Spectrum Preserving Short Cycle Removal on Regular Graphs

Author Pedro Paredes

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Pedro Paredes
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA


I am very grateful to Ryan O'Donnell for numerous comments and suggestions, as well as very thorough feedback on an earlier draft of this paper.

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Pedro Paredes. Spectrum Preserving Short Cycle Removal on Regular Graphs. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 55:1-55:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We describe a new method to remove short cycles on regular graphs while maintaining spectral bounds (the nontrivial eigenvalues of the adjacency matrix), as long as the graphs have certain combinatorial properties. These combinatorial properties are related to the number and distance between short cycles and are known to happen with high probability in uniformly random regular graphs. Using this method we can show two results involving high girth spectral expander graphs. First, we show that given d ⩾ 3 and n, there exists an explicit distribution of d-regular Θ(n)-vertex graphs where with high probability its samples have girth Ω(log_{d-1} n) and are ε-near-Ramanujan; i.e., its eigenvalues are bounded in magnitude by 2√{d-1} + ε (excluding the single trivial eigenvalue of d). Then, for every constant d ⩾ 3 and ε > 0, we give a deterministic poly(n)-time algorithm that outputs a d-regular graph on Θ(n)-vertices that is ε-near-Ramanujan and has girth Ω(√{log n}), based on the work of [Mohanty et al., 2020].

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Spectra of graphs
  • Theory of computation → Expander graphs and randomness extractors
  • Ramanujan Graphs
  • High Girth Regular Graphs


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