, demonstrating that every simple theta graph with one arbitrary rooted-tree attached through a single bridge edge satisfies s⁺(G) > |V(G)|, using a combination of root-congruence PSD witnesses, local reductions, phase-sign classification, and other advanced techniques, with machine-checkable certificates.

Introduces TopoGuard, a graph theory-based defense against split-knowledge attacks in RAG systems, where multiple individually benign documents combine to produce harmful outputs. The method detects malicious contexts by building a semantic similarity graph, significantly outperforming existing per-document filters like LlamaGuard.

A tweet reports that an AI (likely GPT-5.6 Pro) disproved the Dinitz-Garg-Goemans conjecture, a long-standing open problem in graph theory, using simple prompts like 'do a breakthrough'.

A Chinese professor has discovered a new deterministic shortest path algorithm that beats the 40-year-old sorting barrier by combining Dijkstra and Bellman-Ford with frontier reduction, achieving O(m log^(2/3) n) time on sparse graphs.

Proposes a prompt perturbation framework that generates perturbed prompt variants, filters out structurally inconsistent comparison patterns using graph-level consistency checks, then applies standard ranking methods to yield more reliable LLM rankings.

Five researchers from Tsinghua, Stanford, and Max Planck have developed a new shortest path algorithm that beats Dijkstra's for sparse directed graphs, achieving O(m log^(2/3) n) time complexity, the first improvement since 1987.

The paper introduces GTBench, a curriculum-grounded benchmark for evaluating LLMs as mathematical research assistants in graph theory, containing 63 problems across three difficulty levels. It evaluates five frontier models and finds that performance degrades with difficulty, with GPT-5 achieving near-perfect results on basic problems but only 82% on graduate-level proofs.

The article explores extending rock-paper-scissors to more than three options by allowing ties, revealing richer game dynamics and strategies through graph theory.

This book presents a comprehensive survey of graph theory under uncertainty, covering fuzzy, neutrosophic, and uncertain graph models, their properties, extensions, and applications in decision-making, graph neural networks, and knowledge graphs.

Sensor Tower's engineering team used graph theory analysis and profiling tools to identify bottlenecks in a slow backend endpoint, achieving a fourfold speed improvement by optimizing Protobuf decoding and encoding steps.

This article explores the mathematical underpinnings of Sudoku, explaining how the puzzle can be modeled as a graph theory vertex coloring problem. It details how algorithms like greedy search and backtracking are applied to solve these structures.