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1. Hosoya entropy of fullerene graphs

   Ghorbani, Modjtaba; Dehmer, Matthias; Rajabi-Parsa, Mina; Emmert-Streib, Frank; Mowshowitz, Abbe (January 2019, Elsevier)

   Entropy-based methods are useful tools for investigating various problems in mathemati- cal chemistry, computational physics and pattern recognition. In this paper we introduce a general framework for applying Shannon entropy to fullerene graphs, and used it to in- vestigate their properties. We show that important physical properties of these molecules can be determined by applying Hosoya entropy to their corresponding graphs. 

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2. Solving Unique Games over Globally Hypercontractive Graphs

   https://doi.org/10.4230/LIPICS.CCC.2024.3  

   Bafna, Mitali; Minzer, Dor (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Santhanam, Rahul (Ed.)

   We study the complexity of affine Unique-Games (UG) over globally hypercontractive graphs, which are graphs that are not small set expanders but admit a useful and succinct characterization of all small sets that violate the small-set expansion property. This class of graphs includes the Johnson and Grassmann graphs, which have played a pivotal role in recent PCP constructions for UG, and their generalizations via high-dimensional expanders. We show new rounding techniques for higher degree sum-of-squares (SoS) relaxations for worst-case optimization. In particular, our algorithm shows how to round "low-entropy" pseudodistributions, broadly extending the algorithmic framework of [Mitali Bafna et al., 2021]. At a high level, [Mitali Bafna et al., 2021] showed how to round pseudodistributions for problems where there is a "unique" good solution. We extend their framework by exhibiting a rounding for problems where there might be "few good solutions". Our result suggests that UG is easy on globally hypercontractive graphs, and therefore highlights the importance of graphs that lack such a characterization in the context of PCP reductions for UG. 

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3. Graph representation of local environments for learning high-entropy alloy properties

   https://doi.org/10.1088/2632-2153/adc0e1  

   Zhang_张, Hengrui_恒睿; Huang, Ruishu; Chen, Jie; Rondinelli, James_M; Chen, Wei (April 2025, Machine Learning: Science and Technology)

   Abstract Graph neural networks (GNNs) have excelled in predictive modeling for both crystals and molecules, owing to the expressiveness of graph representations. High-entropy alloys (HEAs), however, lack chemical long-range order, limiting the applicability of current graph representations. To overcome this challenge, we propose a representation of HEAs as a collection of local environment graphs. Based on this representation, we introduce the LESets machine learning model, an accurate, interpretable GNN for HEA property prediction. We demonstrate the accuracy of LESets in modeling the mechanical properties of quaternary HEAs. Through analyses and interpretation, we further extract insights into the modeling and design of HEAs. In a broader sense, LESets extends the potential applicability of GNNs to disordered materials with combinatorial complexity formed by diverse constituents and their flexible configurations. 

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4. Entropic independence: optimal mixing of down-up random walks

   https://doi.org/10.1145/3519935.3520048  

   Anari, Nima; Jain, Vishesh; Koehler, Frederic; Pham, Huy Tuan; Vuong, Thuy-Duong (June 2022, Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing)

   We introduce a notion called entropic independence that is an entropic analog of spectral notions of high-dimensional expansion. Informally, entropic independence of a background distribution $$\mu$$ on $$k$$-sized subsets of a ground set of elements says that for any (possibly randomly chosen) set $$S$$, the relative entropy of a single element of $$S$$ drawn uniformly at random carries at most $O(1/k)$ fraction of the relative entropy of $$S$$. Entropic independence is the analog of the notion of spectral independence, if one replaces variance by entropy. We use entropic independence to derive tight mixing time bounds, overcoming the lossy nature of spectral analysis of Markov chains on exponential-sized state spaces. In our main technical result, we show a general way of deriving entropy contraction, a.k.a. modified log-Sobolev inequalities, for down-up random walks from spectral notions. We show that spectral independence of a distribution under arbitrary external fields automatically implies entropic independence. We furthermore extend our theory to the case where spectral independence does not hold under arbitrary external fields. To do this, we introduce a framework for obtaining tight mixing time bounds for Markov chains based on what we call restricted modified log-Sobolev inequalities, which guarantee entropy contraction not for all distributions, but for those in a sufficiently large neighborhood of the stationary distribution. To derive our results, we relate entropic independence to properties of polynomials: $$\mu$$ is entropically independent exactly when a transformed version of the generating polynomial of $$\mu$$ is upper bounded by its linear tangent; this property is implied by concavity of the said transformation, which was shown by prior work to be locally equivalent to spectral independence. We apply our results to obtain (1) tight modified log-Sobolev inequalities and mixing times for multi-step down-up walks on fractionally log-concave distributions, (2) the tight mixing time of $$O(n\log n)$$ for Glauber dynamics on Ising models whose interaction matrix has eigenspectrum lying within an interval of length smaller than $$1$$, improving upon the prior quadratic dependence on $$n$$, and (3) nearly-linear time $$\widetilde O_{\delta}(n)$$ samplers for the hardcore and Ising models on $$n$$-node graphs that have $$\delta$$-relative gap to the tree-uniqueness threshold. In the last application, our bound on the running time does not depend on the maximum degree $$\Delta$$ of the graph, and is therefore optimal even for high-degree graphs, and in fact, is sublinear in the size of the graph for high-degree graphs. 

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5. Plasmonic high-entropy carbides

   https://doi.org/10.1038/s41467-022-33497-1  

   Calzolari, Arrigo; Oses, Corey; Toher, Cormac; Esters, Marco; Campilongo, Xiomara; Stepanoff, Sergei P.; Wolfe, Douglas E.; Curtarolo, Stefano (December 2022, Nature Communications)

   Abstract Discovering multifunctional materials with tunable plasmonic properties, capable of surviving harsh environments is critical for advanced optical and telecommunication applications. We chose high-entropy transition-metal carbides because of their exceptional thermal, chemical stability, and mechanical properties. By integrating computational thermodynamic disorder modeling and time-dependent density functional theory characterization, we discovered a crossover energy in the infrared and visible range, corresponding to a metal-to-dielectric transition, exploitable for plasmonics. It was also found that the optical response of high-entropy carbides can be largely tuned from the near-IR to visible when changing the transition metal components and their concentration. By monitoring the electronic structures, we suggest rules for optimizing optical properties and designing tailored high-entropy ceramics. Experiments performed on the archetype carbide HfTa 4 C 5 yielded plasmonic properties from room temperature to 1500K. Here we propose plasmonic transition-metal high-entropy carbides as a class of multifunctional materials. Their combination of plasmonic activity, high-hardness, and extraordinary thermal stability will result in yet unexplored applications. 

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