Diskin, S., & Samotij, W. (2024). Isoperimetry in product graphs. arXiv preprint arXiv:2407.02058v2.
This paper aims to establish a general edge-isoperimetric inequality applicable to arbitrary product graphs, addressing the challenge of determining the minimum edge boundary for subsets of vertices within such graphs.
The authors employ an elegant entropy-based approach, drawing inspiration from Boucheron, Lugosi, and Massart's proof for the hypercube. They leverage properties of convex functions, Jensen's inequality, and Han's inequality to derive their main result.
The paper provides a powerful and versatile tool for analyzing edge-isoperimetric properties in a wide range of product graphs. The derived inequality offers a unified framework for understanding these properties across different graph families and has implications for areas like percolation theory and network design.
This work contributes significantly to the field of discrete isoperimetric inequalities, offering a general result applicable to a broad class of graphs. The findings have implications for understanding network connectivity, designing robust communication networks, and analyzing algorithms on product graphs.
While the paper provides a comprehensive analysis for undirected graphs, extending the results to directed graphs and exploring their implications for other combinatorial problems remain open avenues for future research.
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by Sahar Diskin... at arxiv.org 11-19-2024
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